Periodic discrete-time frames: Design and applications for image restoration

Transcription

1 Periodic discrete-time frames: Design and applications for image restoration Amir Averbuch Pekka eittaanmäki Valery Zheludev School of Computer Science Tel Aviv University, Tel Aviv 69978, Israel Department of Mathematical Information Technology P.O. Box 35 Agora, University of Jyväskylä, Finland Abstract We present diverse families of tight and semi-tight frames in the space of discrete-time periodic signals, which are generated by three- and four-channel perfect reconstruction periodic filter banks. The filter banks are derived from the interpolating and quasi-interpolating polynomial and discrete splines. Each filter bank comprises one interpolating linear phase low-pass filter and one high-pass filter, whose magnitude s response mirrors that of the low-pass filter. Depending on the channels number, these filter banks comprise one or two band-pass filters. In the semi-tight frames case, all the filters have linear phase and antisymmetric impulse response, while in the tight frame case some of band-pass filters are slightly asymmetric. Since the common definition of the vanishing moments is invalid in periodic setting, we introduce the notion of local discrete vanishing moments LDVM. In the tight frame case analysis framelets coincide with their synthesis counterparts. However, in the semi-tight frames, we have options to swap LDVM between the synthesis and the analysis framelets or to add smoothness to the synthesis framelets. The design scheme is generic and makes it possible to design framelets with any number of LDVM. The computational complexity of the framelet transforms, which consists of calculation of direct and inverse fast Fourier transforms and simple arithmetical operations, practically does not depend on the number of LDVM. The designed frames are tested for image restoration, which are degraded by blurring, affected by random noise and parts of pixels are missing. The images were restored by the application of the Split Bregman Iterations method. The frame performances are compared. Introduction Restoration of corrupted and/or damaged and/or noised multidimensional signals D-acoustics, D-images, 3D-video is a major challenge that the signal/image processing community faces nowadays when rich multimedia content is the most popular data being transmitted over any type of networks especially mobile. Quality degradation in multidimensional signals can come from sampling, acquisition, transmission through noisy channels, to name some. Restoration of multidimensional signals includes denoising, deblurring, recovering of missing or damaged samples or fragments impainting in images, resolution enhancement and super resolution. The processing goals are to improve the visual perception of still images and video signals. But no less important are the goals to reveal internal fine structures and details in multidimensional signals by extracting characteristic features of objects or classes of objects. Medical imaging such as CT, FET, X-Ray and MRI tomography are good examples because they recover images from highly incomplete and noisy data. The trend in tomography is to sparsify the data in order to reduce the radiation dose for a patient and to help the radiologist to automatically classify the huge amount of visual data that the tomographic machines generate. Sparse representations is one of the goals that compressed sensing and l minimization try to achieve.

2 Recent developments in mathematics, especially in wavelet and in wavelet frames framelet analysis, provide innovative and powerful tools to process faithfully and robustly the above challenges. The framelets produce redundant expansions for multidimensional signals that, in particular, provides an additional sparsity of the signals representation due to better adaptation abilities of redundant representations. A valuable advantage that redundant representations hold is their ability to restore missing and incomplete information and to represent efficiently and compactly the data. The frame expansions of signals demonstrate resilience to the coefficients disturbances and losses [0, 9, 4, ]. Thus, frames may serve as a tool for error correction to signals transmitted through lossy channels. Implicitly, this resilience is utilized in signal/image restoration, which is based on the prior assumption that a frame expansion of a given signal/image is sparse. In principle, only part of the samples/pixels is sufficient to near perfectly restore the object. This approach, which is a version of the Compressive Sensing methodology [6, 7], for example, proved to be extremely efficient for image restoration. Practically, this approach is implemented via minimization of a parameterized functional, where the representation sparsity is reflected in the l norm of the transform coefficients. Unlike the minimization in the norm, the minimization does not have an explicit solution and can be resolved only by iterative methods. The so-called split Bregman iteration SBI scheme, which was recently used in [8], provided a fast and stable algorithm for this purpose. Variations of this scheme and its application to image restoration using wavelet frames are presented in [3, ], to name a few. A variety of impressive results on image restoration were reported in the last couple of years. A survey can be found in [34] while the recent development is presented in []. Due to applications diversity, it is important to have a library of wavelet frames in order to select a frame that fits a specific task. The direct and inverse transforms in iterative algorithms are repeated many times, therefore, members in this library must have fast and stable implementation of the transforms. Waveforms symmetry with having vanishing moments are also important in order to avoid distortions when thresholding is used. To satisfy these requirements, most of the framelet systems designed so far operate with the compactly supported framelets and the transforms are implemented by finite and short impulse response FIR oversampled filter banks [5,,, 9]. Thus, the number of framelet systems available for applications is very limited. This number is even smaller when the requirement is to have tight frames. A common approach for framelet system construction in the function space L starts from the introduction of a pair of refinable functions or one function, which generates the multiresolution analysis MRA in L. Then, the wavelets are derived from a linear combinations of refinable functions. Many construction schemes are based on Unitary Extension Principle UEP [30] for tight frames and Mixed Extension Principle MEP [9] for bi-frames. These principles reduce the construction of a framelet system to the design of a perfect reconstruction filter bank. The masks of the given refinable functions serve as low-pass filters in the filter bank. The framelet transforms of signals are implemented via multirate filtering. On the other hand, the oversampled perfect reconstruction filter banks by themselves generate wavelettype frames in the space of discrete-time signals [4, 0]. Utilizing infinite impulse response IIR filter banks and relaxation of the tightness requirement provides a number of additional opportunities. Such properties as symmetry, interpolation, flat spectra combined with fine time-domain localization of framelets can be easily achieved as well as a high number of vanishing moments [7, 6]. In these papers, the key point is the design of low-pass filters, which, being applied to the even subarray of a signal, well approximate the odd subarray the prediction filters. A natural source for such filters are the discrete and the polynomial splines. In [7, 6], a number of 3- framelet systems was derived from the discrete splines. The transforms are implemented in a fast way using recursive filtering. on-compactness of the waveforms supports is compensated by their exponential decay as time tends to infinity. In principle, any number of vanishing moments can be achieved but the computational cost of implementation is growing fast. This drawback can be overcome by switching to a periodic setting, which is the subject of this paper. A variety of prediction filters is derived from interpolating and quasi-interpolating polynomial and from discrete periodic splines. Using these prediction filters, we design libraries of 3- framelet and 4- framelet periodic tight and the so-called semi-tight frames with diverse properties. The transforms implementation is reduced to application of the direct and the inverse fast Fourier transforms FFT with simple arithmetical

3 operations. Any number of vanishing moments in the framelets and any flatness of spectra can be produced without computational cost increase. Altogether, more than twenty framelets systems are explicitly designed and described in this paper. However, the design scheme is generic and this framelets library can be easily expanded. Preliminary results in the periodic discrete-time frame design are reported in [36]. Recently, framelets in the space of periodic functions were studied in [3]. The designed framelets libraries were tested for image restoration and demonstrated a high quality. Their diversity enabled us to select a frame, which best fits each specific application. In particular, in a number of experiments, the semi-tight frames were advantageous over tight frames. The paper is organized as follows. The introductory Section recalls the notions of periodic discrete-time signals and of periodic filter p-filter. In Section 3, the perfect reconstruction PR periodic filter banks p-filter banks are presented and characterized via the polyphase representation of their discrete Fourier transform DFT. Section 4 introduces frames in the discrete-time signals space, which originate from PR oversampled p-filter banks and describes the implementation of the framelet transforms. In Section 5., design of three-channel p-filter banks, which generate tight and semi-tight frames, is presented. Section 5. describes the design scheme of four-channel p-filter banks generating tight frames. This scheme is, in a sense, similar to the construction in [9]. Section?? discusses the restoration of sampled polynomials by low-pass p-filters and introduces the notion of the local discrete vanishing moments LDVM. In Section 6, a collection of FIR and IIR prediction p-filters is derived from the interpolating and quasi-interpolating polynomial and discrete splines. In Section 7, a number of three-channel p-filter banks with spline p-filters generating tight and semi-tight frames is presented. In Section 8, these spline p-filters are used to design four-channel p-filter banks generating tight frames. In Section 9, the designed tight and semi-tight frames are used for image restoration, which were degraded by blurring, affected by random noise and parts of pixels are missing. The frames performances are compared. Periodic discrete-time signals and filters. Signals and transforms We call the -periodic real-valued sequences x = {x[k]}, k Z, x[k + ] = x[k], the discrete-time periodic signals. In the sequel, we assume that = j, j is a natural number and ω = e πi/. These signals form an -dimensional vector space, which we denote by Π[], where the inner product and the norm are defined as x, h = k=0 x[k] h[k], x = x = k=0 x[k]. ote that any discrete-time signal y of limited duration L can be embedded into a space Π[], L, by the periodization ỹ = { ỹ[k] = l Z y[k + l]}. The discrete circular convolution of two -periodic signals is an -periodic signaly = h x y[k] = h[k l] x[l]. The discrete-time Fourier exponentials, {e n}, e n [k] = e πink/ = ω kn, n, k = 0,...,, form an orthogonal basis of the space Π[]: e n, e l = k=0 e πink/ e πilk/ = k=0 e πin lk/ = δ[k l].. The symbol δ[k] denotes the periodized Kronecker delta impulse, that is the signal δ[k] is zero for all integers k except for k = l, l Z, where δ[l] =. The expansion of a signal over the exponential basis x[k] = n=0 ˆx[n] ω nk = / n= / ˆx[n] ω nk, where ˆx[n] = k=0 x[k] ω nk, is called the inverse discrete Fourier transform IDFT of the signal x, while the set of coordinates {ˆx[n]}, n Z, is referred to as the discrete Fourier transform DFT of x. Both IDFT and DFT are calculated by the 3

4 fast Fourier transform FFT algorithm [3]. Since the signals are real-valued, the complex conjugate DFT ˆx[n] = ˆx[ n]. Throughout the paper, we use the notation ˆx[n] m = / m k=0 x[k] ω mnk, x[k] = m / m n=0 ˆx[n] ω m nk for the DFT of signals belonging to the space Π[/ m ]. The sequence {ˆx[n] m } is / m -periodic. The following relations between the signals from Π[] and their DFT hold: Parseval identities: Equation. implies that x, h = k=0 x[k] h[k] = / n= / ˆx[n] ĥ[ n], x = k=0 x[k] = / n= / ˆx[n]. Circular convolution: If y = x h then ŷ[n] = ĥ[n] ˆx[n]. Circular shift: If x d = {x[k + d]}, where d is an integer number, then ˆxd [n] = ω nd ˆx[n]. Finite differences The finite differences of a sequence x are defined iteratively: [x] = {x[k + ] x[k]}, k Z, n [x] = [ n [x]]. The central finite difference of second order δ [x] = {x[k + ] x[k] + x[k ]}, k Z, and δ r [x] = δ [δ n [x]]. Proposition. Assume that P n = {Pn k}, k Z, is a sampled polynomial of degree not exceeding n. Then, n [P n ] = 0. If n = r then δ r [P r ] = 0. { Proof: Denote K } l = k l. The difference [K l ][k] = { k + l k l} { l = l m=0 m k m}. Thus, application of the difference reduces by one the degree of a sampled polynomial. Consequently, application of the difference n to a sampled polynomial P n produces a constant, while application of n results in identical zero. If x is a periodic discrete-time signals then their differences are are periodic signals and their DFT coefficients are derived from the shift property of the DFT: m [x][n] = ω n m ˆx[n], δl [x][n] = i sin πn l ˆx[n].. Polyphase representation of the DFT A signal x = {x[k]} Π[] can be decomposed into a superposition of M = m, m < j, signals belonging to Π[/M] = Π[/ m ]: x = M x l,m, x l,m [k] = x[l + km], k Z, which is referred to as the polyphase decomposition of the signal x. The signals x l,m Π[/M], are called the polyphase components of x. The DFT of x can be split into the sum ˆx[n] = k=0 ω nk x[k] = M m=0 /M k=0 ω nm+km x[m + km] = M m=0 ω ln x l,m [n] m. Special case: M =. Then, x 0, = {x[k]} and x, = {x[k + ]} are the even and odd components of the signal x. The DFTs are ˆx[n] = x 0, [n] + ω n x, [n], ˆx[n + /] = x 0, [n] ω n x, [n] = x 0, [n] = ˆx[n] + ˆx[n + /] /, x, [n] = ˆx[n] ˆx[n + /] /ω n..3 4

5 . Periodic filters A linear operator h : Π[] Π[], y = hx, is time-invariant if the shift x d = {x[k + d]} of the input signal produces the same shift y d = {y[k + d]} of the output. Such operators are called digital periodic filters p-filters. If the input signal i = {δ[k]}, then the output signal h i = {h[k]} is called the impulse response IR of the p-filter h. In the sequel, we use the notation h for both a p-filter and its IR {h[k]}. Obviously, the IR h is a signal from Π[]. Any periodic signal x can be represented as a linear combination of impulses x[k] = x[l] δ[k l] = x i. Then, due to the time-invariance of the filter h, the output signal becomes y = hx = x h y[k] = h[k l] x[l]. Thus, application of the p-filter to a periodic signal is implemented via the } discrete circular convolution. The DFT of the IR of the p-filter h is called its frequency response {ĥ[n] ĥ =, ĥ[n] = h[k] ω nk, n Z. Periodic filtering a signal reduces to multiplication in the frequency domain: h x ĥ[n] ˆx[n], n Z. The frequency response of a filter can be represented in a polar form ĥ[n] = ĥ[n] e i argĥ[n], where the positive -periodic sequence ĥ[n] is called the magnitude response of h and the real-valued π-periodic sequence argĥ[n] is called the phase response of h. A p-filter is referred to as a linear phase one if its phase response is linear in n. If the IR of a filter h is symmetric or antisymmetric within the interval k = /,..., / then h is a linear phase filter. A p-filter h is called low-pass if it passes low frequencies of signals but attenuates or completely rejects frequencies higher than the cutoff frequency. Its frequency response ĥ[0] 0, ĥ[/] is close or equal to zero. On the contrary, a high-pass p-filter g attenuates or rejects frequencies lower than the cutoff frequency. Its frequency response ĝ[/] 0, ĥ[0] is close or equal to zero. A band-pass filter passes frequencies within a certain range passband and attenuates rejects frequencies outside that range stopband. 3 Periodic filter banks 3. Multirate p-filtering Assume that x = {x[k]} belongs to Π[], = j, and M = l, l < j. The operation Mx = `x = {x[mk]} Π[/M] is called downsampling the signal x by factor of M. Assume that a signal x belongs to Π[/M]. The operation { x[l], if k = lm; Mx = x, x[k] =, l Z, 0, otherwise. is called upsampling the signal x by factor of M. Obviously, the downsampled signal Mx = x 0,M is the zero polyphase component of the signal x, while the upnsampled signal Mx is a signal, whose all the polyphase components, except for the zero component, vanish. If p-filtering a signal is accompanied by downsampling or upsampling then it is called multirate filtering. Let h = {h[k]} and h } = { h[k] be p-filters. Application of the p-filter h } = { h[ k], which is the timereversed filter h, to the signal x, which is followed by downsampling by factor of M, produces the signal ỹ = {ỹ[k]} Π[/M], ỹ[k] = h[l Mk] x[l], which is the zero polyphase component of the signal y = h x. Application of the p-filter h to the signal y = { y[k]} Π[/M], which is upsampled by factor of M, produces the signal ˇx = {ˇx[k]} Π[], ˇx[k] = /M h[k Ml] y[l] ˆˇx[n] = ĥ[n] ˆ y[mn]. In the rest of the paper we assume that the downupsampling factor M = and denote by x 0 = x 0, and x = x, the even and the odd polyphase components of a signal x. 5

6 M = : Downsampling: The downsampled signal y = y 0 is the even polyphase component of the signal y = {y[k]}, y[k] = h[l k] x[l], ŷ[n] = ˆ h[ n] ˆx[n]. Using Eq..3, we get ˆ h[ n] ˆx[n] = h0 [ n] x 0 [n] + h [ n] x [n] + ω h0 n [ n] x [n] + h [ n] x 0 [n + ] = ˆ y[n] = h0 [ n] x 0 [n] + h [ n] x [n], n Z. 3. M = : Upsampling: Assume that p-filter h is applied to the signal y = { y[k]} Π[/], which is upsampled by factor of. Then, ˇx[k] = / h[k l] y[l] ˆˇx[n] = ĥ[n] ˆ y[n] = ĥ0 [n] + ω n ĥ [n] ˆ y[n] = ˇx 0 [n] = ĥ0[n] ˆ y[n], ˇx [n] = ĥ[n] ˆ y[n]. 3. Interpolating p-filters If the DFT of the even polyphase component of a p-filter h is constant ĥ0[n] C then the p-filter is called interpolating. In that case, Eq. 3. implies that the DFT of the zero polyphase component of the output signal ˇx ˇx 0 [n] = C ˆ y[n]. This means that ˇx[k] = C y[k], k Z. 3. Filter banks The set of p-filters H = { hs }, s = 0,..., S, which, being time-reversed and applied to an input signal x Π[], produces the set of the output signals {ỹ s } S s=0 ỹ s [l] = k=0 downsampled by factor of M, h s [k Ml] x[k], s = 0,..., S, l Z, 3.3 is called the S channel analysis p-filter bank. The set of filters H = {h s }, s = 0,..., S, which, being applied to a set of input signals {y s } Π[/M], s = 0,..., S, that are upsampled by factor of M, produces the output signal ˇx[l] = S /M s=0 k=0 h s [l Mk] y s [k], l Z, is called the S channel synthesis p-filter bank. If the upsampled signals ỹ s, s = 0,..., S, which are defined in Eq. 3.3, are used as an input to the synthesis p-filter bank and the output signal is ˇx = x, then the pair of analysis synthesis p-filter banks form a perfect reconstruction PR p-filter bank. If the number of channels S equals to the downsampling factor M then the p-filter bank is said to be critically sampled. If S > M then the p-filter bank is oversampled. Critically sampled PR p-filter banks are used in the wavelet analysis, while oversampled PR p-filter banks serve as a source for discrete-time wavelet frames design. In the paper, we deal with p-filter banks, whose downsampling factor is M = and h 0 and h 0 are the low-pass filters. 3.3 Characterization of p-filter banks Assume that H = { hs }, s = 0,..., S, is an analysis p-filter bank with the downsampling factor of. Then, its application to a signal x Π[] produces S signals from Π[/]: It follows from Eq. 3. that ỹ s [l] = k=0 h s [k l] x[k], s = 0,..., S, l Z. 3.4 ỹ s [n] = hs 0 [ n] x 0 [n] + hs [ n] x [n], n Z

7 Assume that H = {h s }, s = 0,..., S, is a synthesis p-filter bank with the upsampling factor of. Then, its application to the upsampled signals ỹ s Π[/] produces a signal from Π[]: S ˇx[l] = s=0 / Equation 3. implies that the DFT of the polyphase components are k=0 h s [l k] ỹ s [k], l Z. 3.6 S ˇx 0 [n] = ĥ s 0 [n] ỹ S s [n], ˇx [n] = ĥ s [n] ỹ s [n]. 3.7 s=0 s=0 Equations 3.5 and 3.7 can be represented in a matrix form by ỹ 0 [n]. = x0 [n] P[ n] ˇx, 0 [n] = P[n] x [n] ˇx [n] ỹ S [n] ỹ 0 [n]. ỹ S [n] where the S analysis and the S synthesis polyphase matrices are, respectively, P[n] = h 0 0 [n] h 0 [n] h S 0 [n] hs [n] P[n] = ĥ0 0 [n]... ĥ 0 [n]... h S 0 [n], n Z. h S [n] If the relations P[n] P[ n] = I, 3.8 where I is the identity matrix, holds for all n Z then x0 [n] P[n] P[ n] x0 [n] =. x [n] x [n] { } Thus, Eq. 3.8 is the condition for the analysis synthesis pair H, H of p-filter banks to form a PR p-filter bank. 4 Frames in the periodic signals space Definition 4. A system Φ = { φ l } L, L, of signals from Π[] forms a frame of the space Π[] if there exist positive constants A and B such that for any signal x = {x[k]} Π[] L A x x, φ l B x. If the frame bounds A and B are equal to each other then the frame is said to be tight. If the system Φ is a frame then there exists another frame Φ = {φ l } L of the space Π[] such that any signal x Π[] can be expanded into the sum x = L x, φ l φ l. The analysis Φ and synthesis Φ frames can be interchanged. Together they form the so-called bi-frame. If the frame is tight then Φ can be chosen as Φ = c Φ. 7

8 If the elements { φ l } of the analysis frame Φ are not linearly independent L > then many synthesis frames can be associated with a given analysis frame. In this case, the expansion x = L x, φ l φ l provides a redundant representation of the signal x. It was established in [4] that the PR filter banks operating in the space l of decaying discrete-time signals generate frames of this space. A similar fact was proved in [36] for the p-filter banks operating in Π[]. 4. One-level frame transform Assume an analysis H = { hs }, s = 0,..., S, and a synthesis H = {h s }, s = 0,..., S p-filter banks with downsampling by factor of form a PR p-filter bank. Denote ψ [] s = { ψs [] [l] = h } { } s [l], ψ[] s = ψ[] s [l] = hs [l], s = 0,..., S, 4. where { hs [l]} and {h s [l]} are the impulse responses of the p-filters h s and h s, respectively. Then, the relations in Eqs. 3.4 and 3.6 can be rewritten as S x[k] = x s [] [k], xs [] [k] = s=0 y[] s [l] = κ=0 S S x = x s [] = s=0 / ψ [] s [κ l] x[κ] = x, ψ [] s [ l], s=0 / y[] s [l] ψs [][k l], 4. x, ψ [] s [ l] ψs [][ l]. 4.3 } Thus, the signal x Π[] becomes expanded over the system {ψ s[] [ l], s = 0,..., S, l = 0,...,. The expansion coefficients are the inner products { x, ψ } s[] [ l]. Theorem 4. [36] If the polyphase matrices P and P of the} p-filter banks H } and H satisfy the PR conditions Eq. 3.8 then the -sample circular shifts { ψs [] [ l] and {ψ s[] [ l], s = 0,..., S, l = 0,..., / of the signals ψ [] s and ψs [], s = 0,..., S, defined in Eq. 4. form a pair of the analysis and the synthesis frames of the space Π[], respectively. These frames are interchangeable. If the relations hold then the frame is tight. P[n] T P[ n] = ci, n Z, 4.4 The notation T means matrix transposition. If the condition in Eq. 4.4 is satisfied, then the synthesis filter bank can be chosen to be equal to the analysis one up to a constant factor. If the number of channels in the p-filter banks H and H is S =, then the number of elements } in the frames is that equals the dimension of the space Π[]. Thus, in that case the sets { ψs [] [ l] and } {ψ s[] [ l], s = 0,, l = 0,..., /, constitute a biorthogonal pair of bases of the space Π[]. If S > then the representation in Eq. 4.3 of signals from Π[] is redundant. The redundancy ratio for one-level frame transform is ρ = S/ :. The signals ψ [] s and ψs [], s = 0,..., S, are called the analysis and synthesis discrete-time framelets of the first decomposition level, respectively. 8

9 4. Multi-level frame transform To increase the { redundancy } of a signal representation, the frame transform is applied to the low frequency signal y[] 0 = y[] 0 [l], y[] 0 [l] = ψ k=0 [] s [k l] x[k], which belongs to Π[/]. Assume that an analysis H } } [] = { hs [], s = 0,..., S, and a synthesis H [] = {h s [], s = 0,..., S, are p-filter banks with downsampling by factor of that form} a PR p-filter } bank, which operates in the subspace Π[/]. It means that the IR of the filters { hs [] [k] and {h s[] [k], k Z, are /-periodic signals. Remark 4. Assume P[n] and P[n], n Z, are the polyphase matrices of the p-filter banks H and H, respectively. It is natural to define the p-filter banks H [] [n] and H [] [n] via their polyphase matrices as P [] [n] = P[n], P [] [n] = P[n]. Surely, the PR conditions P [] [n] P [] [ n] = I are satisfied. However, different p-filter banks are possible. Similarly to the whole signal x, the low-frequency signal y[] 0 can be represented as S y[] 0 [k] = s=0 / y[] s [l] = = = κ=0 x[κ] ν=0 /4 / ν=0 y[] s [l] hs [][k l], 4.5 / h s [] [ν l] y0 [] [ν] = ν=0 h s [] [ν l] ψ [] 0 [κ ν] = h s [] [ν l] κ=0 x[κ] κ=0 / ν=0 x, ψ / s[] [ 4l], where ψ [] s [κ] = h s [] [ν] ψ [] 0 [κ ν]. ν=0 By substituting Eq. 4.5 into Eq. 4. with s = 0 we get. The signals / x 0 [] [k] = S = s=0 /4 λ=0 where ψ s [] [k] = / y[] 0 [l] ψ0 [] [k l] = / y[] s [λ] / S ψ[] 0 [k l] s=0 S ψ[] 0 [k l] hs [] [l λ] = s=0 ψ[] 0 [k l] hs [][l], s = 0,..., S. ψ [] 0 [κ ν] x[κ] h s [] [ν] ψ [] 0 [κ ν 4l] /4 λ=0 /4 λ=0 y[] s [λ] hs [][l λ] y[] s [λ] ψs [][k 4λ], } { } { ψs [] and ψ[] s are called the analysis and the synthesis discrete time periodic framelets of the second decomposition level, respectively. As a result from the two-level framelet transform, we have the signal x expanded over the 4-sample shifts of the second level framelets and the -sample shifts of the first level high-frequency framelets: x = S s=0 xs [] + S s= xs [], where x s [] [k] = /4 x, ψ s[] x, [ 4l] ψ s[] [ l] x s [] [k] = / ψ[] s [k 4l], s = 0,..., S, ψ[] s [k l], s =,..., S. 9

10 Assume that the analysis H } } [µ] = { hs [µ], s = 0,..., S, and the synthesis H [µ] = {h s [µ], s = 0,..., S, p-filter banks with downsampling by factor of form PR p-filter banks, which operates in the subspaces Π[/ µ ] and P [µ] [n], µ = 0,...m, and P [µ] [n] are their polyphase matrices, respectively. Then, the iterated transform with the polyphase matrices P [µ] [n] and P [µ] [n] leads to the frame expansion of the signal x = x 0 [m] + m S µ= s= xs [µ], where x s [m] [k] = l Z x, ψ [m] s [ m l] ψ[m] s [k m l], s = 0,..., S, x s [µ] [k] = / µ x, ψ 4.6 [µ] s [ µ l] ψ[µ] s [k µ l], µ < m, s =,..., S. The synthesis and analysis framelets are derived iteratively ψ s [µ] [k] = / µ h s [µ ] [l] ψ0 [µ ] [k µ l], ψ [µ] s [k] = / µ h s [µ ] [l] ψ [µ ] 0 [k µ l], s = 0,..., S Implementation of the multi-level frame transform For uniformity, denote temporarily the PR pair of p-filter banks H and H from the initial level as H [0] and H [0], respectively. Their polyphase matrices are denoted as P [0] [n] and P [0] [n]. Assume that x = {x[k]} Π[] is a signal to be operated by the m-level frame transform with the PR pairs H [µ] and H [µ], µ = 0,..., m. For simplicity, assume that the polyphase matrices of the p-filter banks from the adjacent levels are linked as P [µ+] [n] = P [µ] [n], P [µ+] [n] = P [µ] [n]. 0

11 Algorithm : Direct transform : Initialization: y = x. Split y into the even and the odd polyphase components y = y 0 + y and calculate the DFT of {ŷ 0 [n] } and {ŷ [n] }. : for µ = to m do Implement the one-level transform Calculate the IDFT ŷ 0 [µ] [n] µ. y S [µ] [n] µ = P [µ ] [ n] ŷ0 [n] µ ŷ [n] µ / µ y[µ] s µ [k] = ω µ kn ŷ [µ] s [n] µ, s =,..., S, n= 0 } and store the transform coefficients y[µ] s = {y s[µ] [k], k = 0,.../ µ, s =,..., S of the µ-th decomposition level Put ŷ[n] µ = ŷ[µ] 0 [n] µ, n Z. Compute the DFT {ŷ 0 [n] µ+ } and {ŷ [n] µ+ } of the even and the odd polyphase components of the signal y:, Calculate the IDFT ŷ 0 [n] µ+ = ŷ[n] µ + ŷ[n + / µ+ ] µ ŷ [n] = ŷ[n] µ ŷ[n + / µ+ ] µ ω µ k y 0 [m] [k] = m / m n=0 and store the transform coefficients y 0 [m] ω m kn ŷ 0 [m] [n] m } = {y 0[m] [k], k = 0,.../ m

12 Algorithm : Inverse transform 3: Initialization: Calculate the DFT ŷ s [m] [n] m = / m k=0 ω m kn y s [m] [k], k = 0,.../m, s = 0,...S for µ = m to 0 do Implement the one-level inverse transform ŷ [µ+] 0 [n] µ+ ŷ0 [n] µ = P ŷ [n] [µ] [n] µ. y S [µ+] [n] µ+ Calculate the IDFT Compute the DFT ŷ s [µ] [n] µ = / µ k=0 ω µ kn y s [µ] [k], k = 0,..., / µ, s =,..., S Calculate the IDFT ŷ 0 [µ] [n] µ = ŷ 0 [n] µ+ + ω µ knŷ [n] µ+, Put x[k] = y 0 [k], x[k + ] = y [k] y 0 [k] = y [k] = / n=0 / n=0 ω kn ŷ 0 [n] ω kn ŷ [n] 5 Design of the linear phase p-filter banks for frames generation In this section, we discuss the design of three- and four-channel linear phase p-filter banks, which generate the tight and the so-called semi-tight frames in the periodic signals space Π[]. 5. Interpolating three-channel p-filter banks generating frames 5.. Interpolating p-filter banks for frame generation Assume that H = { h0, h, h } and H = { h 0, h, h } are the analysis and the synthesis p-filter banks that operate in the space Π[]. Assume that the low-pass p-filters h 0 and h 0 are interpolating, that is their frequency responses can be represented as h 0 [n] = + ω n f[n], ĥ 0 [n] = + ω n f[n], 5.

13 where f[n] and f[n] are / periodic sequences. Denote W [n] = f[ n] f[n]. We assume that the following conditions are satisfied: Conditions: A. f[n] and f[n] are rational functions of ω n = e πin/ that have no poles for n Z. A. f[0] = f[0] =. A3. Symmetry: ω n f[n] = ω n f[ n], ω n f[n] = ω n f[ n]. 5. Then, the p-filters h 0 and h 0 have linear phase. The polyphase matrices for filter banks comprising the interpolating low-pass filters are / f[n] / P[n] = h 0 [n] h [n], P[n] / ĥ = 0 [n] ĥ 0 [n] h 0 [n] h [n] f[n] / ĥ [n] ĥ [n]. The PR condition Eq. 3.8 leads to where P [n] P [ n] = Q[n], 5.3 P [n] = h 0 [n] h [n], P h 0 [n] h [n] ĥ = 0 [n] ĥ 0 [n] [n] ĥ [n] ĥ [n], Q[n] = f[ n]. 5.4 f[n] f[n] f[ n] Therefore, design of a PR p-filter bank starting from the interpolating low-pass filters h 0 and h 0 reduces to factorization of the matrix Q[n] as in Eq A straightforward option is the triangular factorization of Q[n]: P [n] = 0 w[n] f[n], P [n] = 0, 5.5 w[n] f[n] where w[n] w[ n] = W [n]. Thus, to complete { } the design, the function W [n] should be factorized. As soon as it is done, the PR p-filter bank H, H, where H = { h0, h, h }, and H = { h 0, h, h }, whose frequency responses are h 0 [n] = + ω n f[n], ĥ 0 [n] = + ω n f[n], h [n] = ω n w[n] /, ĥ [n] = ω n w[n] / 5.6 h [n] = ω n f[n] = h0 [n + /], ĥ [n] = ω n f[n] = ĥ0 [n + /], comes to existence. In this case, the filters h and h are interpolating. The condition f[0] = f[0] = implies that h [0] = h [0] = 0 and these filters are high-pass. The filters h and h do not have even polyphase component. The PR p-filter bank Eq. 5.6 generates a bi-frame in the periodic signals space Π[]. The 3

14 framelets of the first decomposition level are equal to the IR of the corresponding filters: ψs [] [k] = h s [k], ψ[] s [k] = hs [k], s = 0,,, k Z. The framelets of the subsequent decomposition levels can be derived by an iterated application of the formulas in Eq However, a simpler way to derive, for example, a synthesis framelet ψ[m] s of the level µ is to launch the m-level inverse transform using the polyphase matrices P [µ] as it is described in Section 4.3, provided that all the transform coefficients except for the single coefficient y[µ] s [] are zero and ys [µ][] =. In order to derive an analysis framelet ψ s [µ], we can apply the same procedure using the analysis polyphase matrices P [µ] instead of the synthesis matrices P [µ]. This is justified by the fact that the synthesis and the analysis frames are interchangeable. 5.. Tight and semi-tight frames When the sequence satisfies f[n] = f[n], we have ĥ0 [n] = h0 [n], ĥ [n] = h [n] = h 0 [n + /] and W [n] = f[n] = ĥ0 [n] ĥ [n]. 5.7 The condition A and the symmetry condition A3 imply that, in this case, the numerator and the denominator of the sequence W [n] are trigonometric polynomials with real coefficients containing only cosines of the form cos πνn/. Tight frames; If the inequality f[n] as n Z 5.8 holds, then, the sequence W [n] can be factored as W [n] = w[n] w[ n] /. This factorization is not unique. However, the following Riesz s lemma holds. Lemma 5. [8] Let A be a positive trigonometric polynomial containing only cosines, Av = M ν= a ν cos νv, with real coefficients a ν. Then, there exists a trigonometric polynomial Bv = M ν= b ν e iνv order M with real coefficients b ν such that Bv = Av. Thus, a rational factorization W [n] = w[n] w[ n] is possible and we have h [n] = h [n]. analysis p-filter bank coincides with the synthesis p-filter bank The ĥ 0 [n] = h0 [n] = + ω n f[n] /, ĥ [n] = h [n] = ω n f[n] /, ĥ [n] = h = ω n w[n] /, that generates a tight frame. ote that the symmetry of the sequence W [n] does not guarantee the antisymmetry of the sequence w[n] = W [n]. Semi-tight frames: If the condition in Eq. 5.8 is not satisfied, t is still possible to generate frames that are very close to tight frames. In this case, the sequence W [n] can be factored as W [n] = w[n] w[ n] and the frequency responses of the filter bank are 5.9 ĥ 0 [n] = h0 [n] = + ω n f[n] /, ĥ [n] = h [n] = ω n f[n] /, h [n] = ω n w[n] /, ĥ [n] = ω n w[n] /. 5.0 Such a p-filter bank generates a frame, which is natural to refer to as to a semi-tight frame. In the nonperiodic setting, this notion was introduced in [6]. Due to the symmetry of W [n], its antisymmetric factorization W [n] = w[n] w[ n] is always possible. Therefore, even when Eq. 5.8 holds, sometimes it is preferable to construct a semi-tight rather than a tight frame. 4

15 5. Four-channel p-filter banks for tight frames generation According to Theorem 4., if the polyphase matrix P[n] of an analysis filter bank H is paraunitary, that is it satisfies the condition P T [n] P[ n] = I, n Z, then the p-filter bank H generates a tight frame of the signal space Π[]. In other words, the analysis filter bank should coincide with the synthesis one. Thus, we drop and denote ĥ 0 0 [n] ĥ 0 [n] P[n] = ĥ 0 [n] ĥ [n] ĥ 0 [n] ĥ [n]. ĥ 3 0 [n] ĥ 3 [n] The design begins from a linear phase low-pass filter h 0, whose frequency response ĥ0 [n] = ĥ0 0 [n] +ω nĥ0 [n] is a rational of ω n = e πin/ function with real coefficients that has no poles for n Z. Assume ĥ0 [n] is symmetric about swap n n, which implies that ĥ0 0 [n] = ĥ0 0 [ n] and ω nĥ0 [n] = ω n ĥ0 [ n]. addition, assume that the following inequality holds h 0 0[n] + h 0 [n]. 5. The paraunitarity of the matrix P[n] implies that its columns are orthogonal to each other while their norms are equal to. The orthogonality of the whole columns can be achieved by separate orthogonalization of the first and the last pairs of rows: In ĥ 0 0 [n] ĥ0 [ n] + ĥ 0 [n] ĥ [ n] = 0, ĥ 0 [n] ĥ [ n] + ĥ3 0 [n] ĥ3 [ n] = An obvious way to satisfy Eqs. 5. is to select ĥ 0 [n] = ĥ0 [ n], ĥ [n] = h 0 0[ n], ĥ 3 0 [n] = ĥ [ n], ĥ3 [n] = ĥ 0 [ n]. 5.3 Thus, due to the symmetry of ĥ0 [n], the frequency response ĥ [n] = ĥ0 [ n] ω nĥ0 0 [ n] = ω n ĥ 0 0 [n] ω nĥ0 [n] = ω n ĥ 0 [n + /]. 5.4 The sequence ω nĥ [n] = ĥ0 [n + /], which is the DFT of the sequence { h [k + ] }, is symmetric about change n n. Thus, the IR { h [k] } is symmetric about k =. Subject to Eq. 5.3, the columns norms will be equal to one if ĥ 0 [n] + ĥ [n] = ĥ0 0 [n] + ĥ0 [n]. 5.5 Thus, the design of the paraunitary matrix P[n] is reduced to finding two rational functions of ω n ĥ 0 [n] and ĥ [n], which satisfy Eq The task becomes even simpler if we choose ĥ [n] = ĥ 0 [ n] = A[n]. 5.6 A consequence of this choice is the fact that the p-filters h and h 3 have a linear phase: ĥ [n] = A[ n] + ω n A[n], ĥ 3 [n] = A[ n] + ω n A[n], ĥ [ n] = A[n] + ω n A[ n] = ω n h [n], ĥ 3 [ n] = A[n] + ω n A[ n] = ω n ĥ 3 [n]. The IR of the p-filter h is symmetric about -/, while the IR of h 3 is antisymmetric about -/. 5

16 By substituting Eq. 5.6 in Eq. 5.5 we get A[n] = ĥ0 0 [n] + ĥ0 [n] The solution of Eq. 5.5 completes the design of the linear phase p-filter bank H, which generates a tight frame in the signal space Π[]. The frequency responses of the p-filters are ĥ 0 [n] = ĥ0 0 [n] + ω nĥ0 [n], ĥ [n] = ĥ0 [ n] + ω nĥ0 0 [ n], ĥ [n] = A[ n] + ω n A[n], ĥ 3 [n] = A[ n] + ω n A[n]. 5.8 Proposition 5. Assume that h 0 is a low-pass p-filter, whose frequency response ĥ0 [n] = ĥ0 0 [n] +ω nĥ0 [ n] is a rational function of ω n, which does not have poles for n Z. If the inequality Eq. 5. holds then there exists a rational solution of ω n A[n] of Eq. 5.7, which does not have poles for n Z. Consequently, the frequency responses of the four-channel p-filter bank H, which are given in Eq. 5.8, are rational functions of ω n, which do not have poles for n Z. If the sequence h 0 [n] is symmetric about the swap n n, then the IR of h 0 is symmetric about zero, the IR of h is symmetric about -, the IR of h is symmetric about -/ and the IR of h 3 is antisymmetric about -/. Proof: The rational function of ω n h 0 0[n] = h 0 0[n] h 0 0[ n] is symmetric about inversion ω n ω n. Therefore, it is a fraction of two positive cosine polynomials. The same is true for the function h 0 0[n]. Then, Eq. 5. implies that the sequence T [n] = ĥ0 0 [n] + ĥ0 [n] = P cos πn/ Qcos πn/, where Q is strictly positive and P is non-negative polynomials. Due to Riesz Lemma 5., the polynomials can be factorized P cos πn/ = pω n pω n, Qcos πn/ = qω n qω n, where p and q are polynomials with real coefficients and qω n does not have roots for n Z. Thus, the solution to Eq. 5.7 is the rational function of ω n A[n] = pω n /qω n. The symmetry of the p-filters constituting the p-filter bank H was discussed above. Corollary 5.3 Under the conditions of Proposition 5., the p-filter bank H, whose frequency responses are given in Eq. 5.8, generates a tight frame of the periodic signals space Π[]. Remark 5. The design scheme of the four-channel p-filter bank generating tight frames described in this section is inspired by the scheme presented in [9]. Certainly, other schemes are possible. 5.. Interpolating p-filter banks Assume the low-pass p-filter h 0 is interpolating and its frequency response is ĥ0 [n] = + ω n f[n] /, where the sequence f[n] satisfies the conditions A A3. Then, Eq. 5.8 and the symmetry condition A3 imply that the frequency response of the high-pass p-filter is ĥ [n] = ω n ω n f[ n] = ω n ω n f[n] = ω n ĥ 0 [n + /]. The sequence A[n] is derived from the equation A[n] = ĥ0 0 [n] + ĥ0 [n] = W [n], W [n] = f[n]. 5.9 Thus, as in the case of three-channel interpolating p-filter banks, design of the four-channel interpolating p-filter bank is reduced to the factorization of the sequence W [n] = A[n] A[ n]. 6

17 5.3 Restoration of sampled polynomials and discrete vanishing moments It is apparent that as the frequency response of a low-pass filter h at the vicinity of zero becomes more flat, it better restores band-limited signals. To be specific, if ĥ[n] c as n n 0 up to a period and the DFT ˆx[n] of a signal x Π[] is zero outside the interval n 0 n n 0, then application of the p-filter h restores the signal x by h[k l] x[l] = c x[k] Restoration of sampled polynomials The flatness of the frequency response of a low-pass filter can be characterized by the difference ĥ[n] ĥ[0]. Under the assumption that the frequency response ĥ[n] is a rational function of ωn = e πin/, which has no poles while n Z, the difference ĥ[n] ĥ[0] = ωn m α[n], where m is some natural number and α[n] is a rational function of ω n, which has no poles while n Z and α[0] 0. As the multiplicity m of the root is higher, the frequency response becomes flatter. Equation. claims that the sequence ω n m is the DFT of the circular finite difference and ω n mˆx[n] = m ˆ[x][n]. The sequence {α[n]} can be regarded as the frequency response of a low-all-pass p-filter a. Then, application of the p-filter h to a signal x Π[] can be represented as h x = ĥ[0] x + a m [x]. 5.0 Certainly, sampled polynomials do not belong to Π[] and p-filters can not be applied to them. However, the following fact holds, which in a sense is a discrete periodic counterpart of the classical Fix-Strang condition [35]. Proposition 5.4 Assume that the frequency response of the low-pass filter h can be represented as ĥ[n] = ĥ[0] + ω n m α[n], where m is some natural number and α[n] is a rational function of ω n, which has no poles for n Z and α[0] 0. Assume p is a signal from Π[], which coincides with a sampled polynomial P m of degree m at some interval p[k] = P m [k] as k = k 0,..., k m, where m < k m k 0 <. Then, h[k l] p[l] = ĥ[0] P m [k], as k = k 0,..., k m m. Proof: Under the conditions of the proposition, application of the p-filter h to a signal x Π[] can be represented as in Eq The finite difference of the signal p m m m [p][k] = l p[k + l] = m [P m ][k], as k = k 0,..., k m m. l According to Proposition., m [P m ][k] = 0. Thus, the statement follows from Eq Definition 5.5 If a low-pass p-filter h meets the conditions of Proposition 5.4 then we say that the p-filter h locally restores sampled polynomials of degree m. Proposition 5.6 Assume that Rz is the rational function, which does not have a root at z =, Rz = P ν z/q κ z and the polynomials P ν z and Q ν z satisfy P ν z = p + z s P ν s z, Q κ z = q + z r Q κ r z. If p 0 then Rz = R + z mins,r Rz, where Rz is a rational function, which does not have root at z =, Proof: is straightforward Discrete vanishing moments Proposition 5.7 Assume that the frequency response of the highband-pass filter g can be represented as ĝ[n] = ω n m α[n], where m is some natural number and α[n] is a rational function of ω n, which has no poles for n Z and α[0] 0. Assume p is a signal from Π[], which coincides with a sampled polynomial P m of degree m at some interval p[k] = P m [k] as k = k 0,..., k m, where m < k m k 0 <. Then, g[k l] p[l] = 0, as k = k 0,..., k m m. 7

18 Proof: is similar to the proof of Proposition 5.4. Definition 5.8 If a highband-pass p-filter g satisfies the conditions of Proposition 5.7, we say that the p-filter g locally eliminates sampled polynomials of degree m. If a framelet ψ [] [k] = g[k], k Z, we say that the framelet ψ [] has m local discrete vanishing moments LDVM. Assume g is a high-pass p-filter, whose frequency response ĝ[n] is a rational function of ω n, which has no poles for n Z and ĝ[/] 0. Then, in the vicinity of /, the frequency response can be represented as ĝ[n] = ĝ[/] + ω n + l β[n], where l is some natural number and β[n] is a rational function of ω n, which has no poles for n Z and β[/] 0. The power l determines the flatness of the frequency response ĝ[n] in the vicinity of /. Remark 5. The statements of Propositions 5.4 and 5.7 concerning the low-pass h and high-pass g filters, respectively, remain true if the frequency responses are represented as ĥ[n] = ĥ[0] + sinm πn ᾱ[n], πn ĝ[n] = sinm ᾱ[n], where m is some natural number and ᾱ[n] is a rational function of ω n, which has no poles for n Z and ᾱ[0] 0. Restoration of polynomials by low-pass filters coupled with their elimination by respective high- and band-pass filters constituting the p-filter banks is important in signal processing applications. For example, when data compression is implemented by multiscale wavelet or framelet transforms, this property enables us to condense the information on the smooth component of a signal or an image into a small number of low-frequency transform coefficients. 6 Spline based interpolating filters Assume that the frequency response of a high-pass interpolating p-filter g operating in Π[] is presented by ĝ[n] = ω n f[n] /. Then, the polyphase components are g 0 = I/ and g = f/, where I is the identity operator in Π[/] and f is the p-filter in Π[/], whose frequency response is f[n]. Then, the DFT of the signal y = h x, where x Π[], is ŷ[n] = x0 [n] ω n f[n] x [n] + ω n x [n] f[n] x 0 [n]. Denote by x = {x[k ]} the shifted odd polyphase component, where x = {x[k + ]}. Then, the polyphase components of y are y 0 = x 0 f x, y = x f x It is apparent from Eq. 6. that in order for the interpolating high-pass filter to eliminate smooth signals for example, fragments of polynomials, the p-filter f should be predictive in a sense that its application to the even polyphase component of the signal should predict the odd polyphase component and vice versa. Polynomial and discrete splines are natural sources to derive prediction p-filters from. The idea is to construct the spline quasi-interpolating the even samples of a signal and to predict the odd samples of the spline s values in the midpoints between the interpolation points. Such an approach was explored for the design of the biorthogonal wavelet transforms in [, 5, 4] and for the non-periodic interpolating frames [6, 7] with three-channel filter banks. 8

19 6. Prediction p-filters derived from polynomial splines A spline of order p on the grid {t k } is a function that has p continuous derivatives. At the intervals t k, t k+ between the grid points the spline of order p coincides with polynomials P p k of degree p. We deal with splines that are defined on the uniform grid {k}, which are periodic with the period K = / = j. 6.. Periodic interpolating splines Denote the space of splines of order p defined on the uniform grid {k}, which are periodic with the period K = / = j, by S p K. A basis in this space is constituted by shifts of the so-called B-splines. The K-periodic B-spline B t of first order is the periodization of the compactly supported function β t { β t =, if t /, /; B t = β t + Kl = πint/k sin πn/k e 0, otherwise, K πn/k. l Z n Z The B-splines of higher order are defined iteratively via the circular convolutionb p t = B B p t. Consequently, the expansion in Fourier series is B p t = p sin πn/k e πint/k. K πn/k n Z The B-spline B p t is supported on the interval p/, p/ up to periodization, is strictly positive inside this interval and is symmetric about zero, where it has its single maximum. Any spline S p t from S p K is represented as S p t = K k=0 q[k] B p t k. 6. If the spline S p t interpolates the even polyphase component x 0 = {x[l]} of a signal x Π[] then its coefficients q[k] can be explicitly calculated via the DFT S p l = K k=0 K u p [n] = k=0 q[k] B p l k = x 0 [l] ˆq[n] u p [n] = x 0 [n], ω nk B p k > 0 = ˆq[n] = x 0[n] u p [n]. 6.3 The K = /-periodic sequence u p [n], which is the DFT of the sampled B-spline {B p k}, is strictly positive. Due to symmetry and positiveness of the B-spline, u p [n] = u p [ n], thus it is a cosine polynomial with positive coefficients. It is symmetric about K/, where it has its single minimum. The maximal value of u p [n] is being reached when n = 0. Approximation properties of interpolating splines are well investigated. In particular, the non-periodic interpolating spline of order p, which consists of piece-wise polynomials of degree p, restores these polynomials. It means that the spline of order p, which interpolates a polynomial of degree p, coincides with this polynomial. For periodic splines, this property holds locally. This observation justifies the choice of interpolating splines as a source for the design of prediction filters. To be specific, if the spline S p t S p K, given in Eq. 6., interpolates the even polyphase component x 0 = {x[l]} of a signal x Π[] then the odd polyphase component x = {x[l + ]} is predicted as the midpoint values of the spline σ[l] = S p l + / = K k=0 q[k] Bp l + / k, whose DFT is ˆσ[n] = ˆq[n] v p [n] = vp K [n] u p x 0 [n], where v p [n] = ω nk B p l + [n] 9 k=0

20 is the DFT of the B-spline {B p k + /} sampled at midpoints between grid points. The continuous counterparts of the sequences u p [n] and v p [n], which are the discrete-time Fourier transforms of the non-periodic B-splines, were introduced and studied in [3]. Some additional properties of the sequences u p [n] and v p [n] are established in [38, ]. Denote Obviously, the sequence f p c [n] satisfies the conditions A A3. Proposition 6. [] If the spline order is p = r or p = r then f p c [n] = v p [n] u p [n]. 6.4 ĥ[n] = + ω n f p c [n] = r πn ĥ[0] + sin χp [n] 6.5 ĝ[n] = ω n f p c [n] = sin r πn γp [n], 6.6 where r is a natural number and χ p [n] and γ p [n] are rational functions of ω n, which have no poles for n Z and no root at n = 0. This proposition coupled with Propositions 5.4 and 5.7 and Remark 5. imply the following Corollary. Corollary 6. The low-pass h and the high-pass g p-filters, whose frequency responses are given in Eqs. 6.5 and 6.6,respectively, locally restore and eliminate sampled polynomials of degree r, respectively. Therefore, the p-filter f p c, whose frequency response f p c [n] is defined by Eq. 6.4, is a proper candidate to be utilized as a prediction p-filter. Remark 6. It is emphasized that the p-filters derived from the odd order r splines, restore eliminate sampled polynomials of the same r degree as the the p-filters derived from the even order r splines. This is a consequence of the so-called super-convergence property of the odd order interpolating splines [38], which claims that the approximation order of such splines at the midpoints between the interpolation points is higher than those at the remaining points of the intervals between the interpolation points. 6.. Examples of spline based p-filters The sequences {v p [n] } and {u p [n] }, whose ratio constitute the frequency response f p c [n] of a prediction p-filter f p c, are calculated explicitly via the DFT of the B-splines sampled at the points {k + /} and {k}, respectively. Table presents the sequences {B p k} and {B p k + /} for different p values. k B k B 3 k B 4 k B 5 k B 6 k B 7 k B k + / B 3 k + / B 4 k + / B 5 k + / B 6 k + / B 7 k + / Table : Sampled B-splines {B p k} and {B p k + /}. 0

21 Linear spline p = : u [n] =, v [n] = + ωn, fc [n] = + ωn, ĥ[n] = + ω n f c [n] = cos πn, ĝ[n] = sin πn. Quadratic spline p = 3: Cubic spline p = 4: u 3 [n] = ωn ω n 8 fc 3 + ω n [n] = 4 ĥ[n] =, v 3 [n] = + ωn, ω n ω n = ωn cos4 πn/ sin 4 πn/ cos 4 πn/ + sin 4 πn/, 6.7 cos 4 πn/ cos 4 πn/ + sin 4 πn/, ĝ[n] = sin 4 πn/ cos 4 πn/ + sin 4 πn/. fc 4 [n] = ω4n + 3ω n ω n 8ω n ω n, 6.8 ĥ[n] = cos4 πn/ + cos πn/ + cos 4πn/, ĝ[n] = sin4 πn/ + cos πn/. + cos 4πn/ Comment We observe that the high-pass p-filters g derived either from the quadratic or from the cubic splines locally eliminate sampled cubic polynomials. However, the structure of the quadratic frequency response is much simpler than the cubic frequency response. Therefore, in many applications the p-filters derived from quadratic spline are advantageous over the cubic spline p-filters. The p-filters, originated from higher order splines, are designed by the application of the DFT to the sampled B-splines. 6. Prediction p-filters derived from discrete splines Another source for designing prediction p-filters is provided by the so-called discrete interpolating splines, which are the discrete-time counterparts of the polynomial splines [33, 7, 5, 4]. The scheme of the p-filters design is similar to the scheme that utilizes polynomial splines. 6.. Periodic discrete interpolating splines The centered non-periodic discrete B-spline b of second order is the three-impulse signal: b [ ] = b [] = /4, b [0] = /. The -periodic discrete B-spline b of second order is produced by -periodization of b by b [k] = l Z b [k + l]. The DFT is b [k] = cos πn/. The discrete B-splines of a high even order are defined iteratively via the discrete circular convolution: b r = b b r = b r r πn [n] = cos. 6.9 The -periodic discrete splines s r are defined as linear combinations of the B-splines: s r [k] = / σ[l]b r [k l] = ŝr r πn [n] = ˆσ[n] cos. 6.0

22 Since the sequence ˆσ[n] is /-periodic, the DFT of the even s r 0 and odd s r polyphase components of the discrete spline s r are ŝ r 0 [n] = ŝr [n] + ŝr [n + /] = ˆσ[n] U r [n], ŝ r [n] = ŝr [n] ŝr [n + /] ω n = ˆσ[n] V r [n], U r [n] = cos r πn + sinr πn, V r [n] = ω n cos r πn πn sinr. 6. Mimicking the p-filter design scheme based on polynomial splines, we assume that the discrete spline s r interpolates the even polyphase component x 0 of a signal x Π[] on the grid {k} and predict the component x via the spline values on the grid {k + }. Thus, s r 0 = x 0 = ˆσ[n] = x 0[n] U r [n], ŝ r [n] = V r [n] U r [n] x 0 [n]. 6. Like the sequence u p [n] defined in Eq. 6.3, the sequence U r [n] is strictly positive, symmetric about K/ = /4 and U r [0] =. Similarly to Eq. 6.4, define the prediction p-filters fd r via the frequency responses: fd r V r [n] [n] = U r = ωn cos r πn/ sin r πn/ [n] cos r πn/ + sin r πn/. 6.3 The sequence fd r[n] satisfies the conditions A A3. Then, interpolating the low-pass h and the high-pass g p-filters can be represented explicitly via their frequency responses: ĥ[n] = + ω n f r d [n] = cos r πn/ cos r πn/ + sin r πn/, 6.4 ĝ[n] = ω n f r d [n] = sin r πn/ cos r πn/ + sin r πn/. 6.5 Obviously, the low-pass h and the high-pass g p-filters locally restore and eliminate sampled polynomials of degree r, respectively. Remark 6. The sequences ĥ[n] and ĝ[n] are the frequency responses of the periodized half-band low- and high-pass Butterworth filters, respectively. The Butterworth filters are widely used in signal processing [6]. 6.3 Examples of p-fir p-filters The IR of most p-filters introduced in Sections 6. and 6. are infinite, that is they occupy the whole set Z. In this section, we introduce a few p-filters, whose IR are finite up to periodization p-fir p-filters. One such p- filters is originated from the linear interpolating spline. Another way to design the p-fir prediction p-filters is to construct the splines, which quasi-interpolate rather that interpolate the even polyphase component of a signal x and to predict the odd polyphase component by the spline s values at the midpoints between the grid nodes. The quasi-interpolating splines are studied in [37, 3]. Linear spline: The frequency response of the p-filters h and g are ĥ[n] = cos πn, ĝ[n] = sin πn. Their IR comprise only three terms up to periodization: h[] = h[ ] =, g[] = g[ ] =, g[0] = h[0] = 4 4.

23 The low-pass h and the high-pass g p-filters locally restore and eliminate sampled linear polynomials, respectively. Quadratic quasi-interpolating spline: The frequency response of the prediction p-filter f 3 c, derived from a quadratic interpolating spline, is given as f 3 c [n] = ωn cos πn/ sin πn/ = ωn cos πn k πn l sin. 6.6 Two initial terms in the series in Eq. 6.6 are taken and define fq 3 [n] = ω n cos πn + πn sin = ω n ω n ω 4n The frequency responses of the corresponding p-filters h and g are ĥ[n] = ω 3n + 9ω n ω n ω 3n 6 ĝ[n] = ω 3n 9ω n + 6 9ω n + ω 3n 6 = cos 4 πn + sin πn = sin 4 πn + cos πn., 6.8 Their IR comprise five terms up to periodization. The low-pass h and the high-pass g p-filters locally restore and eliminate sampled cubic polynomials, respectively. Like the interpolating spline, the quadratic quasi-interpolating spline possesses the super-convergence property at midpoints between grid nodes. This is the reason why the p-filters h and g locally restore and eliminate sampled cubic polynomials, respectively. Upgraded quadratic quasi-interpolating spline: It is possible to upgrade the quadratic quasi-interpolating spline in a way that enhances the super-convergence property [3]. The frequency response of the prediction p-filter f uq originating from the upgraded spline is fuq[n] 3 = 3ω 4n 5ω n ω n 5ω 4n + 3ω 6n Then, the frequency responses of the p-filters h and g are ĥ[n] = cos 6 πn ĝ[n] = sin 6 πn + 9 sin πn 3 πn sin,. + 9 cos πn 3 πn sin Their IR comprise seven terms up to periodization. The low-pass h and the high-pass g p-filters locally restore and eliminate sampled polynomials of fifth degree, respectively. Pseudo-spline p-filters: A family of linear phase FIR low-pass filters is introduced in [9] in non-periodic setting. In the periodic case, the design starts from the obvious identity = cos πn πn m+l + sin, m, l. 6.0 The frequency response of the low-pass p-filter h is derived by summation of the first l + terms of the binomial expansion in Eq. 6.0: ĥ[n] = m πn cos l m + l sin j j=0 j πn cosl j πn. 6. 3

24 Example m =, l = ĥ[n] = cos 4 πn cos πn πn + 3 sin. 6. We observe that the filter h defined by Eq. 6. is the same as the p-filter derived from the quadratic quasi-interpolating spline, whose frequency response is given by Eq Example from [9]: m = 3, l = ĥ[n] = cos 6 πn + 3 sin πn, ĝ[n] = sin 6 πn 3 cos 6 πn + 3 cos πn ĥ[n] ĥ[0] = sin πn πn πn + cos4 + cos Then, Proposition 5.4 implies that the low-pass p-filter h locally restores only first degree sampled polynomials. However, the high-pass p-filter g locally eliminates fifth degree sampled polynomials. Unlike all the previous examples, the p-filters h and g are not interpolating. The DFTs of the polyphase components of h are ĥ 0 [n] = 3ωn + ω n ω n 3ω n 8 The IR of the filters h and g comprise nine terms up to periodization., ĥ [n] = ωn + 9ω n + 9 ω n Interpolating three-channel p-filter banks using spline filters In this section, we present a few examples of PR three-channel p-filter banks, which generate tight and semi-tight frames in the space Π[] of -periodic discrete-time signals. In all the forthcoming examples, the low-pass and the high-pass p-filters are interpolating. The design scheme is described in Section p-filter banks derived from polynomial splines Linear polynomial spline: The prediction p-filter is f c [n] = + ω n /. Thus, the sequence W [n] = f c [n] f c [ n] = ωn + ω n can be factorized to be W [n] = w[n] w[ n], where w[n] = ω n /. The analysis p-filter bank, which coincides with the synthesis p-filter bankh = { h 0, h, h }, generates a tight frame in Π[]. The frequency responses of the p-filters are ĥ 0 [n] = + ω n f c [n] = cos πn, ĥ [n] = ω n w[n] = ω n ω n, 7. ĥ [n] = ω n f c [n] = sin πn. The p-filter h 0 locally restores the sampled polynomials of the first degree while the p-filter h locally eliminates them. The p-filter h eliminates only constants. Thus, the framelets ψ[] and ψ[], which are the IR of the p-filters h and h, respectively, have one and two LDVM, respectively. The impulse and the magnitude responses of the filters h s are displayed in Fig

25 Quadratic interpolating spline p = 3: Denote Ω 3 c[n] = cos 4 πn/ + sin 4 πn/. Then, f 3 c [n] = ω n cos4 πn/ sin 4 πn/ Ω 3 c[n], W [n] = ĥ 0 [n] = cos4 πn/ Ω 3 c[n], ĥ [n] = sin4 πn/ Ω 3 c[n] sin πn/ 7. Ω 3 c[n] ĥ [n] = ω n sin πn/ Ω 3 c[n]. The p-filter h 0 locally restores the sampled cubic polynomials while the p-filter h locally eliminates them. The p-filter h eliminates the first degree polynomials. Thus, the framelets ψ[] and ψ [] have two and four LDVM, respectively. The impulse and the magnitude responses of the filters h s are displayed in Fig Cubic interpolating spline: p = 4: Denote Ω 4 c[n] = ω n ω n, Γ 4 c[n] = ω n + 4 ω n. 7.3 Then, f 4 c [n] = ω4n + 3ω n ω n 8Ω 4 c[n], W [n] = ω n + ω n 8Ω 4 c[n] Γ 4 c[n], 7.4 ĥ 0 [n] = cos4 πn/ + cos πn/ Ω 4 c[n], ĥ [n] = sin4 πn/ + cos πn/ Ω 4 c[n]. The factorization W [n] = w[n] w[ n], which results in a tight frame, provides the p-filter h, whose frequency response is ĥ [n] = ω n ωn + ω n 8 q Ω 4 c[n] qω n, 7.5 where q = The frequency response is not symmetric about swap n n and, consequently, the framelet ψ[] is slightly asymmetric. It has two LDVM, whereas the framelet ψ [] has four LDVM. Figure 7. displays the IR of the filters h s, which are the discrete time framelets of the first level, and their magnitude responses. Observe that the IR of the filter h is non-symmetric. Figure 7.: Impulse left and magnitude responses right of the p-filter bank derived from cubic interpolating splines tight frame. Left picture: Left to right h 0 h h. Right picture: Magnitude responses h 0 and h dashed lines and h solid line Framelets symmetry can be achieved by factorization W [n] = w[n] w[ n] with the unequal factors w[n] and w[n], which leads to a semi-tight frame with ψ [] ψ []. Such a factorization is not unique that provides additional adaptation abilities. For example, either the analysis h or the synthesis h p-filter can be chosen to be p-fir. The local discrete vanishing moments can be transferred from the synthesis ψ[] to the analysis ψ [] framelet and vice versa. Following are examples of the p-filters h and h that result from three different factorization modes. 5

26 . Symmetric factorization with equal number two of LDVM in the analysis ψ [] and synthesis ψ[] framelets: h [n] = ω n ωn + ω n ω Ω 4, h [n] = ω n n + ω n Γ 4 c[n] c[n] 3 Ω c[n]. Symmetric factorization with equal number two of LDVM in the analysis ψ [] and synthesis ψ [] framelets. The analysis filter h is p-fir and, consequently, the framelet ψ [] is compactly supported: h [n] = ω n ωn + ω n 4, h [n] = ω n ω n + ω n Γ 4 c[n] 6 Ω 4 c[n] Antisymmetric factorization with three LDVM in the analysis ψ [] and one LDVM in synthesis ψ[] framelets: h [n] = ω n ω4n 3ω n + 3 ω n ω 4 Ω 4, h [n] = ω n n Γ 4 c[n] c[n] 6 Ω c[n] Figure 7. displays the IR and the magnitude responses of the p-filters h and h, which stem from different factorizations of the sequence W [n] given in Eq The IR of the p-filters h are either symmetric or antisymmetric. Figure 7.: Top left: IR of the synthesis p-filters h for the semi-tight frames, which are defined in Eqs. left to right: Bottom left: IR of the corresponding analysis p-filters h. Top right: Magnitude responses of the synthesis p-filters h, which are defined in Eqs.: 7.6 dashed line, 7.7 dashdot line and 7.8 solid line. Bottom right: Magnitude responses of the corresponding analysis p-filters h Quadratic quasi-interpolating spline: Equation 6.7 implies that In this case, all the p-filters have finite IR up to periodization. fq 3 [n] = ω n ω n ω 4n, W [n] = 6 ω n + ω n 6 Γ 4 c[n], 7.9 where the sequence Γ 4 c[n] is defined in Eq The frequency responses of the low- and the high-pass p-filters are ĥ 0 [n] = cos 4 πn + sin πn, ĥ [n] = sin 4 πn + cos πn, 7.0 6

27 respectively. The following factorization modes W [n] = w[n] w[ n] are similar to the previous modes of the cubic interpolating spline:. on-symmetric factorization W [n] = w[n] w[ n], which results in the frequency responses h [n] = h [n] = ω n ω n + ω n q ω n 6, q = q The p-filter bank generates a tight frame. The framelet ψ[] has two LDVM, while ψ [] has four. Figure 7.3 displays the IR of the p-filters h s, which are the discrete time framelets of the first level, and their magnitude responses. The IR of the p-filter h is not symmetric. Figure 7.3: Impulse left and magnitude responses right of the p-filter bank derived from quadratic quasiinterpolating splines tight frame. Left picture: Left to right h 0 h h. Right picture: Magnitude responses h 0 and h dashed lines, and h solid line. Symmetric factorization with equal number two of LDVM in the analysis ψ [] ψ[] framelets: and in the synthesis h [n] = ω n ω n + ω n 4 ω, h [n] = ω n n + ω n Γ 4 c[n] Antisymmetric factorization with three LDVM in the analysis ψ t and one LDVM in the synthesis ψ t framelets: h [n] = ω n ω4n 3ω n + 3 ω n 8 ω, h [n] = ω n n Γ 4 c[n] Symmetric factorization with all four LDVM in the analysis ψ t and no LDVM in the synthesis ψ t framelets: h [n] = ω n ω4n 4ω n + 6 4ω n + ω 4n, h [n] = ω n Γ4 c[n] The p-filter h is all-pass. Figure 7.4 displays the IR and the magnitude responses of the p-filters h and h, respectively, which stem from different factorizations of the sequence W [n] given in Eq The IR of the p-filters are either symmetric or antisymmetric. 7

28 Figure 7.4: Top left: IR of the synthesis p-filters h for the semi-tight frames, which are defined in Eqs. left to right: Bottom left: IR of the corresponding analysis p-filters h. Top right: Magnitude responses of the synthesis p-filters h, which are defined in Eqs. 7. dashed line, 7.3 dashdot line and 7.4 solid line. Bottom right: Magnitude responses of the corresponding analysis p-filters h 7. p-filter banks derived from discrete splines Denote Ω r d [n] = cos r πn/ + sin r πn/. Once the prediction p-filter is derived from the discrete spline of order r, Eqs. 6.3, 6.4 and 6.5 imply that fd r [n] = ωn cos r πn/ sin r πn/ Ω r d [n], 7.5 ĥ 0 [n] = cos r πn/ Ω r d [n], ĥ [n] = sin r πn/ Ω r d [n]. 7.6 The low-pass h 0 and the high-pass h p-filters locally restore and eliminate sampled polynomials of degree r, respectively, thus the framelet ψ [] has r LDVM. The sequence W [n] W [n] = 8 sinr πn/ 8 r ω nr ω n r r Ω r d [n] = 4 r Ω r d [n] = w[n] w[ n], ω np ω n r 8 sin r πn/ w[n] = 4 r Ω r d [n] r is odd, w[n] = r Ω r d [n] r is even. Here p can be chosen to be any integer. The band-pass p-filter h, whose frequency response is h [n] = ω n w[n] /, eliminates sampled polynomials of degree r, thus, the framelet ψ [] has r LDVM. The triple { h 0, h, h } forms a PR p-filter bank, which generates a tight frame in the space Π[]. Discrete spline of sixth order: r = 3. ĥ [n] = ω n In this case, ω n ω n 3 ω n 3 ω n = ω n ω n 3 + 3ω n ω 4n 3 ω n ω n The p-filter h locally eliminates sampled quadratic polynomials. Thus, the framelet ψ[] is antisymmetric and has two LDVM. The impulse and the magnitude responses of the filters h s are displayed in Fig

29 Discrete spline of eighth order: r = 4. In this case, ĥ [n] = ω n ω n + ω n ω 4n + 8 ω n ω n ω 4n The p-filter h locally eliminates sampled cubic polynomials. Thus, the framelet ψ[] is symmetric and has three LDVM. The impulse and magnitude responses of the filters h s are displayed in Fig Remark 7. One of advantages of the semi-tight frames is the option to swap the LDVM between the analysis and the synthesis framelets. Typically, it is preferable to have more LDVM in the analysis rather than in synthesis framelets even in cases when the tight frame p-filters have linear phase. As an example, we introduce the semi-tight frame originated from the quadratic interpolating spline, where the frequency response of the p-filters h 0 and h are given in Eq. 7. and h [n] = ω n sin 4 πn/ Ω 3, ĥ c[n] [n] = ω n Ω 3 c[n], 7.0 Ω 3 c[n] = cos 4 πn/ + sin 4 πn/. The analysis framelet ψ [] has four LDVM while the synthesis framelet ψ [] has none. One more example is the semi-tight frame originating from the sixth order discrete spline, where the frequency response of the p-filters h 0 and h are given in Eq. 7.6 and h [n] = ω n sin 4 πn/ Ω 6 d [n], ĥ [n] = ω n sin πn/ Ω 6 d [n] 7. where Ω 6 d [n] = cos 6 πn/ + sin 6 πn/. The analysis framelet ψ [] has four LDVM while the synthesis framelet ψ[] has two. Figure 7.5 displays the IR of the p-filters h s, s = 0,,, which are the discrete time framelets ψ s [] of the first level, and their magnitude responses. The frequency responses of the low-pass filters h 0 and the corresponding high-pass filters h mirror each other. 9

30 Figure 7.5: Impulse and magnitude responses of the three-channel p-filter banks that generate tight frames. Left pictures display the impulse responses of p-filters where from left to right we have: h 0 h h. Right pictures display the magnitude responses of these p-filters: h 0 and h dashed lines, and h solid line. Top: linear interpolating spline. Second from top: Quadratic interpolating splinediscrete spline of order 4. Center: discrete spline of order 6. Second from the bottom: discrete spline of order 8. Bottom: discrete spline of order 8 Four-channel p-filter banks using spline filters In this section, we present a few examples of PR four-channel p-filter banks, which generate tight frames in the space Π[] of -periodic discrete-time signals. In almost all the forthcoming examples, the low-pass p-filters are interpolating. The design scheme is described in Section 5.. We utilize the same spline-based prediction p-filters that were used for the design of the three-channel p-filter banks in Section 7. If the low-pass p-filter is interpolating then the frequency responses of the four-channel PR p-filter bank are ĥ 0 [n] = + ω n f[n] /, ĥ [n] = ω n ω n f[n] /, ĥ [n] = A[ n] + ω n A[n], ĥ 3 [n] = A[ n] + ω n A[n], where A[n] = W [n] /4. Thus, in this case, once the prediction p-filter f is available, the design of the p-filter bank is reduced to the factorization of the same sequence W [n] = f[n] as in the three-channel case. 30

31 The analysis polyphase matrix is P[n] = The synthesis polyphase matrix is P[n] = P[n] T. / f[ n] / f[n] / / A[ n] A[n] A[ n] A[n] 8. Four-channel p-filter banks with p-fir p-filters In this section, a few examples of p-filter banks are given.. Linear spline: The prediction p-filter is f c [n] = + ω n /. Thus, W [n] /4 = ωn + ω n 6 The frequency responses of the p-filters are ĥ 0 [n] = cos πn, ĥ [n] = ω n 4 = A[n] = ωn. 4 ĥ [n] = ω n sin πn, 8. n ωn + ω 4, ĥ 3 [n] = ω n 4 n ωn + ω. 4 The p-filter h 0 locally restores the sampled polynomials of the first degree while the p-filters h and h locally eliminate them. The p-filter h 3 eliminates only constants. The framelets ψ[] and ψ [], which are the IR of the p-filters h and h, respectively, have two LDVM. Either of them is symmetric. The framelet ψ[] 3 is antisymmetric and has one LDVM. The IR of the p-filters h s, s = 0,,, 3, and their magnitude responses are displayed in Fig. 8.. Although both magnitude responses ĥ v and ĥ3 v are equal to zero when v = 0, the magnitude response ĥ v is flatter in the zero vicinity in comparison to ĥ3 v. This geometrical phenomenon reflects the fact that ψ has two LDVM in comparison to one LDVM in ψ 3. Quadratic quasi-interpolating spline: The frequency response of the prediction p-filter f and the sequence W [n] are given in Eq The sequence W [n] /4 can be factorizized as W [n] /4 = A[n] [n] A[n] [ n], where Then, the frequency responses of the p-filters are ĥ 0 [n] = cos 4 πn + sin πn A[n] = ω n + ω n q ω n 3, q = q, ĥ [n] = ω n sin 4 πn + cos πn, 8.3 ĥ [n] = A[ n] + ω n A[n], ĥ 3 [n] = A[ n] + ω n A[n], 8.4 where A[n] is given in Eq. 8.. The framelet ψ[] has two LDVM while ψ [] has LDVM. The IR of the p-filters h s, s = 0,,, 3, and their magnitude responses are displayed in Fig. 8.. Unlike the filters derived from the linear spline, the antisymmetric framelet ψ[] 3 has more LDVM than the symmetric framelet ψ[]. 3

32 Quadratic quasi-interpolatory spline, alternative design: 7.9, can be represented as the sum W [n] 4 = 3ωn + ω n 56 The addends of the above sum can be factorized separately to be Define W z 4 = A [n] A [ n] + A [n] A [ n], A [n] = 3ω n + ω n 8 The sequence W [n] /4, defined in Eq. ωn + ω n ω 4n ω n 3, A [n] = 6. h [n] = A [ n] + ω n A [n], h 3 [n] = A [ n] + ω n A [n]. 8.5 The filters h and h 3, whose transfer functions are given in Eq. 8.5, do not have linear phase. They locally annihilate sampled polynomials of the first degree, thus the framelets ψ[] and ψ3 [] have two LDVM. The IR of the p-filters h s, s = 0,,, 3, and their magnitude responses are displayed in Fig. 8.. They hold the same mirror relations between the magnitude responses of the filters as in the previous example. Upgraded quadratic quasi-interpolating spline: The frequency responses of the prediction p-filters f is given in Eq Then, the frequency responses of the low- and high-pass p-filters h 0 and h, respectively, are ĥ 0 [n] = cos 6 πn + 9 sin πn 3 πn sin, 8.6 ĥ [n] = sin 6 πn + 9 cos πn 3 πn sin Their IR comprise seven terms up to periodization. The low-pass h 0 and h and the high-pass h p-filters locally restore and eliminate the sampled polynomials of fifth degree, respectively. The sequence W [n] = ωn + ω n 3 56 P [n], P [n] = 9ω 4n 96ω n ω n + 9ω 4n. The Laurent polynomial of ω n, P [n] has four roots: α = 5i i i, 3 3 α = /α, α 3 = α, α 4 = /α, where a is the complex conjugate of a. Thus, the sequence W [n] can be factored. A[n] = 4 W [n] = A[n] = 3 ω n 3 β0 β ω n + β ω 4n, β 0 = α , β = α + α , α β = α The frequency responses of the filters h and h 3 are ĥ [n] = A[ n] + ω n A[n], ĥ 3 [n] = A[ n] + ω n A[n]. 8.8 The p-filter h 3 eliminates sampled polynomials of second degree, thus the framelet ψ 3 has three LDVM. The p-filter h eliminates cubic polynomials, thus the framelet ψ has four LDVM. The IR of the p-filters h s, s = 0,,, 3, and their magnitude responses are displayed in Fig

33 Pseudo-spline We start from the non-interpolating low-pass p-filter h 0, whose frequency response is given by Eq. 6.3, and the corresponding high-pass p-filter h. Their frequency responses are ĥ 0 [n] = cos 6 πn + 3 sin πn, ĥ [n] = ω n sin 6 πn + 3 cos πn. 8.9 We have T [n] = h 0 0 [n] h 0 [n] = ω n + ω n P [n] /56 where P [n] = 9ω 4n 8ω n ω n 9ω 4n. The Laurent polynomial P [n] of ω n can be factored A[n] = T [n] = A[n] = 3ω + ω α ω n + α ω n 56 α α 8.0 α = , α = The frequency responses of the p-filters h and h 3 are h [n] = A[ n] + ω n A[n], h 3 [n] = A[ n] + ω n A[n]. 8. The p-filters h and h 3 locally eliminate sampled polynomials of first and second degrees, respectively, thus the framelets ψ[] and ψ3 [] have two and three LDVM, respectively. The IR of the p-filters h s, s = 0,,, 3, and their magnitude responses are displayed in Fig. 8.. Like the p-filters derived from the quadratic quasi-interpolating spline, the antisymmetric p-filter h 3 is closer to a high-pass p-filter three LDVM, while the symmetric filter h is closer to a low-pass p-filter two LDVM. The low-pass p-filter h 0 locally restores only first degree sampled polynomials. However, the high-pass p- filter h locally eliminates fifth degree sampled polynomials. Therefore, the frequency response of the p-filter h 0 is flatter at the stopband near / than at the passband near zero. The IR of the p-fir p-filters h s, s = 0,,, 3, which are the discrete time framelets ψ[] s of the first level, and their magnitude responses are displayed in Fig. 8.. The magnitude responses of the low-pass filters h 0 and the corresponding high-pass filters h mirror each other. The same can be said about the p-filters h and h 3. 33

34 Figure 8.: Impulse and magnitude responses of the p-fir p-filter banks that generate tight frames. Left pictures display impulse responses of the p-filters where left to right displays: h 0 h h h 3. Right pictures display the magnitude responses of these p-filters: h 0 and h dashed lines, h 3 dashdot line and h solid line. In the right pictures, top: linear spline, second from the top: quadratic quasi-interpolating spline, center: quadratic quasi-interpolating spline alternative design, second from the bottom: upgraded quadratic quasi-interpolating spline, bottom: pseudo-spline 8. Four-channel p-filter banks with IIR p-filters In this section, we provide some examples for four-channel p-filter banks with IIR p-filters. Cubic interpolating spline: defined in Eq Thus, The prediction p-filter f and the sequence W [n] to be factorizized are A[n] = W [n] 4 = A[n] = ωn + ω n 6 q Ω 4 c[n] qω n, 8. where q = and Ω 4 c[n] = ω n ω n. Then, the frequency responses of the p-filter bank that generates tight frame are ĥ 0 [n] = cos4 πn/ + cos πn/ Ω 4 c[n], ĥ [n] = ω n sin4 πn/ + cos πn/ Ω 4 c[n] h [n] = A[ n] + ω n A[n], h 3 [n] = A[ n] + ω n A[n]

35 Unlike the three-channel p-filter bank that uses cubic spline prediction p-filter, all the p-filters in the fourchannel p-filter bank have a linear phase. The symmetric framelets ψ[] and ψ[] two and four LDVM, respectively, whereas the anntisymmetric framelet ψ[] 3 has three LDVM. The IR of the filters h s, s = 0,,, 3, and their magnitude responses are displayed in Fig Four-channel p-filter banks derived from discrete splines The prediction p-filter f and the sequence W [n] are given in Eqs. 7.5 and 7.7. The frequency responses of the low- and high-pass p-filters are ĥ 0 [n] = cos r πn/ Ω r d [n], ĥ [n] = sin ω n r πn/ Ω r d [n], 8.4 where Ω r d [n] = cos r πn/ + sin r πn/. A[n] = ω np ω n r 4 r Ω r d [n] r is odd, p Z, A[n] = sinr πn/ r Ω r d [n] r is even h [n] = A[ n] + ω n A[n], h 3 [n] = A[ n] + ω n A[n]. 8.5 The low-pass p-filter h 0 locally restores sampled polynomials of degree r while the high-pass p-filter h eliminates them. If r is even then the band-pass p-filter h locally eliminates sampled polynomials of degree r, while p-filter h 3 eliminates polynomials of degree r and vice versa for r odd. Example: Discrete spline of fourth order: r =. In this case, the prediction p-filter is the same as the p-filter derived from the quadratic polynomial interpolating splinewhere A[n] = sin πn/ 4 cos 4 πn/ + sin πn/ Example: Discrete spline of sixth order: r = 3. In this case, A[n] = ω n 3 + 3ω n ω 4n 4 3 ω n ω n. 8.7 The IR of the IIR p-filters h s, s = 0,,, 3, which are the discrete time framelets ψ[] s, of the first level, and their magnitude responses are displayed in Fig. 8.. The magnitude responses of the low-pass filters h 0 and the corresponding high-pass filters h mirror each other. The same can be said about the p-filters h and h 3. 35

36 Figure 8.: Impulse and magnitude responses of the IIR p-filter banks that generate tight frames. Left pictures display impulse responses of the p-filters. Left to right in the left picture have: h 0 h h h 3. Right pictures display the magnitude responses of these p-filters: h 0 and h dashed lines, h 3 dashdot line and h solid line. In the right pictures, top: cubic interpolating spline, second from the top: quadratic interpolating splinediscrete spline of order 4, center: discrete spline of order 6, second from the bottom: discrete spline of order 8, bottom: discrete spline of order 9 Application of periodic frames to image restoration In this section, we present the results for image restoration. The images were degraded by blurring aggravated by random noise and random loss of significant number of pixels. The images are transformed by periodic frames designed in Sections 7 and 8, which are extended to the D setting in a standard tensor product way. The goal of our experiments is to compare between the performances of different tight and semi-tight frames in identical conditions. 9. Outline of the restoration scheme The images are restored by the application of the split Bregman iteration SBI scheme [8] that uses the so-called analysis based approach see for example []. Denote by u = {u[κ, ν]} the original image array to be restored from the degraded array f = K u + ε, where K denotes the D discrete convolution operator of the array u with a kernel k = {k[κ, ν]} and 36

37 ε = {e k,n } is the random error array. K denotes the conjugate operator of K, which implements the discrete convolution with the transposed kernel k T. If some number of pixels are missing then the image u should be restored from the available data P Λ f = P Λ K u + ε, 9. where the symbol P Λ denotes the projection on the remaining set of pixels. The solution scheme is based on the assumption that the original image u has a redundant representation that can be sparsely represented in a frame domain. Denote by F the operator of frame expansion of the image u where C = F u, C = {c[κ, ν]}, is the set of the frame coefficients. Denote by F the reconstruction operator for the image u from the set of the frame coefficients. We get F C = u, F F = I, where I is the identity operator. We assumed that the approximated solution to Eq. 9. is derived via minimization of the functional min u P Λ K u f F + λ u, 9. where and are the l and l norms of the sequences, respectively. If x = {x[κ, ν]}, κ = 0,..., k, ν = 0,..., n, then k n x = x[κ, ν], x = k n x[κ, ν]. κ=0 ν=0 Denote by T ϑ the operator for soft thresholding: κ=0 ν=0 T ϑ x = {x ϑ [κ, ν]}, x ϑ [κ, ν] = sgnx[κ, ν] max {0, x[κ, ν] ϑ}. Following [], we solve the minimization problem in Eq. 9. by an iterative algorithm. We begin with the initialization u 0 = 0, d 0 = b 0 = 0. Then, u k+ := K P Λ K + µ I u = K P Λ f + µ F d k b k, d k+ = T λ/µ F u k+ + b k, b k+ = b k + F u k+ d k The linear system in the first line of Eq. 9.3 is solved by the application of the conjugate gradient algorithm. The operations in the second and third lines are straightforward. The choice of the parameters λ and µ depends on experimental conditions. 9. Experimental results The restoration results were done for Window, Barbara, Lena and Fingerprint images. These images were blurred by convolution with the motion or Gaussian kernels, affected by zero mean random noise and degraded by the fact that a large number of pixels were missing. In the experiments, we compare between the performance of tight frames TF and semi-tight frames STF defined in Sections 7 and 8. The framelets in table participated in the experiments and they are denoted by the following notation: 37

38 T 4 LS TF, Eq. 8., T 3 LS TF, Eq. 7., T 4 QqIS TF, Eqs. 8.3,8.4, T 3 QIS TF, Eq. 7., T 4 3 QqISAD TF, Eqs. 8.3,8.5, S 3, QIS STF, Eqs. 7.,7.0, T 4 4 QqISU TF, Eqs. 8.6,8.8, T 3 3 DS6 TF, Eqs. 7.6, 8.7, T 4 5 PS TF, 8.9,8., S 3 3, DS6 STF, Eqs. 7.6, 7., T 4 6 CIS TF, Eq. 8.3, T 3 4 DS8 TF, Eqs. 7.6, 7.9, T 4 7 QIS TF, Eqs. 8.4,8.5, T 3 6 CIS TF, Eqs. 7.4, 7.5, T 4 8 DS6 TF, Eqs. 8.4, 8.5, S 3 6, CIS STF, Eqs. 7.4, T 4 9 DS8 TF, Eqs. 8.4, 8.5, T 3 7 QqIS TF, Eqs. 7.0,7., 7.7, T 4 0 DS TF, Eqs. 8.4, 8.5, S 3 7, QqIS STF, Eqs. 7.0,7., Table : Four- and three-channel p-filter banks. LS linear spline, QIS quadratic interpolating spline, QqIS the quadratic quasi-interpolating spline, QqISAD the quadratic quasi-interpolating spline alternative frame design, QqISU upgraded quadratic quasi-interpolating spline, CIS cubic interpolating spline, PS pseudo-spline, DS6, DS8, DS mean the discrete splines of orders 6,8,, respectively. The proximity between an image ũ and the original image u is evaluated visually and by the Peak-Signalto-oise ratio PSR P SR = M 55 0 log 0 M k= x db. 9.4 k x k where M is the number of pixels in the image in our experiments, M = 5, {x k } M k= pixels of the image u, while { x k } M k= are the pixels of the image ũ. are the original 9.. Restoration experiments for the Window image This image was taken from []. Different experimental scenarios took place in order to evaluate the performance of different framelets. o random noise: In the first round of experiments, the image was blurred by convolution with the motion kernel MATLAB function fspecial motion,5,45 and its PSR becomes PSR=3.56 db. Then, 30% of pixels were randomly removed. This reduces the PSR to 0, db. o random noise was added. The image was restored by 50 SBI using the parameters λ = 0.003, µ = 0.00 in Eq The tight and semi-tight frames listed in Table were used. The decomposition is implemented down to the fourth level. The restored PSR results are given in Table 3. Frame T 3 T 3 S 3, T 3 6 S 3 6, T 3 3 S 3 3, T 3 4 T 3 7 S 3 7, PSR Frame T 4 T 4 T 4 3 T 4 4 T 4 5 T 4 6 T 4 7 T 4 8 T 4 9 T 4 0 PSR Table 3: PSR results after the restoration of the Window image that was a blurred input where of 30% of its pixels were randomly removed The best PSR result 4.8 db is achieved by using the three-channel semi-tight frame S 3, derived from the quadratic interpolating spline. Figure 9. displays the restoration result. Visually, the restored image hardly can be distinguished from the original one. 38

39 Figure 9.: Source input Window image top left, blurred top right:psr=3.56 db, after random removal of 30% of its pixels bottom right:psr=0. db. The best restored image was achieved by the frame S 3, displayed in bottom left PSR=4.8 db The SBI algorithm that uses the spline-based frames was able to restore images even when most of the pixels are missing. To achieve a satisfactory restoration, more iterations were required. Figure 9. displays the restoration result of the same blurred image as in Fig. 9., where 90% of the pixels were randomly removed. 50 iterations were applied with the four-channel tight frame derived from the cubic interpolating spline T 4 6. The decomposition is implemented down to the second level while using parameters λ = 0.006, µ = 0.00 in Eq

40 Figure 9.: Left: The restored Window image, PSR=8.76 db. Right: The source input image that was blurred PSR=3.56 db after random removal of 90% of its pixels PSR=5.59 db. The image was restored after 50 SBI using the frame S 4 6. Weak random noise: In this series of experiments, the image was degraded by convolution with the motion kernel MATLAB function fspecial motion,5,45 and a random zero-mean Gaussian noise, whose STD is σ = 5, was added and the PSR was 3.0 db. Then, 30% of pixels were randomly removed and the PSR was reduced to 0, db. The image is restored by 50 SBI using the tight and semi-tight frames listed in Table. The parameters were λ = 0.5, µ = 0. for the three-channel frames and λ = 0.4, µ = 0. for the four-channel frames. The decomposition is implemented down to the second level. The restored PSR results are given in Table 4. Frame T 3 T 3 S 3, T 3 6 S 3 6, T 3 3 S 3 3, T 3 4 T 3 7 S 3 7, PSR Frame T 4 T 4 T 4 3 T 4 4 T 4 5 T 4 6 T 4 7 T 4 8 T 4 9 T 4 0 PSR Table 4: Restoration of the Window image from a blurred and noised input after random removal of 30% of the pixels The best PSR result 8.76 db was achieved by using the four-channel tight frame T 4 7 derived from the quadratic interpolating spline. Figure 9.3 displays the restored result. 40

41 Figure 9.3: Source input Window image top left, blurred and noised top right:psr=3.0 db, after random removal of 30% of its pixels bottom right:psr=0. db. The restored image by the frame S47 is displayed in bottom left PSR=8.76 db 9.. Restoration experiments for the Barbara image o random noise: In these experiments, the image was blurred by convolution with the Gaussian kernel MATLAB function fspecial gaussian,[5,5],5, then 50% of its pixels were randomly removed. o random noise was added. The image was restored by 50 SBI with the parameters λ = 0.005, µ = 0.00, using the tight and semi-tight frames listed in Table. The decomposition is implemented down to the second level. The PSR results are given in Table 5. Frame PSR T T S3, 9.47 T S36, 9.4 T S33, 9.57 T T S37, 9.6 Frame PSR T T4 9. T T T45 9. T T T T T Table 5: The restoration results for Barbara image that was blurred and 50% of its pixels were randomly removed The best PSR result was 9.7 db that was achieved by the application of the four-channel tight frame originating from the discrete spline of twelfth order, where the low-pass p-filter h0 locally restores sampled polynomials of eleventh degree while the high-pass p-filter h eliminates them. The band-pass p-filter h eliminates polynomials of sixth degree, while p-filter h3 eliminates polynomials of fifth degree. We see in Table 5 that the PSR for the four-channel tight frames grow in line when the degrees of the sampled 4

42 polynomials, which are eliminated restored by the frame p-filters, grow. A similar observation is true for the three-channel frames. ote that the semi-tight frames performed at least not worse than their tight counterparts. Figure 9.4: Source Barbara image top left, blurred top right:psr=3.06 db, after random removal of 50% of its pixels bottom right:psr=7.55 db. The restored image by the T40 frame is displayed in the bottom left PSR=9.7 db Moderate random noise: The Barbara image was blurred by convolution with a Gaussian kernel MATLAB function fspecial gaussian,[5,5],5. Random zero-mean Gaussian noise, whose STD σ = 0, was added achieving PSR=.89 db. Then, 50% of its pixels were randomly removed reducing the PSR to 7.53 db. The image is restored by 50 SBI with the parameters λ =.8, µ = 0.8 for the three-channel frames and with the parameters λ =.08, µ = 0.07 for the four-channel frames listed in Table, respectively. The decomposition is implemented down to the fourth level. The PSR results are given in Table 6. Frame PSR T T3 4. S3, 4. T S36, 4.03 T S33, 4.06 T T37 4. S37, 4. Frame PSR T4 4. T4 4.6 T T44 4. T T T T T T Table 6: Restoration of the Barbara image from blurred and noised input after random removal of 50% of its pixels The best PSR result 4.0 db was achieved by using the four-channel tight frame T45 derived from the pseudo-spline, however other frames produce similar results. Figure 9.5 displays the restoration result. 4

43 Figure 9.5: Source Barbara image top left, blurred and noised top right:psr=.89 db, after random removal of 50% its pixels bottom right:psr=7.5 db. The restored image by the T45 frame is displayed on the bottom left PSR=4.0 db 9..3 Restoration experiments for the Lena image o random noise: The source image was convolved with a Gaussian kernel MATLAB function fspecial gaussian,[8,8],5 PSR=5.4 db and the blurred image is distorted by randomly drawn curves PSR=6.68 db. o noise was added. The image is restored by 50 SBI with the parameters λ = 0., µ = 0.35 for the three-channel tight and semi-tight frames and with the parameters λ = 0.095, µ = 0.35 for the four-channel tight frames listed in Table. The decomposition was implemented down to the fifth level. The PSR results are given in Table 7. Frame PSR T T S3, T S36, T S33, 30.3 T T S37, Frame PSR T4 30. T T T T T T T T T Table 7: Restoration of Lena image from a blurred input distorted by randomly drawn curves The best PSR result db was achieved by using the three-channel semi-tight frame S36, derived from the cubic interpolating spline. Almost the same PSR = db was produced by the tight frame T45 derived from the pseudo-spline, however other frames produced similar results. Figure 9.6 displays the best restoration result. 43

44 Figure 9.6: Source Lena image top left, blurred top right:psr=5.4 db distorted by randomly drawn curves bottom right:psr=6.68 db. The restored image by the S36, semi-tight frame is displayed in the bottom left PSR=30.38 db Denoising experiment: The source image was corrupted by a strong zero-mean Gaussian noise with STD σ = 30 PSR=8.59 db. Then, it was distorted by randomly drawn curves PSR=4.99 db. The image is restored by 50 SBI with the parameters λ = 5, µ =.9 for the three-channel tight and semi-tight frames and with the parameters λ = 0, µ =. for the four-channel tight frames listed in Table. The decomposition is implemented down to the third level. The PSR results are given in Table 8. Frame PSR T3 7.4 T3 7.4 S3, 6.08 T S36, 6.63 T S33, 6.40 T T S37, 7.9 Frame PSR T4 7.3 T T T T T T47 7. T T T Table 8: Restoration results for Lena image after it was noised and distorted by randomly drawn curves The best PSR result 7.5 db was produced by the tight frame T45 derived from the pseudo-spline. Figure 9.7 displays the restoration result. We observe that noise was completely suppressed but some oversmoothing remains. 44

45 Figure 9.7: Source Lena image top left, noised top right:psr=8.59 db and distorted by randomly drawn curves bottom right:psr=4.93 db. The restored image by the T45 tight frame is displayed in bottom left PSR=7.5 db 9..4 Restoration experiments for the Fingerprint image o random noise: In the first round of experiments, the image was blurred by convolution with the motion kernel MATLAB function fspecial motion,5,45 producing PSR=7.79 db. Then, 50% of its pixels are randomly removed reducing to PSR=8.93 db. o random noise was added. The image is restored by 50 SBI with the parameters λ = 0.5, µ = 0.05, using the tight and semi-tight frames listed in Table. Only one decomposition level is implemented. The PSR results are given in Table 9. Frame PSR T T S3, 4.90 T S36, 4.99 T S33, 5.03 T T S37, 4.77 Frame PSR T4 4.7 T T T T T T T T T40 5. Table 9: PSR results for Fingerprint image restoration from blurred input followed by random loss of 50% of its pixels The best PSR result 4.59 db was achieved by using the four-channel tight frame T40 derived from the discrete spline of twelfth order. Figure 9.8 displays the restoration result. As in the experiments with the Barbara image described in Table 5, the PSR grows in line when the degrees of the sampled polynomials, which are eliminated restored by the frame p-filters, grow. Figure 9.9 displays the restoration result. Visually, the restoration is almost perfect. 45

46 Figure 9.8: Restoration of Fingerprint image top left from blurred input top right:psr=7.79 db after random loss of 50% pixels bottom right:psr=8.83 db. Image restored by the T 4 0 frame bottom left:psr=5. db Random noise added: The image was blurred by convolution with the motion kernel MATLAB function fspecial motion,5,45, and the zero-mean Gaussian noise whose STD σ = 0, was added to yield PSR=6.43 db. Then, 50% of its pixels were randomly removed and this produces PSR=8.83 db. The image was restored by 50 SBI with the parameters λ =., µ = 0.38, using the tight and semi-tight frames listed in Table. The decomposition is implemented down to the second level. The PSR results are given in Table 0. Frame T 3 T 3 S 3, T 3 6 S 3 6, T 3 3 S 3 3, T 3 4 T 3 7 S 3 7, PSR Frame T 4 T 4 T 4 3 T 4 4 T 4 5 T 4 6 T 4 7 T 4 8 T 4 9 T 4 0 PSR Table 0: Restoration results in PSR for the Fingerprint image when it was blurred, noised and 50% of its pixels were randomly removed The best PSR result 9.94 db was achieved by using the four-channel tight frame T 4 8 derived from the discrete spline of sixth order. A similar result with PSR=9.79 db was produced by the three-channel tight frame T 3 3, which also was derived from the sixth order discrete spline. 46

47 Figure 9.9: Source Fingerprint image top left, blurred and noised top right:psr=6.43 db, after random removal of 50% of its pixels bottom right:psr=8.83 db. The image restored by the T48 frame displayed in bottom left PSR=9.94 db 9..5 Comments on the experiments In the above experiments, the designed three- and four-channel frames proved to be highly efficient for images restoration. It is important that the computational cost does not depend on the spline order that makes the framelets with high number of LDVM available for utilization. In some cases, such as in the experiments with Barbara Table 5 and Fingerprint Table 9, the best results were produced by tight frames derived from the discrete spline of twelfth order, where the framelets ψ[m] have LDVM and the 3 framelets ψ[m] and ψ[m] have 7 and 6 LDVM, respectively. ote that both images have fine texture, which is satisfactorily restored after strong blurring and random removable of 50% of their pixels. When images are relatively smooth, as Window and Lena, and no additive noise is present, they are restored close to perfection Figs. 9. and 9.6. In both cases, the best results were produced by the three-channel semi-tight frames, which are derived from the interpolating quadratic for Window and cubic for Lena polynomial splines. ote that in many cases the semi-tight frames performed better than their tight frames counterparts. If the majority, up to 90%, of the pixels in a blurred image is randomly removed, the image still can be restored by an increased SBI number Fig. 9.. However, in more regular situations, increase of the SBI number above 50 does not contribute into the restoration quality. When images are corrupted by noise, the frames derived from the quadratic interpolating and quasiinterpolating splines and from the pseudo-spline are advantageous. Choice of the regularization parameters λ and µ is of crucial importance. The restoration quality is sensitive to them. Even a small change in the parameters significantly affect the restoration quality. 47