Recovery of Piecewise Smooth Images from Few Fourier Samples

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1 Recovery of Piecewise Smooth Imaes from Few Fourier Samples Gre Onie Department of Mathematics University of Iowa Iowa City, Iowa 54 Mathews Jacob Department of Electric and Computer Enineerin University of Iowa Iowa City, Iowa 54 arxiv: v1 [cs.cv] 3 Feb 015 Abstract We introduce a Prony-like method to recover a continuous domain -D piecewise smooth imae from few of its Fourier samples. Assumin the discontinuity set of the imae is localized to the zero level-set of a trionometric polynomial, we show the Fourier transform coefficients of partial derivatives of the sinal satisfy an annihilation relation. We present necessary and sufficient conditions for unique recovery of piecewise constant imaes usin the above annihilation relation. We pose the recovery of the Fourier coefficients of the sinal from the measurements as a convex matrix completion alorithm, which relies on the liftin of the Fourier data to a structured lowrank matrix; this approach jointly estimates the sinal and the annihilatin filter. Finally, we demonstrate our alorithm on the recovery of MRI phantoms from few low-resolution Fourier samples. I. INTRODUCTION The recovery of continuous domain parametric representations from few measurements usin harmonic retrieval/linear prediction has received considerable attention in sinal processin since Prony s seminal work [1], []. Extensive research has been devoted to the recovery of finite linear combination of exponentials with unknown continuous frequencies as well as linear combination of Diracs and non-uniform 1-D splines with unknown locations/knots [3] [5]. Recently, convex alorithms that minimize atomic norm were also introduced to recover such continuous sinals [6], [7]; these methods are off-therid continuous eneralizations of compressed sensin theory, which can avoid discretization errors and hence are potentially more powerful. However, the direct extension of the above Prony-like and convex off-the-rid methods to -D piecewise smooth imaes is not straihtforward. Specifically, the partial derivatives of piecewise smooth imaes can be thouht of as a linear combination of a continuum of Diracs supported on the curves separatin the reions, which current methods are not desined to handle. Recently, Pan et al. [8] introduced a complex analytic sinal model for continuous domain -D imaes, such that the complex derivatives of the imae are supported on a curve. Under the assumption that the curve is the zero-set of a bandlimited function, the authors show the Fourier transform of the complex derivative of such a sinal is annihilated by convolution with the Fourier coefficients of the band-limited function. This property is used to extend the finite-rate-ofinnovation model [3] to this class of -D sinals. This work has several limitations. One problem is that a complex analytic sinal model is not realistic for natural imaes (e.., the only real-valued analytic functions are constant functions). In addition, if we are only measurin a finite number of Fourier samples, one can choose analytic functions such that the all of these coefficients vanish; i.e. the recovery of the sinal from few Fourier coefficients of an arbitrary sinal in this class is ill-posed. In this work, we address the above limitations by proposin an alternative sinal model based on the a more realistic class of piecewise smooth functions. We show that we can eneralize the annihilation property in [8] to this class of functions, such that it is possible to recover an exact continuous domain representation of the ede set from few Fourier samples. We also determine necessary and sufficient conditions on the number of Fouier samples required for perfect recovery of the ede set in the caseof piecewise constant imaes. To recover the full sinal, we propose a sinle-step convex alorithm which can be thouht of as jointly estimatin the ede set and imae amplitudes. This is fundamentally different than the two stae approach proposed in [8], which we also investiated for super-resolution MRI in [9]. Motivated by recently proposed alorithms for calibration-free parallel MRI recovery [10], [11], our new approach is based on the observation that the ideal Fourier samples of the sinal lift to a structured low-rank matrix, which allows us to pose the recovery as a low-rank structured matrix completion problem. We demonstrate our alorithm on the recovery of MR phantoms from low-resolution Fourier samples. A. Piecewise smooth sinals II. SIGNAL MODEL In this paper, we consider the eneral class of -D piecewise smooth functions: f(r) = i (r) χ Ωi (r), r = (x, y) [0, 1], (1) where χ Ω denotes the characteristic function of the set Ω: { 1 if r = (x, y) Ω χ Ω (r) = () 0 else.

2 Here we assume each Ω i [0, 1] is a simply connected reion with piecewise smooth boundary Ω i. The functions i in (1) are smooth functions that vanish under of a collection of constant coefficient differential operators D = {D 1,..., D N } within the reion Ω i : D j i (r) = 0, r Ω i ; j = 1,.., N. (3) We now show that the above class of functions is fairly eneral and includes many well-understood imae models by appropriately choosin the set of differential operators D. 1) Piecewise constant imaes: We set D to D = = { x, y }. Note that D = 0 if and only if = c i for c i C. Hence, (1) reduces to the well-known piecewise constant imae model: f(r) = c i χ Ωi (r), r = (x, y) [0, 1], (4) This case will be the primary focus of this work due to its simplicity and provable uarantees. ) Piecewise analytic imaes: Choosin D = z = x + j y, then D = 0 if and only if is complex analytic. Hence this model is equivalent to the one proposed in [8]. As described above, this sinal model is not very realistic for natural imaes. 3) Piecewise harmonic: Both the above cases consider only first-order differenial operators. One choice of a second-order differential operator is the Laplacian D = = xx + yy. Then D = 0 if and only if is harmonic. 4) Piecewise linear imaes: If we consider all second order partial derivatives D = { xx, xy, yy}, then D = 0 if and only if is linear, i.e. (r) = a, r + b, for a C, b C, and so f has the expression f(r) = ( a i, r + b i ) χ Ωi (r), r = (x, y) [0, 1]. (5) 5) Piecewise polynomial: Generalizin the above case we may consider all nth order partial derivatives D = { α } α =n where α is a multi-index. then D = 0 if and only if is a polynomial of deree at most (n 1). We will show that under certain assumptions on the ede set C = n Ω i, the Fourier transform of derivatives of a piecewise smooth sinal specified by (1) satisfies an annihilation property. This will enable us to recover an exact continuous domain representation the ede set C of a piecewise smooth sinal from finitely many of its Fourier samples by solvin a linear system. B. Trionometric polynomials and curves Followin [8], we will assume the ede set C to be the zero-set of a band-limited periodic trionometric polynomial µ(r) = k Λ c[k] e jπ k,r, r [0, 1], (6) where c[k] C and Λ is any finite subset of Z ; we call any function µ described by (6) a trionometric polynomial, and the zero-set C : {µ = 0} a trionometric curve. We also define the deree of a trionometric polynomial µ to be the dimensions of the smallest rectanle that contains the frequency support set Λ, denoted as de(µ) = (K, L). For trionometric polynomials µ and ν, we say ν divides µ or ν µ if µ = ν γ where γ is another trionometric polynomial. Usin elementary results from alebraic eometry, we may show there is a unique minimal deree trionometric polynomial associated with any trionometric curve C, which we call the minimal polynomial for C: Proposition 1. For every trionometric curve C there is a unique (up to scalin) trionometric polynomial µ 0 with C : {µ 0 = 0} such that for any other trionometric polynomial µ with C : {µ = 0} we have de(µ 0 ) de(µ) and µ 0 µ. The followin property of minimal polynomials is also important for our uniqueness results: Proposition. Let C be the zero set of a trionometric polynomial with minimal polynomial µ 0. Suppose ν is a trionometric polynomial such that ν = 0 and ν = 0 on C, then µ 0 ν. In particular, µ 0 = 0 for at most finitely many points on C. III. ANNIHILATION PROPERTY We now show that the Fourier transform of the partial derivatives of piecewise smooth sinals (1) satisfy an annihilation property. 1) First order partial derivative operators: First we consider the case of a sinle characteristic function χ Ω. Note that since χ Ω is non-smooth at the boundary, its derivatives are only defined in a distributional sense. Lettin ϕ denote any test function we have: x χ Ω, ϕ = χ Ω, x ϕ = x ϕ dr = ϕ dy (7) Ω Ω where the last step follows by Green s theorem. Likewise, y χ Ω, ϕ = ϕ dx (8) Hence x χ Ω and y χ Ω can be interpreted as a continuous stream of weihted Diracs supported on Ω. In particular, if ψ is any smooth function that vanishes on Ω then Ω ψ x χ Ω = ψ y χ Ω = 0 (9) where equality holds in the distributional sense. Assumin ψ = µ is a trionometric polynomial, takin Fourier transforms of (9) yields the followin annihilation relation: Proposition 3. Let f = χ Ω with boundary Ω iven by the trionometric curve C : {µ = 0}. Let D be any first order differential operator. Then the Fourier transform of Df is annihilated by convolution with the Fourier coefficients c[k], k Λ of µ, that is c[k] Df(ω πk) = 0, for all ω R. (10) k Λ

3 Due to the above property, we call µ an annihilatin polynomial for Df. It is straihtforward to extend the above proposition to piecewise constant functions f = n c iχ Ωi, provided µ = 0 on the union of the boundaries C = n Ω i. Likewise, if f = Ω where D = 0, then by the product rule Df = D χ Ω + Dχ Ω = Dχ Ω which has support on Ω and so, µ Df = 0, which implies (10) holds for f, and similarly for the linear combination f = n i χ Ωi, where D i = 0 for all i = 1,..., n. ) Second order partial derivative operators: Now consider the case where D is any second-order differential operator. Let f = χ Ω where D = 0. We now show that µ is an annihilatin polynomial for Df, where µ is any trionometric polynomial that annihilates the partial derivatives of χ Ω. Let = 1 where i { x, y }, i = 1,. By the product rule we have: f = χ Ω + 1 χ Ω + 1 χ Ω + χ Ω. Since 1 χ Ω and χ Ω are annihilated by µ, we have µ f = χ Ω µ + µ χ Ω. Aain by the product rule µ χ Ω = (µ 1 χ Ω ) µ µ 1 χ Ω = 0, which implies and so by linearity µ f = χ Ω µ µ Df = χ Ω µ D = 0. The above shows that µ is always sufficient to annihilate Df, where D is second-order. However, usin Prop., we may show that when does not vanish on Ω, then µ is also necessary for annhilation of Df, in the sense that if ν is any other tri polynomial satisfyin ν Df = 0, then µ 0 ν, where µ 0 is the minimal polynomial for Ω. If de(µ 0 ) = (K, L), this implies any annihilatin polynomial ν for Df has de(ν) (K 1, L 1). 3) Partial derivative operators of arbitrary order: A similar arument shows that when D is any nth order differential operator, and f = χ Ω where D = 0, then µ n Df = 0. (11) This yields the followin annihilation relation for hiher-order differential operators: Proposition 4. Let D be any nth order differential operator. Let f = χ Ω with D = 0 and Ω {µ = 0} for some trionometric polynomial µ. Then the Fourier transform of Df is annihilated by convolution with the Fourier coefficients d[k], k Γ of µ n, that is d[k] Df(ω πk) = 0, for all ω R. (1) k Γ Likewise, by linearity, (1) is valid for linear combinations f = i i χ Ωi, where D i = 0 and µ = 0 on i Ω i. IV. RECOVERY FROM FINITE FOURIER SAMPLES We now investiate necessary and sufficient conditions for the recovery of the filter coefficients describin the ede set from finitely many Fourier samples of the oriinal sinal f. For these results we restrict our attention to piecewise constant sinals. A. Necessary conditions For a piecewise constant sinal f, from the annihilation condition (10) we may form the linear system of equations: { k Λ d[k] f x (π[l k]) = 0, k Λ d[k] f l Γ. (13) y (π[l k]) = 0, where f x and f y may be computed from samples of f by f x (ω) = jω x f(ω), and f y (ω) = jω y f(ω). Supposin the samplin rid Ω is a rectanular of dimensions (K, L ), and the minimal polynomial for C has deree (K, L), with coefficients c[k] supported in Λ, then we may form at most M = (K K + 1) (L L + 1) valid equations from (13). Therefore to solve for the at most K L unknowns c[k], k Λ, we require at least M = K L equations. This ives the followin necessary condition for recovery of C: Proposition 5. Let f be piecewise constant such that the ede set C has minimal polynomial µ of deree (K, L). A necessary condition to recover the ede set C from (13), is to collect samples of f on a (K, L ) rectanular rid such that (K K + 1) (L L + 1) K L. To illustrate this bound, suppose the minimal polynomial has deree (K, K), and we take Fourier samples from a square reion. Then this requires at least 1.71K 1.71K Fourier samples to recover the ede set C. Our numerical experiments on simulated data (see Fi. 1) indicate the above necessary condition miht also be sufficient for unique recovery; that is, we hypothesize the minimal filter coefficients c[k] are the only non-trivial solution to the system of equations (13). B. Sufficient conditions We now focus on sufficient conditions for the recovery of the ede set. Here we will use mλ to denote a dilation of the set Λ by a factor of m: if Λ = {(k, l) : k K, l L}, then mλ = {(k, l) : k m K, l m L}. Theorem 6. Let f = χ Ω be the characteristic function of a simply connected reion Ω with boundary Ω havin minimal polynomial µ with coefficients c[k], k Λ. Then the c[k] can be uniquely recovered (up to scalin) as the only non-trivial solution to the equations { k Λ c[k] f x (π[l k]) = 0, k Λ c[k] f l Λ. (14) y (π[l k]) = 0, The proof entails showin any other trionometric polynomial η(r) havin coefficients d[k], k Λ satisfyin (6) must vanish on Ω, from which it then follows that η is a scalar multiple of the minimal polynomial µ by deree

4 (a) Oriinal sinal (c) Ede set {µ = 0} lo(σ) (b) Recovered µ sinular values σ (d) Sinular values (lo scale) Fi. 1. Exact recovery of ede set of a piecewise constant sinal from the minimum necessary number of Fourier samples. The oriinal piecewise constant sinal is shown in (a), and was enerated to have ede set C : {µ = 0} where the deree of µ is known to be (9, 9). Here the minimal polynomial and ede set C is shown to be recovered from the system (13) havin dimensions correspondin to the necessary minimum number of Fourier samples predicted by Prop. 5. The sinular values of this system are plotted in (d), indicatin the recovered µ is the only non-trivial solution. considerations. We also are able to show similar result holds for piecewise constant sinals, provided the characteristic functions do not intersect: Theorem 7. Let f(r) = n a iχ Ωi (r) be piecewise constant, where the boundaries Ω i are described by nonintersectin trionometric curves {µ i = 0}, where µ i is the minimal polynomial for Ω i. Then, the coefficients d[k],k Λ, of µ = µ 1 µ n, and equivalently the ede set C = n Ω i, can be uniquely recovered (up to scalin) as the only nontrivial solution of { k Λ d[k] f x (π[l k]) = 0, k Λ d[k] f l Λ. (15) y (π[l k]) = 0, Note that to form the equations in (14) and (15) requires access to Fourier samples f[k] for all k 3Λ, which is reater than necessary number of samples iven in Theorem 5. We conjecture that the uniqueness results in Theorems 6 and 7 can in fact be sharpened to the necessary number of samples, and extended to piecewise constant sinals where the boundaries of the reions intersect. V. RECOVERY ALGORITHMS Up to now we have only considered the problem of recoverin the ede set of a piecewise constant sinal f = n a iχ Ωi from finite Fourier samples. In analoy with Prony s method, once the ede set is determined it is theoretically possible to recover the sinal amplitudes a i by substutin f back into (15) and solvin a full rank system. However, this is not feasible in practice since it requires factorin a hih deree multivariate polynomial into its irreducible factors. Instead we pursue approaches that allow us to pose the recovery as the solution to a convex optimization problem. A. Curve-aware recovery Supposin we have access to an annihilatin polynomial µ of the sinal (equivalently, the ede set C : {µ = 0}), one approach is to pose the recovery as the weihted total variation minimization problem: f = ar min µ(r) (r) dr subject to ĝ[k] = f[k], k Γ (16) Here, since µ = 0 on the ede set C, the radient of the imae not penalized alon the curve, which allows for the recovery of an imae with sharp edes alon C. A version of this approach was investiated in an earlier work for the super-resolution recovery of MR sinals from few Fourier samples [9]. However, we found this scheme to have certain drawbacks, namely that it requires discretization onto a spacial rid, the optimization of many parameters, and is very sensitive to the estimate µ. Hence we consider an alternate approach which does not rely on an explicit estimate of the annihilatin polynomial µ, but instead jointly recovers the imae and the annihilatin polynomial in a sinle stae alorithm. B. Low-rank recovery The annhilation equations specified by (13) can be represented in the matrix form as [ T x [ f] ] T y [ f] d = 0 (17) }{{} T[ f] where d is a vectorized version of the Fourier coefficients d[k], k Λ, and T x and T y are block Toeplitz matrices correspondin to the -D convolution of d[k] with the discrete samples of jω x f(ω) and jωy f(ω), respectively, for ω = πl, l Ω. Specifically, if the coefficient support set Λ has dimensions K L, each row of T x ( ˆf) is the vectorized version of an K L patch of jω x f(ω), and likewise for Ty ( ˆf). The number of rows is equal to the number of distinct patches, which correspond to the number of annihilation equations. Note that for a iven piecewise constant sinal, a priori we do not know the deree of the minimal polynomial describin the ede set, which is needed to specify the size of T. However, if d[k], k Λ correspondin to the trionometric polynomial µ is a solution to (17) whose support set Λ is strictly smaller than the assumed support set Λ, then any multiple ν = µ γ havin coefficients e[k] = (d )[k] supported within Λ, is also a solution. This implies that if we consider a larer filter size than required by the minimal

5 polynomial, i.e. more columns in T than the number of coefficients in d, the matrix T will be low-rank. The precedin discussion suests we may pose the recovery of the sinal as a structured low-rank matrix completion problem, entirely in the Fourier domain: fb = ar min rank(t[b ]) subject to b[k] = fb[k], k Γ b (18) We note this approach is still off-the-rid in the sense that we may recover a discrete imae at any desired resolution by extrapolatin fb to this resolution in Fourier space, and applyin an inverse DFT. To address the case of noisy measurements and model mismatch we propose solvin the convex relaxation of (18): fb = ar min kt[b ]k + λkpγ (b fb)k b (a) Fully sampled (b) Zero-padded (c) Low-rank (19) Fi.. Recovery of Shepp-Loan phantom on a 56x56 rid from where k k denotes the nuclear norm, i.e. the absolute sum of sinular values, λ is a tunable parameter, and PΓ is the projection onto the samplin set Γ. A standard approach to solvin (19) is by an iterative sinular value soft-thresholdin alorithm, which requires an SVD of the estimate T [b ] at each step. Due to the size of T [b ], such an alorithm is computationally prohibitive in this case. Instead we use the SVD-free alorithm proposed in [1], which involves introducin auxiliary variables U CM r and V CN r via the well-known relation kxk = minx=uvh kukf + kvkf, and enforcin the constraint T[b ] = UVH, with the ADMM alorithm. In Fi. we demonstrate the ability of the proposed alorithm to recover a piecewise constant sinal from few of its uniform low-resolution Fourier samples. We experiment on simulated data obtained from analytical MRI phantoms derived in [13]. We extrapolate from = 3185 analytical Fourier samples of the Shepp-Loan phantom to a rid ( 0-fold undersamplin), and recover the sinal by performin a inverse DFT. Note that the rinin artifacts observed in the recovery are to be expected due to fact we are recoverin exact Fourier coefficients of the sinal, and could be removed with mild post-processin. VI. C ONCLUSION We propose an extension of the annihilatin filter method to a wide class of -D piecewise smooth functions whose edes are supported on level set of a band-limited function. This enables us to recover an exact continous domain representation of the ede set from few low-frequency Fourier samples. In the case of piecewise constant sinals, we derive conditions of the necessary and sufficient number of Fourier samples to ensure exact recovery of the ede set. Lastly, we prosed one-stae alorithm to recover piecewise constant sinals by extrapolatin the sinal in Fourier domain. We demonstrate that we may accurately recover MRI phantoms from few of their low-resolution Fourier samples. 65x49=3185 Fourier samples ( 0 fold undersamplin). 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