BOUNDED SPARSE PHOTON-LIMITED IMAGE RECOVERY. Lasith Adhikari and Roummel F. Marcia

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1 BONDED SPARSE PHOTON-IMITED IMAGE RECOVERY asith Adhikari and Roummel F. Marcia Department of Applied Mathematics, niversity of California, Merced, Merced, CA 9533 SA ABSTRACT In photon-limited image reconstruction, the behavior of noise at the detector end is more accurately modeled as a Poisson point process than the common choice of a Gaussian distribution. As such, to recover the original signal more accurately, a penalized negative Poisson log-likelihood function and not a least-squares function is minimized. In many applications, including medical imaging, additional information on the signal of interest is often available. Specifically, its maximum and minimum amplitudes might be known a priori. This paper describes an approach that incorporates this information into a sparse photon-limited recovery method by the inclusion of upper and lower bound constraints. We demonstrate the effectiveness of the proposed approach on two different low-light deblurring examples. Index Terms Photon-limited imaging, Poisson noise, bounded sparse approximation, convex optimization, wavelets.. INTRODCTION Signal recovery under low-light conditions arises in many applications, including medical imaging [, ], astronomy [3, ], and security and defense [5, ]. In low photon context, the arrival of photons at the detector is typically modeled by the inhomogeneous Poisson process: y Poisson(Af ), where y Z m + is the vector of observed photon counts, A R m n + is the linear projection matrix, and f R n + is the nonnegative true signal of interest (see e.g., [7, ]). If the signal to be reconstructed is known to be sparse, then it can be approximated by solving a Poisson inverse problem with a sparsity-promoting penalty (see e.g., [9,,, ]). In addition to sparsity and nonnegativity, other intensity information about the true signal may be known. In particular, its maximum and minimum amplitudes at specific regions might be known a priori. In medical imaging, structural information such as tissue geometries are used to improve the accuracy of tomography. For example, in near infrared diffuse optical tomography, structural priors from magnetic resonance imaging are incorporated to limit smoothing across This work was supported by National Science Foundation Grant CMMI asith Adhikari s research is supported by the C Merced Graduate Student Opportunity Fellowship Program. their shared boundaries and adjust the image smoothness [3,, 5]. These structural priors can be expressed as bounds on the signal intensity, and as such, they can be incorporated into the photon-limited image recovery problem to enhance the quality of the reconstruction. This paper describes an optimization method (based on the SPIRA approach []) that includes upper and lower bound constraints that model additional signal intensity information. We demonstrate the effectiveness of the proposed approach on two different low-light deblurring examples.. PROBEM FORMATION When the maximum and minimum signal intensity information is known, the sparsity-promoting Poisson intensity reconstruction problem has the following constrained minimization form: ˆf = arg min f R n Φ(f) F (f) + τ f subject to b f b, where τ > is the regularization parameter and F (f) is the negative Poisson log-likelihood function F (f) = T Af m y i log(e T i Af + β), i= where is an m-vector of ones, e i is the ith canonical basis unit vector, β > (typically β to avoid the singularity at f = ) (see e.g., []). Our proposed approach builds upon the the SPIRA framework in [], which defines a sequence of quadratic subproblems, where F (f) is approximated by a second-order Taylor series expansion, where the Hessian matrix is replaced by a scaled identity matrix I, where > (see e.g., [7, ]). In our proposed approach, these quadratic subproblems are of the form f k+ = arg min f R n f sk + τ f subject to b f b, where s k = f k F (f k ). If the signal of interest is sparse in some orthonormal basis W, then the penalty term f is

2 replaced by θ, where θ = W T f. Then the minimization subproblem becomes θ k+ = arg min θ R n φ k (θ) = θ sk + τ θ, subject to b W θ b. () We note that typically, b =, but we do not make that assumption here. We can solve this minimization problem by solving its agrangian dual. The discussion below follows [] very closely. Our main contribution is the extension of the constraints to general bounds and the inclusion of a convergence proof for the subproblem minimization. First, we introduce u, v R n with u, v and write θ = u v so that φ k (θ) in () is differentiable [, ]: (u k+, v k+ ) = arg min u,v R n u v sk + τ T (u + v) subject to u, v, b W (u v) b. () Note, however, that the new problem now has twice as many parameters and has additional nonnegativity constraints on the new parameters. The last constraints can be expressed as W (u v) b and b W (u v). The agrangian function corresponding to () is given by (u, v, λ, λ, λ 3, λ ) = u v sk + τ T (u + v) λ T u λ T v λ T 3 (W (u v) b ) λ T (b W (u v)), where λ, λ, λ 3, λ R n are the agrange multipliers corresponding to the constraints in (). Differentiating with respect to u and v and setting the derivatives to zero yields u v = s k + λ τ + W T λ 3 W T λ, and (3) λ = τ λ. Then it follows that τ T (u+v) λ T u λ T v = τ T (u v) λ T (u v) in. Therefore (u, v, λ, λ, λ 3, λ ) = u v + sk (u v) T (s k + λ τ + W T λ 3 W T λ ) + λ T 3 b λ T b. Substituting u v from (3) in, we obtain the agrangian dual function independent of the primal variables, u and v: g(λ, λ 3, λ ) = sk + λ τ + W T (λ 3 λ ) + λ T 3 b λ T b + sk. Next, let γ = λ τ. For the agrange dual problem corresponding to (), the agrange multipliers λ i for i {,, 3, }. Since λ = τ λ = τ γ and λ = γ + τ, then γ satisfies τ γ τ. The agrange dual problem associated with () is thus given by minimize γ,λ 3,λ R n h(γ, λ 3, λ ) = sk + γ + W T (λ 3 λ ) λ T 3 b + λ T b sk subject to λ 3, λ, τ γ τ. () At the dual optimal values γ, λ 3, and λ, the primal iterate θ k+ is given by θ k+ = u k+ v k+ = s k +γ +W T (λ 3 λ ). We note that the duality gap for () and its dual () is zero, i.e., φ k (θ k+ ) = h(γ, λ 3, λ ) because () satisfies (a weakened) Slater s condition [9]. In addition, the function h(γ, λ 3, λ ) is a lower bound on φ k (θ) at any dual feasible point. We note that the objective function h(γ, λ 3, λ ) can be written as h(γ, λ 3, λ ) = { γ + γ T s k } + γ T W T (λ 3 λ ) +{ λ 3 λ + (λ 3 λ ) T W s k λ T 3 b + λ T b }. We minimize the objective function h(γ, λ 3, λ ) by solving for γ, λ 3, and λ alternatingly, which is done by taking the partial derivatives of h(γ, λ 3, λ ) and setting them to zero. Each component is then constrained to satisfy the bounds in (). We now describe each step more explicitly. Step. Given λ (j ) 3 and λ (j ) from the previous iterate, solve γ (j) = arg min γ + γ T s k + γ T W T (λ (j ) 3 λ (j ) ) γ R n subject to τ γ τ. (5) The solution to (5) is obtained via thresholding: { ( ) } γ (j) = mid τ, s k W T λ (j ) 3 λ (j ), τ, () where the operator mid{a, b, c} chooses the middle value of the three arguments component-wise. Step. Given γ (j), solve (λ (j) 3, λ(j) ) = arg min (λ 3, λ ) λ 3 λ λ 3,λ R n + λ T ( 3 W (s k + γ (j) ) ) b + λ T ( b W (s k + γ (j) ) ) subject to λ 3, λ. (7) The minimization problem (7) has the following solution. Noting λ 3 λ = λ 3 λ T 3 λ + λ, and letting r (j) = W (sk +γ (j) ) b and r (j) = b W (s k +γ (j) ),

3 then (7) can be written as (λ (j) 3, λ(j) ) = arg min λ 3 + λ T 3 r (j) λt 3λ λ 3,λ R n + λ + λ T r (j) subject to λ 3, λ. () Note that if r (j) and r(j), i.e., b W (s k + γ (j) ) b, then (λ 3, λ ) for λ 3, λ, and is therefore minimized at λ 3 = λ =. We now assume otherwise. Computing the gradient of (λ 3, λ ) with respect to λ 3 and λ yields λ3 (λ 3, λ ) = λ 3 + r (j) λ and λ (λ 3, λ ) = λ +r (j) λ 3. For each i, unless (b ) i = (b ) i, both ( λ3 ) i and ( λ ) i cannot be simultaneously (since this implies (r (j) ) i + (r (j) ) i =, or equivalently, (b ) i (b ) i = ). Therefore, the components of the gradient of the minimizer must be or the corresponding components of the minimizer must lie on the boundary, i.e., ( λ3 ) i = and (λ ) i =, or (λ 3 ) i = and ( λ ) i =. These conditions define the values of the solutions λ (j) 3 and λ (j) : λ (j) 3 = [ r (j) ] + = [ W (s k + γ (j) ) + b ] + λ (j) = [ r (j) ] + = [ W (s k + γ (j) ) b ] +. where the operator [ ] + = max{, } component-wise. Convergence. We prove the convergence of this alternating minimization strategy from techniques found in []. et f(γ, λ 3, λ ) = γ T W T (λ 3 λ ) g (γ) = γ + γ T s k g (λ 3, λ ) = λ 3 λ + (λ 3 λ ) T W s k λ T 3 b + λ T b so that h(γ, λ 3, λ ) = f(γ, λ 3, λ ) + g (γ) + g (λ 3, λ ). Note the following: (A) Both functions g and g are continuous functions whose domains are closed. Consequently, they are closed (see Sec. A.3.3 in [9]). (B) f is bilinear in γ and in (λ 3, λ ). Therefore, it is a continuously differentiable convex function. (C) The gradient of f with respect to γ is constant, and therefore γ f is ipschitz continuous. (D) The gradient of f with respect to (λ 3, λ ) is constant, and therefore λ3,λ f is ipschitz continuous. (E) Since the primal problem () has a continuous objective function and has a closed and bounded domain, it must have a minimum by the Extreme Value Theorem. Because the duality gap is zero, i.e., φ k (θ k+ ) = h(γ, λ 3, λ ), the dual problem () must have a solution. In addition, the subproblems (5) and (7) have explicit minimizers. With these, the assumptions needed to apply emma 3. in [] are satisfied. In particular, we obtain the following convergence result: Theorem : et {(γ (j), λ (j) 3, λ(j) )} j be the sequence generated by the proposed alternating minimization method. Any accumulation point of {(γ (j), λ (j) of problem (). 3, λ(j) )} is a stationary point Feasibility. We now show that at the end of each iteration j, the approximate solution θ (j) = s k + γ (j) + W T (λ (j) 3 λ(j) ) to () is feasible with respect to the constraint b W θ b. First, note that W θ (j) = W s k + W γ (j) + λ (j) 3 λ (j) [ ] = W (s k + γ (j) ) + b W (s k + γ (j) ) + ] [W (s k + γ (j) ) b. (9) We note that (9) is equivalent to W θ (j) = mid{b, W(s k + γ (j) ), b }. Thus, we can terminate the iterations for the dual problem early and still obtain a feasible point. 3. NMERICA RESTS We investigate the effectiveness of the proposed bounded SPIRA-l (B-SPIRA-l ) method by solving two image deblurring problems. In both experiments, the blurry observations are obtained from Af, where the signal f is convolved with a 5 5 blur matrix, whose action is represented by the matrix A. The MATAB s poissrnd function is used to add Poisson noise. Here, we used the Daubechies- (DB-) wavelet basis for W. We implemented the B-SPIRA-l algorithm by including constraints to the existing SPIRA approach [] to solve subproblem (). The algorithm is initialized using the lower and upper bound information incorporated A T y and terminates if the relative difference between consecutive iterates converged to f k+ f k / f k. Similar to the SPIRA approach, we define 3 as the minimum number of iterations to avoid any issues with premature termination. Finally, we compare the results with nonnegatively constrained SPIRA-l method based on RMSE (%) = ˆf f / f. The final SPIRA-l reconstructions are thresholded using the same bounds used in B-SPIRA-l. The regularization parameters (τ) for both experiments are optimized to get the minimum RMSE value. 3.. QR code deblurring In this experiment, we wish to recover a Quick Response (QR) code of size 5 5 (see Fig. (a)) from the Poissonnoise corrupted blurry image (see Fig. (b) and the red zoomed region in Fig. (c)). We set b as the peak intensity of f, i.e., b = 3e+, and b as the zero intensity. The SPIRA-l method took.5 sec (3 iterations) to converge, and its reconstruction ( ˆf S ) has RMSE =.9%. In contrast, the proposed B-SPIRA-l method took 5. +

4 (b) Observation y (c) Zoomed region Fig.. Experimental setup: (a) True QR code image f, (b) noisy and blurry observation y with mean photon count 5.7, (c) a zoomed region of y (a) True image f (b) Observation y (c) Mask Fig. 3. Experimental setup: (a) True phantom image f, (b) noisy and blurry observation y with mean photon count 5., (c) mask with prior structural information. (b) log( + f fˆb ) (RMSE =. %) Fig.. (a) og magnitude of error between the true image, f, and the SPIRA-` reconstruction fˆs, (b) og magnitude of error between f and the proposed B-SPIRA-` reconstruction fˆb. RMSE (%) = kfˆ f k /kf k. Note the lower RMSE for the proposed method s reconstruction, fˆb, whose log error is closer to zero (represented in blue) than the original method. sec (3 iterations) to converge, but its reconstruction (fˆb ) has RMSE =.%. The B-SPIRA-` improvements can be best seen in the magnitude of the log error between the true signal f and the reconstructions (see Figs. (a) and (b)). Note that the fˆb reconstruction more closely matches the original signal f than the fˆs reconstruction by the prevalence of blue regions in Fig. (b). 3.. Shepp-ogan phantom image deblurring In the reconstruction of optical images, anatomical information (the tissue shape and/or structure) from x-ray computed tomography (CT) or magnetic resonance imaging (MRI) can be used to improve the spatial resolution [, 3]. In this experiment, we wish to apply a similar approach to recover the Shepp-ogan phantom image of size from the observed image (see Fig. 3(a) and (b) respectively), when the tissue outer boundary is known. More specifically, we incorporate that outer boundary as a structural information (see Fig. 3(c)), where the lower and upper bound intensities are known based on the region (i.e., and e+ are outside and inside tissue maximum intensities respectively). (a) log( + f fˆs ) (RMSE =.9%) (a) True image f For this problem, the SPIRA-` method took.9 sec (39 iterations) to converge, and its reconstruction (fˆs ) has RMSE =.57%. The proposed B-SPIRA-` method took.9 sec (3 iterations) to converge, and its reconstruction (fˆb ) has a lower RMSE =.5% (see Figs. (a) and (b)). Note the more accurate reconstruction along the top edges as well as the overall improved accuracy within the body (represented in yellow) in comparison to the fˆs reconstruction. (a) log( + f fˆs ) (RMSE =.57%) (b) log( + f fˆb ) (RMSE =.5%) Fig.. (a) og magnitude of error between the true image, f, and fˆs, and (b) og magnitude of error between f and fˆb. RMSE (%) = kfˆ f k /kf k. Note the proposed method s log error is lower on the whole (represented in yellow) in contrast to the mostly orange in (a).. CONCSION In this paper, we formulated a sparsity-promoting boundconstrained photon-limited image recovery method by solving the dual problem based on an alternating minimization strategy. The utilization of any available prior image information has proven very successful for accurately recovering images. We demonstrate that the proposed B-SPIRA-` method leads to more accurate reconstructions than the simply thresholded solutions from the nonnegatively constrained minimization method. Acknowledgments. We thank Prof. Rebecca Willett for fruitful discussions related to this work.

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