images are not smooth. But can be compressed e ciently. Shannon interpolation: if Supp( ˆ ˜ f) [ N , N ] 0,1,0,. . . h(t) = sin( t) t Sample and compress simultaneously? Pointwise Sampling and Smoothness ˜ f L2 f RN f[i] = ˜ f(i/N) JPEG-2k
f L2 f RN y RP micro mirrors array resolution CS hardware , ... f Operator K K CS Hardware Model CS is about designing hardware: input signals ˜ f L2(R2). , ,
dim(ker(K)) = N P is huge. Prior information: f is sparse in a basis { m }m . J (f) = Card {m \ | f, m | > } is small. Inversion and Sparsity f Operator K f, m f
Kf0 + w = K (x0 ) + w = K drawn from the Gaussian matrix ensemble Ki,j N(0, P 1/2) i.i.d. drawn from the Gaussian matrix ensemble Sparse CS Recovery x0 RN f0 RN
Kf0 + w = K (x0 ) + w = K drawn from the Gaussian matrix ensemble Ki,j N(0, P 1/2) i.i.d. drawn from the Gaussian matrix ensemble Sparse recovery: min || x y|| ||w|| ||x||1 min x 1 2 || x y||2 + ||x||1 ||w|| Sparse CS Recovery x0 RN f0 RN
k )||x||2 Restricted Isometry Constants: 1 recovery: CS with RIP [Candes 2009] x⇥ argmin || x y|| ||x||1 where y = x0 + w ||w|| Theorem: If 2k 2 1, then where xk is the best k-term approximation of x0 . ||x0 x || C0 ⇥ k ||x0 xk ||1 + C1
Stronger result: 2 = µ( ) RIP for Gaussian Matrices k C log(N/P)P Theorem: If then 2k 2 1 with high probability. µ( ) = max i=j | i, j ⇥| µ( ) log(PN)/P
⇥2 (A))|| ||2 Stability constant of A: Upper/lower RIC: i k = max |I|=k i ( I ) k = min( 1 k , 2 k ) k ˆ2 k ˆ2 k Monte-Carlo estimation: ˆ k k smallest / largest eigenvalues of A A N = 4000, P = 1000 Numerics with RIP
Random partial orthogonal matrix: { } orthogonal basis. Fast measurements: (e.g. Fourier basis) Mutual incoherence: µ = ⌅ Nmax ,m |⇥⇥ , m ⇤| [1, ⌅ N] Kf = (h'!, fi)!2⌦ where | ⌦ | = P uniformly random. Structured Measurements Theorem: with high probability on , If M CP µ2 log(N)4 , then 2M 2 1 [Rudelson, Vershynin, 2006] = K
Compressed sensing ideas: The devil is in the constants: Worse case analysis is problematic. Designing good signal models. CS is about designing new hardware. dictionary Conclusion Sparsity: approximate signals with few atoms.
on a generic database of natural images, with two different sizes of patches. Note the large number of color-less atoms. have negative values, the vectors are presented scaled and shifted to the [0,255] range per channel: (a) 5 5 3 patches; (b) 8 8 3 patches. color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( in the new metric). duced with our proposed technique ( in our proposed new metric). Both images have been denoised with the same global dictionary. a bias effect in the color from the castle and in some part of the water. What is more, the color of the sky is piecewise constant when ch is another artifact our approach corrected. (a) Original. (b) Original algorithm, dB. (c) Proposed algorithm, dB. MAIRAL et al.: SPARSE REPRESENTATION FOR COLOR IMAGE RESTORATION 57 Fig. 2. Dictionaries with 256 atoms learned on a generic database of natural images, with two different sizes of patches. Note the large number of color-less atoms. Since the atoms can have negative values, the vectors are presented scaled and shifted to the [0,255] range per channel: (a) 5 5 3 patches; (b) 8 8 3 patches. Fig. 3. Examples of color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( in the new metric). Color artifacts are reduced with our proposed technique ( in our proposed new metric). Both images have been denoised with the same global dictionary. In (b), one observes a bias effect in the color from the castle and in some part of the water. What is more, the color of the sky is piecewise constant when (false contours), which is another artifact our approach corrected. (a) Original. (b) Original algorithm, dB. (c) Proposed algorithm, dB. color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( in the new metric). uced with our proposed technique ( in our proposed new metric). Both images have been denoised with the same global dictionary. bias effect in the color from the castle and in some part of the water. What is more, the color of the sky is piecewise constant when ch is another artifact our approach corrected. (a) Original. (b) Original algorithm, dB. (c) Proposed algorithm, dB. ing Image; (b) resulting dictionary; (b) is the dictionary learned in the image in (a). The dictionary is more colored than the global one. EPRESENTATION FOR COLOR IMAGE RESTORATION Fig. 7. Data set used for evaluating denoising experiments. TABLE I DENOISING ALGORITHM WITH 256 ATOMS OF SIZE 7 7 3 FOR AND 6 6 3 FOR . EACH CASE IS DIVID RESULTS ARE THOSE GIVEN BY MCAULEY AND AL [28] WITH THEIR “3 3 MODEL.” THE TOP-RIGHT RESULTS ARE THOSE O CALE K-SVD ALGORITHM [2] ON EACH CHANNEL SEPARATELY WITH 8 8 ATOMS. THE BOTTOM-LEFT ARE OUR RESULTS RAINED DICTIONARY. THE BOTTOM-RIGHT ARE THE IMPROVEMENTS OBTAINED WITH THE ADAPTIVE APPROACH WITH 20 IT OR IMAGE RESTORATION 61 g. 7. Data set used for evaluating denoising experiments. TABLE I TH 256 ATOMS OF SIZE 7 7 3 FOR AND 6 6 3 FOR . EACH CASE IS DIVIDED IN FOUR Y MCAULEY AND AL [28] WITH THEIR “3 3 MODEL.” THE TOP-RIGHT RESULTS ARE THOSE OBTAINED BY 2] ON EACH CHANNEL SEPARATELY WITH 8 8 ATOMS. THE BOTTOM-LEFT ARE OUR RESULTS OBTAINED OTTOM-RIGHT ARE THE IMPROVEMENTS OBTAINED WITH THE ADAPTIVE APPROACH WITH 20 ITERATIONS. H GROUP. AS CAN BE SEEN, OUR PROPOSED TECHNIQUE CONSISTENTLY PRODUCES THE BEST RESULTS
f= x ||x||1 Some Hot Topics with 256 atoms learned on a generic database of natural images, with two different sizes of patches. Note the large number of color-less atoms. have negative values, the vectors are presented scaled and shifted to the [0,255] range per channel: (a) 5 5 3 patches; (b) 8 8 3 patches. color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( in the new metric). duced with our proposed technique ( in our proposed new metric). Both images have been denoised with the same global dictionary. a bias effect in the color from the castle and in some part of the water. What is more, the color of the sky is piecewise constant when ch is another artifact our approach corrected. (a) Original. (b) Original algorithm, dB. (c) Proposed algorithm, dB. MAIRAL et al.: SPARSE REPRESENTATION FOR COLOR IMAGE RESTORATION 57 Fig. 2. Dictionaries with 256 atoms learned on a generic database of natural images, with two different sizes of patches. Note the large number of color-less atoms. Since the atoms can have negative values, the vectors are presented scaled and shifted to the [0,255] range per channel: (a) 5 5 3 patches; (b) 8 8 3 patches. Fig. 3. Examples of color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( in the new metric). Color artifacts are reduced with our proposed technique ( in our proposed new metric). Both images have been denoised with the same global dictionary. In (b), one observes a bias effect in the color from the castle and in some part of the water. What is more, the color of the sky is piecewise constant when (false contours), which is another artifact our approach corrected. (a) Original. (b) Original algorithm, dB. (c) Proposed algorithm, dB. color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( in the new metric). uced with our proposed technique ( in our proposed new metric). Both images have been denoised with the same global dictionary. bias effect in the color from the castle and in some part of the water. What is more, the color of the sky is piecewise constant when ch is another artifact our approach corrected. (a) Original. (b) Original algorithm, dB. (c) Proposed algorithm, dB. ing Image; (b) resulting dictionary; (b) is the dictionary learned in the image in (a). The dictionary is more colored than the global one. EPRESENTATION FOR COLOR IMAGE RESTORATION Fig. 7. Data set used for evaluating denoising experiments. TABLE I DENOISING ALGORITHM WITH 256 ATOMS OF SIZE 7 7 3 FOR AND 6 6 3 FOR . EACH CASE IS DIVID RESULTS ARE THOSE GIVEN BY MCAULEY AND AL [28] WITH THEIR “3 3 MODEL.” THE TOP-RIGHT RESULTS ARE THOSE O CALE K-SVD ALGORITHM [2] ON EACH CHANNEL SEPARATELY WITH 8 8 ATOMS. THE BOTTOM-LEFT ARE OUR RESULTS RAINED DICTIONARY. THE BOTTOM-RIGHT ARE THE IMPROVEMENTS OBTAINED WITH THE ADAPTIVE APPROACH WITH 20 IT OR IMAGE RESTORATION 61 g. 7. Data set used for evaluating denoising experiments. TABLE I TH 256 ATOMS OF SIZE 7 7 3 FOR AND 6 6 3 FOR . EACH CASE IS DIVIDED IN FOUR Y MCAULEY AND AL [28] WITH THEIR “3 3 MODEL.” THE TOP-RIGHT RESULTS ARE THOSE OBTAINED BY 2] ON EACH CHANNEL SEPARATELY WITH 8 8 ATOMS. THE BOTTOM-LEFT ARE OUR RESULTS OBTAINED OTTOM-RIGHT ARE THE IMPROVEMENTS OBTAINED WITH THE ADAPTIVE APPROACH WITH 20 ITERATIONS. H GROUP. AS CAN BE SEEN, OUR PROPOSED TECHNIQUE CONSISTENTLY PRODUCES THE BEST RESULTS Image f = x Coe cients x
f||1 Js (f) = min f= x ||x||1 Some Hot Topics with 256 atoms learned on a generic database of natural images, with two different sizes of patches. Note the large number of color-less atoms. have negative values, the vectors are presented scaled and shifted to the [0,255] range per channel: (a) 5 5 3 patches; (b) 8 8 3 patches. color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( in the new metric). duced with our proposed technique ( in our proposed new metric). Both images have been denoised with the same global dictionary. a bias effect in the color from the castle and in some part of the water. What is more, the color of the sky is piecewise constant when ch is another artifact our approach corrected. (a) Original. (b) Original algorithm, dB. (c) Proposed algorithm, dB. MAIRAL et al.: SPARSE REPRESENTATION FOR COLOR IMAGE RESTORATION 57 Fig. 2. Dictionaries with 256 atoms learned on a generic database of natural images, with two different sizes of patches. Note the large number of color-less atoms. Since the atoms can have negative values, the vectors are presented scaled and shifted to the [0,255] range per channel: (a) 5 5 3 patches; (b) 8 8 3 patches. Fig. 3. Examples of color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( in the new metric). Color artifacts are reduced with our proposed technique ( in our proposed new metric). Both images have been denoised with the same global dictionary. In (b), one observes a bias effect in the color from the castle and in some part of the water. What is more, the color of the sky is piecewise constant when (false contours), which is another artifact our approach corrected. (a) Original. (b) Original algorithm, dB. (c) Proposed algorithm, dB. color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( in the new metric). uced with our proposed technique ( in our proposed new metric). Both images have been denoised with the same global dictionary. bias effect in the color from the castle and in some part of the water. What is more, the color of the sky is piecewise constant when ch is another artifact our approach corrected. (a) Original. (b) Original algorithm, dB. (c) Proposed algorithm, dB. ing Image; (b) resulting dictionary; (b) is the dictionary learned in the image in (a). The dictionary is more colored than the global one. EPRESENTATION FOR COLOR IMAGE RESTORATION Fig. 7. Data set used for evaluating denoising experiments. TABLE I DENOISING ALGORITHM WITH 256 ATOMS OF SIZE 7 7 3 FOR AND 6 6 3 FOR . EACH CASE IS DIVID RESULTS ARE THOSE GIVEN BY MCAULEY AND AL [28] WITH THEIR “3 3 MODEL.” THE TOP-RIGHT RESULTS ARE THOSE O CALE K-SVD ALGORITHM [2] ON EACH CHANNEL SEPARATELY WITH 8 8 ATOMS. THE BOTTOM-LEFT ARE OUR RESULTS RAINED DICTIONARY. THE BOTTOM-RIGHT ARE THE IMPROVEMENTS OBTAINED WITH THE ADAPTIVE APPROACH WITH 20 IT OR IMAGE RESTORATION 61 g. 7. Data set used for evaluating denoising experiments. TABLE I TH 256 ATOMS OF SIZE 7 7 3 FOR AND 6 6 3 FOR . EACH CASE IS DIVIDED IN FOUR Y MCAULEY AND AL [28] WITH THEIR “3 3 MODEL.” THE TOP-RIGHT RESULTS ARE THOSE OBTAINED BY 2] ON EACH CHANNEL SEPARATELY WITH 8 8 ATOMS. THE BOTTOM-LEFT ARE OUR RESULTS OBTAINED OTTOM-RIGHT ARE THE IMPROVEMENTS OBTAINED WITH THE ADAPTIVE APPROACH WITH 20 ITERATIONS. H GROUP. AS CAN BE SEEN, OUR PROPOSED TECHNIQUE CONSISTENTLY PRODUCES THE BEST RESULTS Image f = x Coe cients x c = D f D
f||1 Js (f) = min f= x ||x||1 Some Hot Topics with 256 atoms learned on a generic database of natural images, with two different sizes of patches. Note the large number of color-less atoms. have negative values, the vectors are presented scaled and shifted to the [0,255] range per channel: (a) 5 5 3 patches; (b) 8 8 3 patches. color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( in the new metric). duced with our proposed technique ( in our proposed new metric). Both images have been denoised with the same global dictionary. a bias effect in the color from the castle and in some part of the water. What is more, the color of the sky is piecewise constant when ch is another artifact our approach corrected. (a) Original. (b) Original algorithm, dB. (c) Proposed algorithm, dB. MAIRAL et al.: SPARSE REPRESENTATION FOR COLOR IMAGE RESTORATION 57 Fig. 2. Dictionaries with 256 atoms learned on a generic database of natural images, with two different sizes of patches. Note the large number of color-less atoms. Since the atoms can have negative values, the vectors are presented scaled and shifted to the [0,255] range per channel: (a) 5 5 3 patches; (b) 8 8 3 patches. Fig. 3. Examples of color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( in the new metric). Color artifacts are reduced with our proposed technique ( in our proposed new metric). Both images have been denoised with the same global dictionary. In (b), one observes a bias effect in the color from the castle and in some part of the water. What is more, the color of the sky is piecewise constant when (false contours), which is another artifact our approach corrected. (a) Original. (b) Original algorithm, dB. (c) Proposed algorithm, dB. color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( in the new metric). uced with our proposed technique ( in our proposed new metric). Both images have been denoised with the same global dictionary. bias effect in the color from the castle and in some part of the water. What is more, the color of the sky is piecewise constant when ch is another artifact our approach corrected. (a) Original. (b) Original algorithm, dB. (c) Proposed algorithm, dB. ing Image; (b) resulting dictionary; (b) is the dictionary learned in the image in (a). The dictionary is more colored than the global one. EPRESENTATION FOR COLOR IMAGE RESTORATION Fig. 7. Data set used for evaluating denoising experiments. TABLE I DENOISING ALGORITHM WITH 256 ATOMS OF SIZE 7 7 3 FOR AND 6 6 3 FOR . EACH CASE IS DIVID RESULTS ARE THOSE GIVEN BY MCAULEY AND AL [28] WITH THEIR “3 3 MODEL.” THE TOP-RIGHT RESULTS ARE THOSE O CALE K-SVD ALGORITHM [2] ON EACH CHANNEL SEPARATELY WITH 8 8 ATOMS. THE BOTTOM-LEFT ARE OUR RESULTS RAINED DICTIONARY. THE BOTTOM-RIGHT ARE THE IMPROVEMENTS OBTAINED WITH THE ADAPTIVE APPROACH WITH 20 IT OR IMAGE RESTORATION 61 g. 7. Data set used for evaluating denoising experiments. TABLE I TH 256 ATOMS OF SIZE 7 7 3 FOR AND 6 6 3 FOR . EACH CASE IS DIVIDED IN FOUR Y MCAULEY AND AL [28] WITH THEIR “3 3 MODEL.” THE TOP-RIGHT RESULTS ARE THOSE OBTAINED BY 2] ON EACH CHANNEL SEPARATELY WITH 8 8 ATOMS. THE BOTTOM-LEFT ARE OUR RESULTS OBTAINED OTTOM-RIGHT ARE THE IMPROVEMENTS OBTAINED WITH THE ADAPTIVE APPROACH WITH 20 ITERATIONS. H GROUP. AS CAN BE SEEN, OUR PROPOSED TECHNIQUE CONSISTENTLY PRODUCES THE BEST RESULTS Other sparse priors: Image f = x Coe cients x c = D f D |x1 | + |x2 | max(|x1 |, |x2 |)
f||1 Js (f) = min f= x ||x||1 |x1 | + (x2 2 + x2 3 )1 2 Some Hot Topics with 256 atoms learned on a generic database of natural images, with two different sizes of patches. Note the large number of color-less atoms. have negative values, the vectors are presented scaled and shifted to the [0,255] range per channel: (a) 5 5 3 patches; (b) 8 8 3 patches. color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( in the new metric). duced with our proposed technique ( in our proposed new metric). Both images have been denoised with the same global dictionary. a bias effect in the color from the castle and in some part of the water. What is more, the color of the sky is piecewise constant when ch is another artifact our approach corrected. (a) Original. (b) Original algorithm, dB. (c) Proposed algorithm, dB. MAIRAL et al.: SPARSE REPRESENTATION FOR COLOR IMAGE RESTORATION 57 Fig. 2. Dictionaries with 256 atoms learned on a generic database of natural images, with two different sizes of patches. Note the large number of color-less atoms. Since the atoms can have negative values, the vectors are presented scaled and shifted to the [0,255] range per channel: (a) 5 5 3 patches; (b) 8 8 3 patches. Fig. 3. Examples of color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( in the new metric). Color artifacts are reduced with our proposed technique ( in our proposed new metric). Both images have been denoised with the same global dictionary. In (b), one observes a bias effect in the color from the castle and in some part of the water. What is more, the color of the sky is piecewise constant when (false contours), which is another artifact our approach corrected. (a) Original. (b) Original algorithm, dB. (c) Proposed algorithm, dB. color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( in the new metric). uced with our proposed technique ( in our proposed new metric). Both images have been denoised with the same global dictionary. bias effect in the color from the castle and in some part of the water. What is more, the color of the sky is piecewise constant when ch is another artifact our approach corrected. (a) Original. (b) Original algorithm, dB. (c) Proposed algorithm, dB. ing Image; (b) resulting dictionary; (b) is the dictionary learned in the image in (a). The dictionary is more colored than the global one. EPRESENTATION FOR COLOR IMAGE RESTORATION Fig. 7. Data set used for evaluating denoising experiments. TABLE I DENOISING ALGORITHM WITH 256 ATOMS OF SIZE 7 7 3 FOR AND 6 6 3 FOR . EACH CASE IS DIVID RESULTS ARE THOSE GIVEN BY MCAULEY AND AL [28] WITH THEIR “3 3 MODEL.” THE TOP-RIGHT RESULTS ARE THOSE O CALE K-SVD ALGORITHM [2] ON EACH CHANNEL SEPARATELY WITH 8 8 ATOMS. THE BOTTOM-LEFT ARE OUR RESULTS RAINED DICTIONARY. THE BOTTOM-RIGHT ARE THE IMPROVEMENTS OBTAINED WITH THE ADAPTIVE APPROACH WITH 20 IT OR IMAGE RESTORATION 61 g. 7. Data set used for evaluating denoising experiments. TABLE I TH 256 ATOMS OF SIZE 7 7 3 FOR AND 6 6 3 FOR . EACH CASE IS DIVIDED IN FOUR Y MCAULEY AND AL [28] WITH THEIR “3 3 MODEL.” THE TOP-RIGHT RESULTS ARE THOSE OBTAINED BY 2] ON EACH CHANNEL SEPARATELY WITH 8 8 ATOMS. THE BOTTOM-LEFT ARE OUR RESULTS OBTAINED OTTOM-RIGHT ARE THE IMPROVEMENTS OBTAINED WITH THE ADAPTIVE APPROACH WITH 20 ITERATIONS. H GROUP. AS CAN BE SEEN, OUR PROPOSED TECHNIQUE CONSISTENTLY PRODUCES THE BEST RESULTS Other sparse priors: Image f = x Coe cients x c = D f D |x1 | + |x2 | max(|x1 |, |x2 |)
f||1 Js (f) = min f= x ||x||1 |x1 | + (x2 2 + x2 3 )1 2 Some Hot Topics with 256 atoms learned on a generic database of natural images, with two different sizes of patches. Note the large number of color-less atoms. have negative values, the vectors are presented scaled and shifted to the [0,255] range per channel: (a) 5 5 3 patches; (b) 8 8 3 patches. color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( in the new metric). duced with our proposed technique ( in our proposed new metric). Both images have been denoised with the same global dictionary. a bias effect in the color from the castle and in some part of the water. What is more, the color of the sky is piecewise constant when ch is another artifact our approach corrected. (a) Original. (b) Original algorithm, dB. (c) Proposed algorithm, dB. MAIRAL et al.: SPARSE REPRESENTATION FOR COLOR IMAGE RESTORATION 57 Fig. 2. Dictionaries with 256 atoms learned on a generic database of natural images, with two different sizes of patches. Note the large number of color-less atoms. Since the atoms can have negative values, the vectors are presented scaled and shifted to the [0,255] range per channel: (a) 5 5 3 patches; (b) 8 8 3 patches. Fig. 3. Examples of color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( in the new metric). Color artifacts are reduced with our proposed technique ( in our proposed new metric). Both images have been denoised with the same global dictionary. In (b), one observes a bias effect in the color from the castle and in some part of the water. What is more, the color of the sky is piecewise constant when (false contours), which is another artifact our approach corrected. (a) Original. (b) Original algorithm, dB. (c) Proposed algorithm, dB. color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( in the new metric). uced with our proposed technique ( in our proposed new metric). Both images have been denoised with the same global dictionary. bias effect in the color from the castle and in some part of the water. What is more, the color of the sky is piecewise constant when ch is another artifact our approach corrected. (a) Original. (b) Original algorithm, dB. (c) Proposed algorithm, dB. ing Image; (b) resulting dictionary; (b) is the dictionary learned in the image in (a). The dictionary is more colored than the global one. EPRESENTATION FOR COLOR IMAGE RESTORATION Fig. 7. Data set used for evaluating denoising experiments. TABLE I DENOISING ALGORITHM WITH 256 ATOMS OF SIZE 7 7 3 FOR AND 6 6 3 FOR . EACH CASE IS DIVID RESULTS ARE THOSE GIVEN BY MCAULEY AND AL [28] WITH THEIR “3 3 MODEL.” THE TOP-RIGHT RESULTS ARE THOSE O CALE K-SVD ALGORITHM [2] ON EACH CHANNEL SEPARATELY WITH 8 8 ATOMS. THE BOTTOM-LEFT ARE OUR RESULTS RAINED DICTIONARY. THE BOTTOM-RIGHT ARE THE IMPROVEMENTS OBTAINED WITH THE ADAPTIVE APPROACH WITH 20 IT OR IMAGE RESTORATION 61 g. 7. Data set used for evaluating denoising experiments. TABLE I TH 256 ATOMS OF SIZE 7 7 3 FOR AND 6 6 3 FOR . EACH CASE IS DIVIDED IN FOUR Y MCAULEY AND AL [28] WITH THEIR “3 3 MODEL.” THE TOP-RIGHT RESULTS ARE THOSE OBTAINED BY 2] ON EACH CHANNEL SEPARATELY WITH 8 8 ATOMS. THE BOTTOM-LEFT ARE OUR RESULTS OBTAINED OTTOM-RIGHT ARE THE IMPROVEMENTS OBTAINED WITH THE ADAPTIVE APPROACH WITH 20 ITERATIONS. H GROUP. AS CAN BE SEEN, OUR PROPOSED TECHNIQUE CONSISTENTLY PRODUCES THE BEST RESULTS Other sparse priors: Image f = x Coe cients x c = D f D |x1 | + |x2 | max(|x1 |, |x2 |) Nuclear
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