Harmonic Wavelet Transform and Image Approximation

Transcription

1 Harmonic Wavelet Transform and Image Approximation Zhihua Zhang and Naoki Saito Dept of Math, Univ of California, Davis, California, 95616, USA Abstract In 006, Saito and emy proposed a new transform called Laplace Local Sine Transform LLST) in image processing as follows Let f be a twice continuously differentiable function on a domain Ω First we approximate f by a harmonic function u such that the residual component v = f u vanishes on the boundary of Ω Next, we do the odd extension for v, and then do the periodic extension, ie we obtain a periodic odd function v Finally, we expand v into Fourier sine series In this paper, we propose to expand v into a periodic wavelet series with respect to a biorthonormal periodic wavelet basis with the symmetric filter banks We call this the Harmonic Wavelet Transform ) has an advantage over both LLST and the conventional wavelet transforms On one hand, it removes the boundary mismatches as LLST does On the other hand, the coefficients reflect the local smoothness of f in the interior of Ω So the algorithm approximates data more efficiently than LLST, periodic wavelet transform, folded wavelet transform, and wavelets on interval We demonstrate the superiority of over the other transforms using several standard images Key words: Harmonic wavelet transform, Laplace local sine transform, biorthonormal wavelets, periodic wavelets, folded wavelets, wavelets on interval, periodic wavelet coefficient, symmetry, odd extension 1 Introduction Wavelet analysis is an important tool in image processing In order to approximate or compress data, there are two common wavelet algorithms: the periodic wavelet algorithm, and the folded wavelet algorithm More precisely, let an image f be supported on a square Ω In the periodic wavelet algorithm, one extends f to a periodic function f, and then expands f into a periodic wavelet series using Daubechies wavelets [1] or polyharmonic spline wavelets [9] Since f is discontinuous at the boundary points Ω in general, the decay rate of the corresponding periodic wavelet coefficients is very slow Hence, in order to obtain a good approximation of the image, we need many periodic wavelet coefficients In the folded wavelet algorithm, in order to avoid the boundary mismatches caused by the brute-force periodization, one does an even extension of f, and then extends it to a periodic function f, and finally expands f into a periodic wavelet series with respect to a biorthonormal periodic wavelet basis If f is smooth, then f lip1 on This makes the decay rate of the 1

2 corresponding wavelet coefficients faster than that in periodic wavelets algorithm But, since x f and y f are discontinuous at the boundary points, the decay rate of the periodic wavelet coefficients is still relatively low It is natural to ask how to ensure the continuity of derivatives of the periodized function across the boundary In 006, Saito and emy [8] presented a good method: the Laplace Local Sine Transform LLST) that ensures the derivatives of the periodized functions are continuous across the boundary It decomposes an image f supported on a square Ω into f = u+v, where u is a harmonic component satisfying Laplace s equation u = 0 in Ω and u = f on the boundary After an odd extension, the residual component v is periodized to be v, ie, v is a periodic odd function If f is smooth, then x v lip 1, y v lip 1 on We call this method LLST decomposition When one applies LLST to image approximation, one expands v into Fourier sine series In this paper, combining the LLST decomposition and the wavelet algorithm, we expand v into the periodic wavelet series More precisely, we propose the harmonic wavelet transform ) In the algorithm, the first step is the same as LLST, ie, we decompose f = u + v and obtain a periodic odd function v Next, we choose a pair of real-valued biorthonormal wavelets ψ, ψ of L ) generated by the real-valued even scaling functions ϕ, ϕ Using a method of the tensor product and periodization, one gets a pair of biorthonormal periodic wavelet bases Finally, we expand v into the periodic wavelet series with respect to this pair of biorthonormal periodic wavelet bases Since x v lip 1 and y v lip 1 on, the decay rate of the corresponding periodic wavelet coefficients is faster than that of the periodic wavelet algorithm or the folded wavelet algorithm In general, the decay rate of the Fourier sine coefficients depends on global smoothness, while that of the periodic wavelet coefficients depends on local smoothness Since the global smoothness of a function is determined by its rough part, we need fewer periodic wavelet coefficients than the Fourier sine coefficients in order to approximate the image with the same quality Comparing the above several algorithms, the

3 algorithm compresses data most efficiently among them In the algorithm, we carefully study the symmetry of the periodic wavelet coefficients and show where these coefficients vanish From this, we see that in order to recover the image exactly, the number of the efficient coefficients is just the same as the size of the sample points of f So the is not a redundant transform Our algorithm is quite different from polyharmonic wavelets proposed by Van De Ville et al [9], which are a kind of wavelet bases constructed by polyharmonic functions The has a different strategy: we approximate a target function by a harmonic function such that the residual part vanishes on the boundary, and then expand the residual part into wavelet series This paper is organized as follows In Section, we recall the notions of biorthonormal periodic wavelet bases and the corresponding Mallat algorithm as well as the LLST algorithm In Section 3, we present the algorithm and show that in the one-dimensional case, the periodic wavelet algorithm, the folded wavelet algorithm, and the algorithm generate periodic wavelet coefficients at the level m with the decay rates O m ), O 3m ), and O 5m ), respectively When we use first M periodic wavelet coefficients, the obtained approximation error are o1), O M ), and O M ), respectively In the two-dimensional case, the decay rates of the corresponding periodic wavelet coefficients are O m ), O m ), and O 3m ), respectively When we use first M periodic wavelet coefficients, the obtained approximation error are o1), O M ), and O M ), respectively In Section 4, first, we discuss one-dimensional discrete and the symmetry of the sequences consisting of the corresponding periodic wavelet coefficients From this, we precisely show the efficient number of the periodic wavelet coefficients in order to recover a signal perfectly Second, we discuss twodimensional discrete and the symmetry of the matrices of the corresponding periodic wavelet coefficients From this, we precisely show the efficient number of the periodic wavelet coefficients in order to recover an image perfectly In Section 5, we apply the algorithm to approximate images and show that the algorithm approximates images better than the LLST algorithm, the periodic wavelet algorithm, and the folded 3

4 wavelet algorithm Preliminaries We state the well-known notions [4, 7] of biorthonormal periodic wavelet bases After that, we recall the corresponding Mallat algorithms 1 One-dimensional biorthonormal periodic wavelets It is well-known that using the method of periodization, one can construct biorthonormal periodic wavelet bases with the help of the biorthonormal wavelets [4, 7] Let ψ and ψ be a pair of compactly supported smooth biorthonormal wavelets of L ) generated by compactly supported smooth scaling functions ϕ and ϕ For m Z, n Z, we denote the function family g m,n := m g m n) Its periodization with period 1) is g per m,n := l Zg m,n + l) The families Ψ per 1 := {ϕ per } {ψ per m,n, m Z +, n = 0,, m 1}, Ψ per 1 := { ϕ per } per { ψ m,n, m Z +, n = 0,, m 1} are called a pair of biorthonormal periodic wavelet bases for L [ 1, 1 ]) Let f L [ 1, 1 ]) For m Z+, n Z, we denote the periodic wavelet coefficients c 1) m,n := 1 ft)ϕ per m,nt)dt, d 1) m,n := 1 ft)ψ per m,nt)dt 1) 1 1 Here the sequences {c 1) m,n} and {d 1) m,n} are periodic sequences with period m ) with respect to the index n, ie, c 1) m,n+ m l = c1) m,n, d 1) m,n+ m l = d1) m,n l Z) It is well known that f = c 1) 0,0 + m=0 m 1 d 1) per m,n ψ m,n ) 4

5 in the space L [ 1, 1 ]) For M Z +, its M partial sum is M 1 S Mf) = c 1) 0,0 + m=0 m 1 d 1) per m,n ψ m,n 3) Let the space Ṽm be the span of { ϕ per m,n},, m 1 One can decompose the projection of f in Ṽm as follows [7] P evm f) = m 1 m 1 1 c 1) m,n ϕ per m,n = m 1 1 c 1) m 1,n ϕper m 1,n + d 1) per m 1,n ψ m 1,n 4) Now we assume that ϕ and ϕ are both compactly supported Denote the coefficients of the filter banks a k := ϕt) ϕt k)dt, b k := ψt) ϕt k)dt 5) Without loss of generality, we assume that for some N Z +, a n = b n+1 = 0 n N) 6) Mallat showed that the fast algorithm to compute periodic wavelet coefficients is similar to the fast algorithm of wavelet coefficients [7, Sec 751] Along Mallat s idea, one can obtain the following algorithm for the one-dimensional periodic wavelet coefficients Proposition 1 Let the filters a k, b k be defined in 5) and c 1) m,n, d 1) m,n be defined in 3) Define a n := a n, b n+1 := b n+1 n J 1 ), a n+ J := a n, b n+ J := b n n Z) 7) Then for J 1 N, we have J 1 c 1) J 1,k = a n k c1) J,n, J 1 d1) J 1,k = b n k c1) J,n, k Z 8) Two-dimensional biorthonormal periodic wavelets Let ψ and ψ be a pair of biorthonormal wavelets generated by the scaling functions ϕ and ϕ Take the tensor products of ϕ, ψ Denote ϕ 0 x,y) := ϕx)ϕy), ψ 1 x,y) := ϕx)ψy), 5

6 ψ x,y) := ψx)ϕy), ψ 3 x,y) := ψx)ψy) Similarly, taking the tensor products of ϕ and ψ, we get ϕ 0 x,y), ψ 1 x,y), ψ x,y), and ψ 3 x,y) Then {ψ µ } 3 1 and { ψ µ } 3 1 are a pair of biorthonormal wavelets generated by the scaling functions ϕ 0 and ϕ 0, respectively The families Ψ per = {ϕ per 0 } {ψ per µ,m,n, µ = 1,,3, m Z +, n = n 1,n ), n 1, n = 0,, m 1}, Ψ per = { ϕ per 0 } per { ψ µ,m,n, µ = 1,,3, m Z +, n = n 1,n ), n 1, n = 0,, m 1} [ are called a pair of biorthonormal periodic wavelet bases for L 1, ] ) 1 [ Let f L 1, ] ) 1 For µ = 1,,3, m Z +, n Z, we denote the periodic wavelet coefficients c ) m,n := fx,y)ϕ per 0,m,n x,y)dxdy, d) µ,m,n := fx,y)ψ per µ,m,nx, y)dxdy 9) Here the sequences {c ) m,n} and {d ) µ,m,n} are periodic sequences with respect to n, ie c ) m,n+ m l = c) m,n, d ) µ,m,n+ m l = d) µ,m,n l Z ) It is well known that f = c ) 0,0 + 3 µ=1 m=0 m 1 n 1,n =0 d ) per µ,m,n ψ µ,m,n 10) For M Z +, its M partial sum is S Mf) = c ) 0,0 + 3 [ in the space L 1, ] ) 1, where n = n 1,n ) µ=1 M 1 m=0 m 1 n 1,n =0 d ) per µ,m,n ψ µ,m,n Let the space Ṽm be the span of { ϕ per 0,m,n } n=n 1,n ), n 1, n =0,, m 1 One can decompose the project of f in Ṽm as follows P evm f) = m 1 n 1, n =0 m 1 1 c ) m,n ϕ per 0,m,n = n 1, n =0 3 c ) m 1,n ϕper 0,m 1,n + µ=1 m 1 1 n 1, n =0 d ) per µ,m 1,n ψ µ,m 1,n 11) 6

7 From the Mallat algorithm of the one-dimensional periodic wavelet, one can easily conclude the following Mallat algorithm for two-dimensional tensor product periodic wavelets Proposition Let the filters {a n}, {b n} be stated in 7) and the periodic wavelet coefficients c ) m,n, d ) µ,m,n be stated in 9) Then for J > N and k = k 1,k ) Z, we have i) c ) J 1,k = J 1 n 1,n =0 ii) d ) 1,J 1,k = J 1 n 1,n =0 iii) d ),J 1,k = J 1 n 1,n =0 iv) d ) 3,J 1,k = J 1 n 1,n =0 3 Laplace local sine transforms a n 1 k 1 a n k c ) J,n 1,n, a n 1 k 1 b n k c ) J,n 1,n, b n 1 k 1 a n k c ) J,n 1,n, b n 1 k 1 b n k c ) J,n 1,n First we consider the one-dimensional case Let Ω 1 := [ 0, 1 ] and the function f be defined on Ω1 and f C Ω 1 ) We split the function f into two components fx) = ux) + vx) on Ω 1 The first function ux) = f 1 ) f0) ) x + f0) and the second function satisfies v0) = v ) 1 After an odd extension, the second function v is periodized to be v, ie, v is a periodic odd function Finally we expand v into Fourier sine series Next, we consider the two-dimensional case Let Ω := [ 0, 1 ] and the function f be defined on Ω and f C Ω ) We split the function f into two components = 0 fx,y) = ux,y) + vx,y) on Ω where ux,y) is the harmonic function which satisfies Laplace s equation ux,y) = 0 x,y) Ω ) and ux,y) = fx,y) on the boundary of Ω So the residual satisfies vx,y) = 0 x,y) Ω ) 7

8 After an odd extension, the residual component v is periodized to be v, ie, v is a periodic odd function Finally we expand v into Fourier sine series 3 Harmonic Wavelet Transform Now we present a notion of the Harmonic Wavelet Transform ) We show that for the algorithm, the decay rate of periodic wavelet coefficients is the faster than those generated by the periodic wavelet transform algorithm and the folded wavelet algorithm 31 One-dimensional algorithm We assume that f is defined on [ 0, 1 ] Let ft) = ut) + vt) [ t 0, 1 ]), where ut) = f 1 ) f0) ) t + f0) and the residual component vt) satisfies v0) = v 1 ) = 0 We do the odd extension v odd of the residual component v to [ 1, 1 ] We again do the 1-periodic extension of v odd, denoted by v Finally, we expand v into the biorthonormal periodic wavelet series We call this process the one-dimensional Now we examine the decay rates of the coefficients of various wavelet algorithms in the one-dimensional case i) The 1D periodic wavelet algorithm Let f C [ 1, 1 ]) In the periodic wavelet algorithm, we directly expand f into biorthonormal periodic wavelet series Clearly, the coefficients d 1) m,n = 1 ft)ψm,nt)dt per = f t))ψ m,n t)dt = m f t)ψ m t n)dt, 31) 1 where f is the 1-periodic extension of f Since f L ), ψ L 1 ), we have d 1) m,n = O m ) ψ m t n) dt = O m ) ψt n) dt = O m ) ψt) dt = O m ) 3) 8

9 ii) The 1D folded wavelet algorithm Let f C [ 0, 1 ]) In the folded wavelet algorithm, we do an even extension of f Then ft), t [ 0, ] 1, f even x) = f t), t [ 1,0) [ f even x) C 1, 1 ] and f even 1 ) ) 1 = f even, Again let f t) be a 1-periodic extension of f even t), we have Proposition 31 The periodic wavelet coefficients of f satisfy d 1) m,n = O 3m ) 33) The partial sum S Mf ) of the periodic wavelet expansion of f satisfies f S Mf ) L [ 1, 1 ]) = O M) 34) Proof Since f C [ 0, 1 ], we have f lip 1 on, ie, there is a constant K such that f t) f n m ) K t n m t ) Since f lip 1, ψt)dt = 0, and 31), we have d 1) m,n = m f t)ψ m t n) dt = m So 33) holds K m = K 3m t n ψ m m t n) dt = K 3m tψt) dt = O 3m ) f t) f n ) m ) ψ m t n) dt t n ψt n) dt By ) and 3), we know that the difference between f and the partial sum of its periodic wavelet series f S M+1f ) = m=m m 1 d 1) per m,n ψ m,n 9

10 is in the space L [ 1, 1 ]) Using the Parseval equality and 33), we get f S Mf ) L [ 1, 1 ]) = m=m m 1 d 1) m,n = O1) m=m m 1 3m ) = O1) m = O M) So 34) holds Proposition 31 is proved iii) The 1D algorithm Let f C [ 0, ]) 1 Now we introduce the algorithm m=m a) We first decompose f on [ 0, 1 ] as follows fx) = ux) + vx) [ x 0, 1 ]), where ux) = f 1 ) f0) ) x + f0) So v0) = v 1 ) = 0 b) We do the odd extension of v, ie, let v odd t) = vt) [ t 0, 1 ]) and v odd t) = vt) [ t 0, 1 ]) So v 1 ) odd ) = vodd 1 = vodd 0) = 0 Since v odd x) x = v oddx) x, letting x 0 +, we get v odd ) 0) = v odd ) +0) Hence v odd t) is differentiable at t = 0 c) We do the 1-periodic extension v of v odd, ie, v is a 1-periodic function and v t) = v odd t) t 1 ) Furthermore, we easily prove d dt v lip 1 on Proposition 3 Let v be stated as above, ie, v is a 1 periodic function and dv dt lip 1 on Then the periodic wavelet coefficients of v satisfy d 1) m,n = O 5m ) 35) The partial sum S M+1v ) of the periodic wavelet expansion of v satisfies v S M+1v ) L [ 1, 1 ]) = O M) 36) Proof Since ψt)dt = 0 and tψt) dt = 0, we have d 1) m,n = m v t)ψ m t n) dt = m v t) v n ) m v n ) m t n ) ) m ψ m t n) dt 37) 10

11 Again, since dv dt lip 1 on, we know that there is a constant K such that for any t 0, v t) v t 0 ) K t t 0 t ) This implies that t v t) v t 0 ) v t 0 )t t 0 ) = v t) v t 0 ))dt Kt t 0) t 0 t ) Taking t 0 = n m in this formula, by 37) we have d 1) m,n K m t n ) ψ m m t n) dt = K 5m t ψt) dt = O 5m ) 38) By the similar argument to 34), from 38) we now have v S Mv ) L [ 1, 1 ]) = O M) So 36) holds Proposition 3 is proved 3 Two-dimensional algorithm We assume that f is defined on [ 0, 1 ] Let fx,y) = ux,y) + vx,y) x,y) [ 0, 1 ] ), [0, ] ) 1 where ux,y) is a harmonic function and vx,y) satisfies vx,y) = 0 for x,y) We do the odd extension v odd of the residual component v to [ 1, 1 ], that is, v odd x,y) = vx,y) x,y) [ 0, 1 ] ) and v odd x,y) = v odd x, y) = v odd x,y) x,y) [ 1, 1 ] ) Again we do the 1-periodic extension of v odd, denoted by v Finally, we expand v into a biorthonormal periodic wavelet series We call this process the two-dimensional 11

12 Now we examine the decay rates of the coefficients of various wavelet algorithms in the two-dimensional case [ i) The D periodic wavelet algorithm Let f C 1, ] ) 1 Denote the 1-periodic extension of f on by f By 9), the periodic wavelet coefficients are So, for each µ, we have d ) µ,m,n d ) µ,m,n = = m f x,y)ψ µ,m,n x,y)dxdy µ = 1,,3) f x,y)ψ µ m x n 1, m y n ) dxdy = O m ) ψ µ x n 1, y n ) dxdy = O m ) ψ µ x, y) dxdy = O m) [0, ] ) ii) The D folded wavelet algorithm Let f C 1, we do an even extension to [, 1, ] 1 and then we do a 1-periodic extension to, denoted by f Then f lip 1 on Proposition 33 Let f be stated as above, ie, f is a 1 periodic function and f lip 1 on Then the periodic wavelet coefficients of f satisfy d ) µ,m,n = O m) 39) The partial sum S M of the periodic wavelet expansion of f satisfies f S Mf ) L [ 1, 1 ] = O M) 310) so Proof Let z = x,y) Since f lip 1 ) and ψz)dz = 0, we have n )) d ) µ,m,n = f z)ψ µ,m,nz) dz = f z) f m ψ µ,m,n z) dz, d ) µ,m,n = O m ) z n ψµ m m z n) dz = O m) ψ µ z) dz = O m ) 1

13 So 39) holds By 10), we know that the difference between f and the partial sum of its periodic wavelet series f S Mf ) = 3 µ=1 m=m m 1 [ is in the space L 1, ] ) 1 Using the Parseval equality, we have d ) per µ,m,n ψ µ,m,n f S Mf ) L [ 1, 1 ] = 3 µ=1 m=m m 1 n 1=0 m 1 d ) µ,m,n n =0, where n = n 1,n ) Z + Again, by d ) µ,m,n = O m), we have f S Mf ) L [ 1, 1 ] = O1) So we get 310) Proposition 33 is proved m=m m = O M) iii) The D algorithm We can easily prove that 1-periodic function v satisfies v x x,y) lip 1 and v y x,y) lip 1 x,y) ) 311) Proposition 34 Let v be stated as above, ie, v is a 1 periodic function and 311) holds Then the periodic wavelet coefficients of v satisfy d ) µ,m,n = O 3m ) 31) The partial sum S Mv ) of the periodic wavelet expansion of v satisfies v S Mv ) L [ 1, 1 ] = O M) 313) 13

14 Proof For z = x,y) and z 0 = x 0,y 0 ), we have v z) v z 0 ) v x z 0)x x 0 ) v y z 0)y y 0 ) = + + x x 0 y y 0 y y 0 ) v x x,y 0) v x x 0,y 0 ) v y ) v x,y) y x,y 0) dy ) v y x,y 0) v y x 0,y 0 ) dx dy 314) =: J 1 + J + J 3 By 311), we get J 1 Kx x 0 ) K z z 0, J Ky y 0 ) K z z 0, J 3 K x x 0 y y 0 K x x0 ) + y y 0 ) ) = K z z 0 Again, by 314), v z) v z 0 ) v x z 0)x x 0 ) v y z 0)y y 0 ) 3K z z 0 315) By the definition of ψ µ x, y) µ = 1,,3), we obtain that for µ = 1,,3, xψ µ z)dz = yψ µ z)dz = By 9) and 316), taking z 0 = x 0,y 0 ) = n n = n m 1,n )), we deduce that d ) µ,m,n = m v z)ψ µ m z n) dz = m ψ µ z)dz = 0 316) v z) v n ) m v n ) x m x n ) 1 m v n ) y m y n )) m ψ µ m z n) dz From this and 315), noticing that x 0 = n1 m and y 0 = n m, we have d ) µ,m,n 3K m z n ψµ m m z n) dz 3K 3m 14 z ψ µ z) dz = O 3m ) 317)

15 So 31) holds By the similar argument to 310) above, from 37) we now have v S Mv ) L [ 1, 1 ] = O M) Proposition 34 is proved 4 Discrete In this section, we will discuss the discrete and study the symmetry property of the coefficients We always assume that ψ and ψ be a pair of biorthonormal wavelets generated by the scaling functions ϕ, ϕ that are compactly supported real-valued even functions We also assume that ψ and ψ are real-valued functions Let the filters {a k } and {b k } be defined in 5) and let us assume the formula 6) holds Since ϕ and ϕ are symmetric at t = 0, by the known result [4], we know that ψ and ψ are symmetric at t = 1 and t = 1, respectively 41 Definition of one-dimensional discrete Let f C [ 0, 1 ]) be a signal of the interval [ 0, 1 ] We are given the discretized version of f sampled at { n J } J 1 : x n = f n J ) n = 0,, J 1 ) Then x 0 = f0) and x J 1 = f 1 ) Using the decomposition, we get ft) = f ) ) 1 f0) t + f0) + vt) = x J 1 x 0 )t + x 0 + vt) 0 t 1 ) So, for n = 0,, J 1, n ) y n := v J n = x n x J 1 x 0 ) J 1 x 0 In particular, y 0 = y J 1 = 0 15

16 Since the odd extension v odd of v to [ 1, 1 ] satisfies v odd t) = v t) t [ 1 ]),0, the sequence {yn odd } J 1 n=, where y odd J 1 n := v n ) odd, satisfies the conditions J y odd n = y odd n J 1 n J 1) and y0 odd = y odd J 1 = yodd J 1 = 0 Let v be a 1-periodic extension of v odd We define the sequence {z n } n Z : z n := v n J ) n Z) 41) So we obtain that for n Z and l Z, z n = z n, z n+ J = z n and z J 1 l = 0 4) Hence {z n } n Z can be determined by J 1 1 different values z 1,, z J The relationships between the coefficients in the representation Let { } { } c 1) m,n and d 1) m,n be periodic wavelet coefficients of v L [ 1, ]) 1 Taking J sufficiently large, the following formula holds [1] and v t) J 1 c 1) J,n ϕper J,n t) c 1) J,n v n J ) = z n 43) By 4), we get J 1 k=0 By Proposition 1, we know that J 1 1 c 1) J,k ϕper J,k = k=0 J 1 1 c 1) J 1,k ϕper J 1,k + k=0 d 1) per J 1,k ψ J 1,k J 1 c 1) J 1,k = J 1 a n kz n and d 1) J 1,k = b n kz n, 44) where {a n} and {b n} are stated in 7) 16

17 Proposition 41 Let c 1) J 1,k = α k k = 1,, J 1) and d 1) J 1,k = β k k = 0,, J 1) Then { i) ii) c 1) J 1,k { d 1) J 1,k } J 1 1 k=0 } J 1 1 k=0 We need a couple of lemmas = {0, α 1,, α J 1, 0, α J 1,, α 1 }; = {β 0,, β J 1, β J 1,, β 0 } Lemma 4 Let {a n} and {b n} be stated in 7) Then, for any n Z, a n = a n and b n = b n Proof Since ϕ, ϕ are both real-valued even functions and a n = ϕt) ϕt n)dt by 5)), we have From this and 7), we get a n = = ϕt) ϕt + n)dt ϕ t) ϕ t + n)dt = ϕt) ϕt n)dt = a n a n = a n, a J +n = a n n Z) 45) By 5), b n = ψt) ϕt n)dt Since ψ 1 t) = ψ 1 + t), we have b n = = ) 1 ψ + t ϕt + 1 n)dt = ψt) ϕ t + n)dt = ) 1 ψ t ϕt + 1 n)dt ψt) ϕt + n)dt = b n So we get b 1+n = b 1 n By 7), we have b 1+n = b 1 n n J 1 ) Again since b n+ J = b n, we have b 1+n = b 1 n for all n Z So we get b n = b n n Z) 46) 17

18 Lemma 43 The periodic wavelet coefficients {c 1) m,n} and {d 1) m,n} satisfy the following relationships c 1) J 1, k = c1) J 1, J 1 k, d1) J 1, k = d1) J 1, J 1 1 k k Z) 47) Proof From 4) and 45), it follows by 44) that c 1) = J 1 a J 1, J 1 k n+k z J n = J 1 a n+k z n = Again thanks to the periodicity of the {a n } and {z n }, we have 0 n= J +1 a n+k z n 0 n= J +1 a n+k z n = J n=1 = J 1 n=1 a n+k z n = J 1 n=1 a n+k z n + a k z 0 = J 1 a n+k z n + a k J z J a n+k z n 48) So we have J 1 c 1) J 1, J 1 k = J 1 a n+kz n = a n kz n = c 1) J 1,k 49) On the other hand, by 44) and 46), d 1) J 1,k = J 1 b n k z n = J 1 b n+k z n = J 1 b n+k z n = 0 n= J +1 b +n+k z n 410) By the similar argument to 48), we have So we get 0 n= J +1 b +n+kz n = J 1 b +n+kz n J 1 d 1) J 1,k = b n J k ) z n = d 1) J 1, J 1 1 k Proof of Proposition 41 By the first formula of 47), we get c 1) J 1, J = c 1) J 1, J, ie, c 1) J 1, J = 0 411) 18

19 Since a n = a n = a J n and z n = z n = z J n, by 44), we have c 1) J 1,0 = J 1 a nz n = J n=1 a nz n So c 1) J 1,0 = 0 = J n=1 a J n z J n = J 1 a nz n = c 1) J 1,0 Let c 1) J 1,k = α k k = 0,, J 1 1) Then, by c 1) J 1,0 = 0, and 49) and 411), we get α 0 = 0, α J = 0, and α k = α J 1 k k = 1,, J 1 ) ie, i) holds Let d 1) J 1,k = β k k = 0,, J 1 1) By the second formula of 47), we have β k = β J 1 1 k k = 0,, J 1 ), ie, ii) holds Proposition 41 is proved From Proposition 41, we see that in order to recover the signal, we only need J 1 1 periodic wavelet coefficients and two boundary values of the signal f, the number of the sampling points {x n } J 1 0 is just J 1 +1 So the 1D is not a redundant transform 43 Two-dimensional discrete We will now discuss two-dimensional discrete and study the symmetry property of the coefficients [0, ] ) Let an image f C 1 For some large J, take the J ) sample points of f x n1,n = f n1 J, n J ) n1, n = 0,, J 1) Using the decomposition, we get fx,y) = ux,y) + vx,y) x,y) [ 0, 1 ] ), where ux,y) is a harmonic function satisfying Laplace s equation u = 0 and u = f on the boundary of [ 0, 1 ] We can efficiently and accurately compute u by using the Averbuch-Israeli-Vozovoi AIV) method [] Let us now discuss the residual component v 19

20 Let y n1,n = v n 1 J, n J ) Then y 0,n = y n1,0 = y J 1,n = y n1, J 1 = 0 n1, n = 0,, J 1) Let v odd be an odd extension of v to [ 1, 1 ], ie, v odd x,y) = vx,y) x,y) [ 0, 1 ] ) and v odd x,y) = v odd x,y) = v odd x, y) = v odd x, y), x,y) [ 1, 1 ] ) Denote z n1,n = v odd n1 J, n J ) Then z n1,n = z n1,n = z n1, n = z n1, n n1, n = 0, ±1,,± J 1) 41) and z 0,n = z n1,0 = z J 1,n = z n1, J 1 = z J 1,n = z n1, J 1 = 0 Let v be a 1-periodic extension of v odd to Denote z n 1,n = v n 1 J, n J ) Then z n 1,n = z n1,n, n1, n = 0, ±1,, ± J 1), z n 1+ J,n = z n 1,n, z n 1,n + J = z n 1,n, n 1 Z, n Z) 413) For µ = 1,,3, m Z, n Z, let c ) m,n 1,n and d ) µ,m,n 1,n be the periodic wavelet coefficients of v [ L 1, ] ) 1 see 9)) Take J sufficiently large such that v x,y) Again, by 11), we have J 1 n 1,n =0 c ) J,n ϕper 0,J,n and c ) J,n v n 1 J, n J ) = z n 1,n, n = n 1,n ) v x,y) J 1 1 n 1,n =0 c ) J 1, n 1, n ϕ per 0, J 1, n 1, n + 3 µ=1 J 1 1 n 1, n =0 d ) µ, J 1, n 1, n ψper µ, J 1, n 1, n 414) 0

21 Now we discuss the symmetry property of the coefficients c ) J 1,k 1,k and d ) µ,j 1,k 1,k From Proposition, Lemma 4, 41), and 413), we can get Proposition 44 For k 1, k Z, we have i) c ) J 1, J 1 k 1, k = c ) J 1, k 1, k, c ) J 1, k 1, J 1 k = c ) J 1, k 1, k, c ) J 1, J, k = c ) J 1, 0, k = c ) J 1, k 1, J = c ) J 1, k 1, 0 = 0; ii) d ) 1, J 1, J 1 k 1, k = d ) 1, J 1, k 1, k, d ) 1, J 1,k 1, J 1 k 1 = d) 1, J 1, k 1, k, d ) 1, J 1, J, k = d ) 1, J 1, 0, k = 0; iii) d ), J 1, J 1 k 1 1, k = d ), J 1, k 1, k, d ), J 1, k 1, J 1 k = d ), J 1, k 1, k, d ), J 1, k 1, J = d ), J 1, k 1, 0 = 0; and iv) d ) 3, J 1, J 1 k 1 1, k = d ) 3, J 1, k 1, k, d ) 3, J 1, k 1, J 1 k 1 = d) 3, J 1, k 1, k as follows: From Proposition 44, the symmetry of the matrix c ) J 1,k 1,k ), k 1, k = 0,1,, J 1 1 is described α 1,1 α 1, J 1 0 α 1, J 1 α 1,1 0 α J 1,1 α J 1, J 1 0 α J 1, J 1 α J 1, α J 1,1 α J 1, J 1 0 α J 1, J 1 α J 1,1 where 0 α 1,1 α 1, J 1 0 α J 1, J 1 α J 1,1 α k1,k = c ) J 1,k 1,k k1, k = 1,, J 1 ), 1

22 Therefore, the matrix c ) J 1,k 1,k ), k 1,k = 0,1,, J 1 1, is determined by J 1 ) values Similarly, we have i) the matrix ii) the matrix iii) the matrix d ) 1,J 1,k 1,k ), k 1,k = 0,, J 1 1, is determined by J J 1 ) values; d ),J 1,k 1,k ), k 1,k = 0,, J 1 1, is determined by J J 1 ) values; and d ) 3,J 1,k 1,k ), k 1,k = 0,, J 1 1, is determined by J 4 values Noticing that J 1 ) + J J 1 ) + J J 1 ) + J 4 = J 1 1 ) we know that in order to recover v, we only need J 1 1 ) periodic wavelet coefficients To obtain the harmonic function u, we need 4 J 1 1 ) [0, ] ) boundary sample points of the image f on Since the number J 1 1 ) + 4 J 1 1 ) + 4 = J ) is exactly equal to the number of sampling points of f, ie, the D is not redundant 5 Image Approximation Experiments via We will examine the approximation performance of the D algorithm The quality of approximation in this paper is measured by PSN or peak signal-to-noise ratio) defined as ) PSN := 0 log 10 max fx) /MSE, x Ω where MSE is the absolute l error between the original and the approximation divided by the square root of the total number of pixels in the original image The unit of PSN is decibel db) 51 Comparison with the periodic and folded wavelet algorithms To approximate an image sampled on a square by the D algorithm, we first decompose the image into the harmonic component u and the residual v see Fig 1) The harmonic component u is determined by the

23 data at boundary of the square Hence in order to approximate u, it suffices to approximate the data on the boundary of the square Since the boundary consists of four segments, we simply apply one-dimensional LLST on each segment For the residual v, we do an odd extension and a periodic extension see Fig ) and expand it into a periodic wavelet series with respect to a biorthonormal periodic wavelet basis with the symmetric filter bank see Fig 3) Our image approximation and reconstruction strategy consists of the following steps: 1) etain the gray values of the four corner pixels of the image that are necessary to compute the 1D harmonic component of each boundary segment of the square; ) Select a certain number of the largest coefficients in terms of energy from the rest of the coefficients both 1D and D); 3) econstruct u and v from these retained coefficients using the AIV algorithm [] and the Mallat algorithm, respectively; and 4) Compute u + v For approximation and reconstruction of the image by the periodic wavelet algorithm and the folded wavelet algorithm, we simply retain a certain number of the largest coefficients in terms of energy from all the coefficients and reconstruct the image from them First, we compare the performance of with that of the periodic wavelet transform PWT) and the folded wavelet transform FWT) We use the 9/7 biorthogonal filter bank see [7, Sec 74] for the actual filter coefficients) The depths of decomposition J we test here are J =,4,log N) for an image of size N N The original image sizes we use are all dyadic, ie, N = n, which are suitable for PWT and FWT Hence, for, we duplicate the last column and row of each image to make it suitable for As for the images, we use the face part of Barbara image with pixels) as well as the standard images Bridge, Truck, and Moon surface, each of which consists of pixels Fig 4 shows the latter three images Fig 5, 6, and 7 show the quality of approximations of these four images when the depth of decomposition J is set to two, four, and maximum seven for the Barbara face image and eight for the other three images), respectively From these figures, the performance of is consistently superior to that of PWT and FWT For J =, we observe that the big performance difference between and that of PWT and FWT particularly 3

24 when the ratio f the retained coefficients is less than about 6% This can be explained as follows For any image of size N N, if the depth of decomposition J is set to, then the number of low-pass wavelet coefficients in PWT and FWT is N /16, ie, very large Since the low-pass wavelet coefficients are not sparse, we need many coefficients to approximate such an image well In other words, we cannot efficiently approximate such an image using a small number of coefficients if J is set to a small number such as In fact, N /16 is 65% of the original image size N, which agrees with our observation On the other hand, decomposes an image into two components: the harmonic component and the residual We can approximate the harmonic component very well using a few 1D LLST coefficients For the residual, we still have the same problem as PWT and FWT if J = However, since the norm of the residual is much smaller than that of the original image thanks to the removal of the harmonic component, the reconstruction error is kept small When J = 4 or J is set to its maximum, the performance of is about 0 05dB better than that of PWT, and the 01 0dB better than that of FWT In general, the smaller the number of the retained coefficients, the clearer the performance difference becomes This is also due to the harmonic component in We now would like to show the reconstructed images because the PSN plots do not tell the whole story Fig 8, 9, and 10 show the reconstructed Barbara face images using top 5% coefficients of PWT, FWT, and for J =,4,7, respectively From these figures, we can see that the algorithm is also perceptually better than the PWT algorithm and the FWT algorithm, particularly for J = We can also notice some difference around the frame boundary of the images This is due to the use of LLST on the boundary pixels in Overall, for each algorithm, the difference between J = 4 and J = 7 is not noticeable 5 Comparison with LLST We now compare the algorithm with the LLST algorithm In general, the decay rate of LLST coefficients depends on global smoothness of the input data while that of coefficients depends on local smoothness Since the global smoothness of a function is determined by its rough part even if that part is localized), we 4

25 need fewer coefficients than LLST coefficients in order to reconstruct the data to the same quality For LLST, we divide the image into several blocks and do LLST on each block The size of block we use in our experiments is 9 9, 17 17, 33 33, 65 65, for all the four images, and in addition, we use block for the Bridge, Truck, and Moon Surface images Note that the largest block size for each image implies that LLST does not divide it into a set of smaller segments We first retain all the corner pixel values of each block, select the certain number of coefficients with the largest energy among all the coefficients both 1D and D) not yet used, reconstruct an approximation from these coefficients, and finally evaluate its quality of approximation For, the depth of decomposition is set to the maximum, and we also use the 9/7 biorthonormal filter bank We apply on the whole image and retain the coefficients with the largest energy Fig 11 shows the quality of approximation by PSN values when we retain %-0% of the original coefficients From this figure, we again observe that consistently outperforms LLST regardless of the block sizes In particular, s performance is significantly better about 1dB) than that of LLST for the Bridge, Truck, and Moon Surface images while its performance on Barbara face image is closely followed by that of LLST with blocks, especially around the ratio of the number of the retained coefficients to that of the original coefficients ranges around 8% to 10% We believe that this is due to the existence of textures in the Barbara face image We also observe that the performance of LLST at a particular ratio of the number of the retained coefficients to that of the whole coefficient strongly depends on the block size For example, for heavy compression ie, the ratio is about % to 8%), the LLST with the larger block sizes perform better than that with the smaller block sizes while the situation is opposite if one can afford to keep a larger number of coefficients eg, the ratio is larger than 10%) does not require the user to choose any such block sizes, which is one of the major advantage of over LLST 53 Comparison with wavelets on the interval It is appropriate to compare with wavelets on the interval WOI) introduced by Cohen, Daubechies and Vial [5] because WOI also tries to overcome boundary effects Compared to, the periodic wavelets, and 5

26 folded wavelets, however, the construction of WOI is quite complicated Their starting point is Daubechies s compactly supported scaling functions and wavelets They first construct scaling functions on the interval consisting of three parts: the left edge scaling function, the interior scaling function, and the right edge scaling function Then, they use these scaling functions on the interval to construct the corresponding wavelets also consisting of three parts: the left edge wavelet, the interior wavelet, and the right edge wavelet Although these wavelets on the interval have high vanishing moments, we cannot apply its discrete version to image approximation immediately This is because their corresponding high pass filter cannot map a simple polynomial sequence to zero So when one uses WOI to approximate images, one has to perform a prefiltering or preconditioning) on the data first; see [5] for the detail We now compare the performance of with that of WOI Since the harmonic component of takes care of linear parts on the boundary, in order to be a fair comparison with, we use WOI with two vanishing moments For, we use Villasenor 5/3 filter bank [10], which also has two vanishing moments Fig 1 shows the quality of approximation measured by PSN values when we retain 1% 10% of the original coefficients The depth of decomposition for both and WOI is set to five for the Barbara face image and six for the other three images since these are the maximal depth of decomposition that the WOI can take From this figure, we observe that that again consistently outperforms WOI Considering the implementation complexity of WOI, should be used if one wants to reduce the boundary effects 6 Conclusion In this paper, we proposed the Harmonic Wavelet Transform ) that is not affected by the boundary of input data The idea of removing the boundary mismatches of input data originally proposed in LLST [8] is quite important since the residual component after odd reflection and periodization, say, the v component, does not contain any artificial discontinuities caused by the boundary mismatches Hence, the expansion coefficients of v with respect to the periodic wavelet basis truly relect the local smoothness of an input image Moreover, 6

27 captures the intrinsic singularities in the interior of the domain more efficiently than LLST that uses the Fourier sine series expansion for v Also, the implementation of is simpler than WOI because does not need any special boundary-dependent filter banks Finally, our image approximation experiments using four standard images demonstrated the superiority of over LLST, PWT, FWT, and WOI We also note that the extension of to a higher dimension is straightforward thanks to the efficient Laplace solver for higher dimension [3] Acknowledgment This research was partially supported by the NSF grant DMS , the ON grants N , N , and N For computing the periodic and folded wavelet transforms as well as the wavelets on interval, we used the WaveLab software developed by Prof David Donoho of Stanford Univ and his team [6] eferences [1] M Antonini, M Barlaud, P Mathieu, and I Daubechies, Image coding using wavelet transform, IEEE Trans Image Proc 1 ) 199) 05 0 [] A Averbuch, M Israeli, and L Vozovoi, A fast Poisson solver of arbitrary order accuracy in rectangular regions, SIAM J Sci Comput ) [3] E Braverman, M Israeli, A Averbuch, L Vozovoi, A fast 3D Poisson solver of arbitrary order accuracy, J Comput Phys ) [4] C K Chui, An Introduction to Wavelets, Academic Press, San Diego, 199 [5] A Cohen, I Daubechies, and P Vial, Wavelet bases on the interval and fast algorithms, Appl Comput Harmon Anal )

28 [6] D Donoho et al, wavelab [7] S Mallat, A Wavelet Tour of Signal Processing Third edition), Academic Press, San Diego, 009 [8] N Saito and J emy, The polyharmonic local sine transform: a new tool for local image analysis and synthesis without edge effect, Appl Comp Harmon Anal ) [9] D Van De Ville, T Blu, and M Unser, Isotropic polyharmonic B-splines: Scaling functions and wavelets, IEEE Trans Image Proc 14 11) 005) [10] J D Villasenor, B Belzer, and J Liao, Wavelet filter evaluation for image compression, IEEE Trans Image Proc 4 8) 1995)

29 a) Original b) The u component c) The v component Figure 1: decomposition of the Barbara face image Each image is displayed using its full dynamic range Figure : The odd extension of the residual component v 9

30 Figure 3: The periodic biorthonormal wavelet coefficients of the odd extension of the residual component v when the depth of decomposition is one a) Bridge b) Truck c) Moon Surface Figure 4: The three more standard images used in our experiments 30

31 PSN 0 PSN PWT FWT 10 PWT FWT atio of retained coefficients %) atio of retained coefficients %) a) Barbara face b) Bridge PSN 0 PSN PWT FWT 10 PWT FWT atio of retained coefficients %) c) Truck atio of retained coefficients %) d) Moon Surface Figure 5: Quality of approximation of the four standard images measured by PSN values as a function of the ratio of the number of the retained coefficients to the total number of the coefficients The depth of decomposition is set to two in each case 31

32 PSN PSN PWT FWT 0 PWT FWT atio of retained coefficients %) a) Barbara face atio of retained coefficients %) b) Bridge PSN PWT FWT 9 8 PWT FWT atio of retained coefficients %) c) Truck d) Moon Surface Figure 6: Quality of approximation of the four standard images measured by PSN values as a function of the ratio of the number of the retained coefficients to the total number of the coefficients The depth of decomposition is set to four in each case 3

33 PSN PSN PWT FWT 0 PWT FWT atio of retained coefficients %) a) Barbara face atio of retained coefficients %) b) Bridge PSN PSN PWT FWT 9 PWT FWT atio of retained coefficients %) c) Truck atio of retained coefficients %) d) Moon Surface Figure 7: Quality of approximation of the four standard images measured by PSN values as a function of the ratio of the number of the retained coefficients to the total number of the coefficients The depth of decomposition is set to the maximum in each case, ie, seven for the Barbara face image and eight for the others 33

34 a) Original Image b) PWT c) FWT d) Figure 8: Approximations of the Barbara face image using the top 5% coefficients when the depth of decomposition is two The PSN values in db) of PWT, FWT, and are 17618, , and 0776, respectively 34

35 a) Original Image b) PWT c) FWT d) Figure 9: Approximations of the Barbara face image using the top 5% coefficients when the depth of decomposition is four The PSN values in db) of PWT, FWT, and are 9791, 31484, and 3651, respectively 35

36 a) Original Image b) PWT c) FWT d) Figure 10: Approximations of the Barbara face image using the top 5% coefficients when the depth of decomposition is seven The PSN values in db) of PWT, FWT, and are 996, 31476, and 3688, respectively 36

37 PSN LLST LLST PHLST PHLST LLST PSN LLST LLST LLST LLST LLST LLST atio of retained coefficients %) a) Barbara face atio of retained coefficients %) b) Bridge PSN LLST LLST LLST LLST LLST LLST PSN LLST LLST LLST LLST LLST LLST atio of retained coefficients %) c) Truck atio of retained coefficients %) d) Moon Surface Figure 11: Quality of approximation of the four standard images measured by PSN values as a function of the ratio of the number of the retained coefficients to the total number of the coefficients These figures compare the performance of with that of LLST with different block sizes For, the depth of decomposition is set to its maximum level 37

38 PSN a) Barbara face WOI atio of retained coefficients %) b) Bridge WOI WOI 8 WOI c) Truck d) Moon Surface Figure 1: Quality of approximation of the four standard images measured by PSN values as a function of the ratio of the number of the retained coefficients to the total number of the coefficients These figures compare the performance of with that of WOI For both and WOI, the depth of decomposition is set to its maximum possible level 38