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Signal Processing Course : Compressed Sensing

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  • 1. Compressive Sensing Gabriel Peyré www.numerical-tours.com
  • 2. Overview•Shannon’s World•Compressive Sensing Acquisition•Compressive Sensing Recovery•Theoretical Guarantees•Fourier Domain Measurements
  • 3. Discretization Sampling: ˜ acquisition N f L ([0, 1] ) 2 d f R device Idealization: ˜ f [n] ⇡ f (n/N )
  • 4. Pointwise Sampling and SmoothnessData aquisition: ˜ f [i] = f (i/N ) Sensors ˜ f L2 f RN
  • 5. Pointwise Sampling and SmoothnessData aquisition: ˜ f [i] = f (i/N ) Sensors ˜ f L2 f RN ˆ ˜Shannon interpolation: if Supp(f ) [ N ,N ]˜ sin( t)f (t) = f [i]h(N t i) h(t) = i t
  • 6. Pointwise Sampling and SmoothnessData aquisition: ˜ f [i] = f (i/N ) Sensors ˜ f L2 f RN ˆ ˜Shannon interpolation: if Supp(f ) [ N ,N ]˜ sin( t)f (t) = f [i]h(N t i) h(t) = i t Natural images are not smooth.
  • 7. Pointwise Sampling and SmoothnessData aquisition: ˜ f [i] = f (i/N ) Sensors JPEG-2k 0,1,0,. . . ˜ f L2 f RN ˆ ˜Shannon interpolation: if Supp(f ) [ N ,N ]˜ sin( t)f (t) = f [i]h(N t i) h(t) = i t Natural images are not smooth. But can be compressed e ciently. Sample and compress simultaneously?
  • 8. Sampling and Periodization(a)(b) 1(c) 0(d)
  • 9. Sampling and Periodization: Aliasing (a) (b) 1 (c) 0 (d)
  • 10. Overview•Shannon’s World•Compressive Sensing Acquisition•Compressive Sensing Recovery•Theoretical Guarantees•Fourier Domain Measurements
  • 11. Single Pixel Camera (Rice)˜f
  • 12. Single Pixel Camera (Rice)˜fy[i] = f, i P measures N micro-mirrors
  • 13. Single Pixel Camera (Rice)˜fy[i] = f, i P measures N micro-mirrors P/N = 1 P/N = 0.16 P/N = 0.02
  • 14. CS Hardware Model ˜CS is about designing hardware: input signals f L2 (R2 ).Physical hardware resolution limit: target resolution f RN . array micro ˜ f L2 f RN mirrors y RP resolution K CS hardware
  • 15. CS Hardware Model ˜CS is about designing hardware: input signals f L2 (R2 ).Physical hardware resolution limit: target resolution f RN . array micro ˜ f L2 f RN mirrors y RP resolution K CS hardware , , ... ,
  • 16. CS Hardware Model ˜CS is about designing hardware: input signals f L2 (R2 ).Physical hardware resolution limit: target resolution f RN . array micro ˜ f L2 f RN mirrors y RP resolution K CS hardware , Operator K , f ... ,
  • 17. Overview•Shannon’s World•Compressive Sensing Acquisition•Compressive Sensing Recovery•Theoretical Guarantees•Fourier Domain Measurements
  • 18. Inversion and Sparsity Operator KNeed to solve y = Kf . More unknown than equations. fdim(ker(K)) = N P is huge.
  • 19. Inversion and Sparsity Operator KNeed to solve y = Kf . More unknown than equations. fdim(ker(K)) = N P is huge.Prior information: f is sparse in a basis { m }m . J (f ) = Card {m | f, m |> } is small. f f, m
  • 20. Convex Relaxation: L1 Prior J0 (f ) = # {m ⇥f, m⇤ = 0} J0 (f ) = 0 null image.Image with 2 pixels: J0 (f ) = 1 sparse image. J0 (f ) = 2 non-sparse image. q=0
  • 21. Convex Relaxation: L1 Prior J0 (f ) = # {m ⇥f, m⇤ = 0} J0 (f ) = 0 null image.Image with 2 pixels: J0 (f ) = 1 sparse image. J0 (f ) = 2 non-sparse image. q=0 q = 1/2 q=1 q = 3/2 q=2 q priors: Jq (f ) = | f, q m ⇥| (convex for q 1) m
  • 22. Convex Relaxation: L1 Prior J0 (f ) = # {m ⇥f, m⇤ = 0} J0 (f ) = 0 null image.Image with 2 pixels: J0 (f ) = 1 sparse image. J0 (f ) = 2 non-sparse image. q=0 q = 1/2 q=1 q = 3/2 q=2 q priors: Jq (f ) = | f, m ⇥| q (convex for q 1) mSparse 1 prior: J1 (f ) = | f, m ⇥| m
  • 23. Sparse CS Recovery f0 RNf0 RN sparse in ortho-basis N x0 R
  • 24. Sparse CS Recovery f0 RNf0 RN sparse in ortho-basis(Discretized) sampling acquisition: y = Kf0 + w = K (x0 ) + w = N x0 R
  • 25. Sparse CS Recovery f0 RNf0 RN sparse in ortho-basis(Discretized) sampling acquisition: y = 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 N x0 R
  • 26. Sparse CS Recovery f0 RNf0 RN sparse in ortho-basis(Discretized) sampling acquisition: y = 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 N Sparse recovery: x0 R ||w|| 1 min ||x||1 min || x y|| + ||x||1 2 || x y|| ||w|| x 2
  • 27. CS Simulation ExampleOriginal f0 = translation invariant wavelet frame
  • 28. Overview•Shannon’s World•Compressive Sensing Acquisition•Compressive Sensing Recovery•Theoretical Guarantees•Fourier Domain Measurements
  • 29. CS with RIP 1 recovery: y= x0 + w x⇥ argmin ||x||1 where || x y|| ||w||Restricted Isometry Constants: ⇥ ||x||0 k, (1 k )||x||2 || x||2 (1 + k )||x||2
  • 30. CS with RIP 1 recovery: y= x0 + w x⇥ argmin ||x||1 where || x y|| ||w||Restricted Isometry Constants: ⇥ ||x||0 k, (1 k )||x||2 || x||2 (1 + k )||x||2Theorem: If 2k2 1, then [Candes 2009] C0 ||x0 x || ⇥ ||x0 xk ||1 + C1 k where xk is the best k-term approximation of x0 .
  • 31. Singular Values DistributionsEigenvalues of I I with |I| = k are essentially in [a, b] a = (1 ) 2 and b = (1 )2 where = k/PWhen k = P + , the eigenvalue distribution tends to 1 f (⇥) = (⇥ b)+ (a ⇥)+ [Marcenko-Pastur] 1.5 2⇤ ⇥ P=200, k=10 P=200, k=10 f ( ) 1.5 1 1 0.5 P = 200, k = 10 0.5 0 0 0.5 1 1.5 2 2.5 0 0 0.5 1 P=200, k=30 1.5 2 2.5 1 P=200, k=30 0.8 1 0.6 0.8 0.4 k = 30 0.6 0.2 0.4 0 0.2 0 0.5 1 1.5 2 2.5 0 0 0.5 1 P=200, k=50 1.5 2 2.5 0.8 P=200, k=50 0.6 0.8 0.6 0.4 Large deviation inequality [Ledoux] 0.2 0.4
  • 32. RIP for Gaussian MatricesLink with coherence: µ( ) = max | i, j ⇥| i=j 2 = µ( ) k (k 1)µ( )
  • 33. RIP for Gaussian MatricesLink with coherence: µ( ) = max | i, j ⇥| i=j 2 = µ( ) k (k 1)µ( )For Gaussian matrices: µ( ) log(P N )/P
  • 34. RIP for Gaussian MatricesLink with coherence: µ( ) = max | i, j ⇥| i=j 2 = µ( ) k (k 1)µ( )For Gaussian matrices: µ( ) log(P N )/PStronger result: CTheorem: If k P log(N/P ) then 2k 2 1 with high probability.
  • 35. Numerics with RIPStability constant of A: (1 ⇥1 (A))|| ||2 ||A ||2 (1 + ⇥2 (A))|| ||2 smallest / largest eigenvalues of A A
  • 36. Numerics with RIPStability constant of A: (1 ⇥1 (A))|| ||2 ||A ||2 (1 + ⇥2 (A))|| ||2 smallest / largest eigenvalues of A AUpper/lower RIC: ˆ2 k i k = max i( I) |I|=k 2 1 ˆ2 k k = min( k, k) 1 2Monte-Carlo estimation: ˆk k k N = 4000, P = 1000
  • 37. Polytopes-based GuaranteesNoiseless recovery: x argmin ||x||1 (P0 (y)) x=y = ( i )i R 2 3 3 2 1 x0 x0 1 y x 3B = {x ||x||1 } 2 (B ) = ||x0 ||1
  • 38. Polytopes-based GuaranteesNoiseless recovery: x argmin ||x||1 (P0 (y)) x=y = ( i )i R 2 3 3 2 1 x0 x0 1 y x 3B = {x ||x||1 } 2 (B ) = ||x0 ||1 x0 solution of P0 ( x0 ) ⇥ x0 ⇤ (B )
  • 39. L1 Recovery in 2-D = ( i )i R 2 3 C(0,1,1) 2 3 K(0,1,1) 1 y x 2-D quadrant 2-D conesKs = ( i si )i R 3 i 0 Cs = Ks
  • 40. Polytope Noiseless RecoveryCounting faces of random polytopes: [Donoho] All x0 such that ||x0 ||0 Call (P/N )P are identifiable. Most x0 such that ||x0 ||0 Cmost (P/N )P are identifiable. Call (1/4) 0.065 1 0.9 Cmost (1/4) 0.25 0.8 0.7 0.6 Sharp constants. 0.5 0.4 No noise robustness. 0.3 0.2 0.1 0 50 100 150 200 250 300 350 400 RIP All Most
  • 41. Polytope Noiseless RecoveryCounting faces of random polytopes: [Donoho] All x0 such that ||x0 ||0 Call (P/N )P are identifiable. Most x0 such that ||x0 ||0 Cmost (P/N )P are identifiable. Call (1/4) 0.065 1 0.9 Cmost (1/4) 0.25 0.8 0.7 0.6 Sharp constants. 0.5 0.4 No noise robustness. 0.3 0.2 Computation of 0.1 “pathological” signals 0 50 100 150 200 250 300 350 400[Dossal, P, Fadili, 2010] RIP All Most
  • 42. Overview•Shannon’s World•Compressive Sensing Acquisition•Compressive Sensing Recovery•Theoretical Guarantees•Fourier Domain Measurements
  • 43. Tomography and Fourier Measures
  • 44. Tomography and Fourier Measures ˆ f = FFT2(f ) kFourier slice theorem: ˆ ˆ p (⇥) = f (⇥ cos( ), ⇥ sin( )) 1D 2D Fourier t RPartial Fourier measurements: {p k (t)}0 k<K Equivalent to: ˆ Kf = (f [!])!2⌦
  • 45. Regularized InversionNoisy measurements: ⇥ ˆ , y[ ] = f0 [ ] + w[ ]. Noise: w[⇥] N (0, ), white noise.1 regularization: 1 ˆ[⇤]|2 + f = argmin ⇥ |y[⇤] f |⇥f, ⇥m ⇤|. f 2 m
  • 46. MRI Imaging From [Lutsig et al.]
  • 47. MRI Reconstruction From [Lutsig et al.] randomizationFourier sub-sampling pattern:High resolution Low resolution Linear Sparsity
  • 48. Radar InterferometryCARMA (USA) Fourier sampling Linear (Earth’s rotation) reconstruction
  • 49. Structured MeasurementsGaussian matrices: intractable for large N .Random partial orthogonal matrix: { } orthogonal basis. Kf = (h! , f i)!2⌦ where |⌦| = P uniformly random.Fast measurements: (e.g. Fourier basis)
  • 50. Structured MeasurementsGaussian matrices: intractable for large N .Random partial orthogonal matrix: { } orthogonal basis. Kf = (h! , f i)!2⌦ where |⌦| = P uniformly random.Fast measurements: (e.g. Fourier basis) ⌅ ⌅Mutual incoherence: µ= N max |⇥⇥ , m ⇤| [1, N] ,m
  • 51. Structured MeasurementsGaussian matrices: intractable for large N .Random partial orthogonal matrix: { } orthogonal basis. Kf = (h! , f i)!2⌦ where |⌦| = P uniformly random.Fast measurements: (e.g. Fourier basis) ⌅ ⌅Mutual incoherence: µ= N max |⇥⇥ , m ⇤| [1, N] ,m Theorem: with high probability on , =K CP If M 2 log(N )4 , then 2M 2 1 µ [Rudelson, Vershynin, 2006] not universal: requires incoherence.
  • 52. ConclusionSparsity: approximate signals with few atoms. dictionary
  • 53. Conclusion Sparsity: approximate signals with few atoms. dictionaryCompressed sensing ideas: Randomized sensors + sparse recovery. Number of measurements signal complexity. CS is about designing new hardware.
  • 54. Conclusion Sparsity: approximate signals with few atoms. dictionaryCompressed sensing ideas: Randomized sensors + sparse recovery. Number of measurements signal complexity. CS is about designing new hardware.The devil is in the constants: Worse case analysis is problematic. Designing good signal models.
  • 55. RESULTS ARE THOSE GIVEN BY MCAULEY AND AL [28] WITH THEIR “3 3 MODEL.” THE TOP-RIGHT RESULTS ARE THOSE O RAINED DICTIONARY. THE BOTTOM-RIGHT ARE THE IMPROVEMENTS OBTAINED WITH THE ADAPTIVE APPROACH WITH 20 IT CALE K-SVD ALGORITHM [2] ON EACH CHANNEL SEPARATELY WITH 8 EPRESENTATION FOR COLOR IMAGE RESTORATION DENOISING ALGORITHM WITH 256 ATOMS OF SIZE 7 7 3 FOR color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( uced with our proposed technique ( in the new metric). in our proposed new metric). Both images have been denoised with the same global dictionary. Some Hot Topics 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 whench is another artifact our approach corrected. (a) Original. (b) Original algorithm, dB. (c) Proposed algorithm, Dictionary learning: dB.with 256 atoms learned on a generic database of natural images, with two differental.: SPARSE REPRESENTATION FOR COLOR IMAGE RESTORATION MAIRAL et sizes of patches. Note the large number of color-less atoms. 57have 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.OR IMAGE RESTORATION 61 Fig. 7. Data set used for evaluating denoising experiments. learning ing Image; (b) resulting dictionary; (b) is the dictionary learned in the image in (a). The dictionary is more colored than the global one. TABLE Ig. 7. Data set used for evaluating denoising experiments. 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. Fig. 2. Dictionaries 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. 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 ( TABLE I our proposed new metric). Both images have been denoised with the same global dictionary. inTH 256 ATOMS OF SIZE castle 7 in3 FOR of the water. What is more, the color of the sky is .piecewise CASE IS DIVIDED IN FOURa bias effect in the color from the 7 and some part AND 6 6 3 FOR EACH constant whench MCAULEY AND approach corrected. (a)HEIR “3(b) Original algorithm, HE TOP-RIGHT RESULTS ARE THOSE OBTAINED BY Y is another artifact our AL [28] WITH T Original. 3 MODEL.” T dB. (c) Proposed algorithm, dB. 8 ATOMS. THE BOTTOM-LEFT ARE OUR RESULTS2] ON EACH CHANNEL SEPARATELY WITH 8 8 ATOMS. THE BOTTOM-LEFT ARE OUR RESULTS OBTAINED AND 6OTTOM-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 6 3 FOR 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. . EACH CASE IS DIVID
  • 56. RESULTS ARE THOSE GIVEN BY MCAULEY AND AL [28] WITH THEIR “3 3 MODEL.” THE TOP-RIGHT RESULTS ARE THOSE O RAINED DICTIONARY. THE BOTTOM-RIGHT ARE THE IMPROVEMENTS OBTAINED WITH THE ADAPTIVE APPROACH WITH 20 IT CALE K-SVD ALGORITHM [2] ON EACH CHANNEL SEPARATELY WITH 8 EPRESENTATION FOR COLOR IMAGE RESTORATION DENOISING ALGORITHM WITH 256 ATOMS OF SIZE 7 7 3 FOR color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( uced with our proposed technique ( in the new metric). in our proposed new metric). Both images have been denoised with the same global dictionary. Some Hot Topics 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 Image f =ch is another artifact our approach corrected. (a) Original. (b) Original algorithm, dB. (c) Proposed algorithm, dB. Dictionary learning:with 256 atoms learned on a generic database of natural images, with two differental.: SPARSE REPRESENTATION FOR COLOR IMAGE RESTORATIONhave negative values, the vectors are presented scaled and shifted to the [0,255] range per channel: (a) 5 MAIRAL et sizes of patches. Note the large number of color-less 5 3 patches; (b) 8 8 atoms. 3 patches. 57 xOR IMAGE RESTORATION 61 Fig. 7. Data set used for evaluating denoising experiments. learning ing Image; (b) resulting dictionary; (b) is the dictionary learned in the image in (a). The dictionary is more colored than the global one. TABLE I Analysis vs. synthesis:g. 7. Data set used for evaluating denoising experiments. 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. Fig. 2. Dictionaries 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. Js (f ) = min ||x||1 color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( TABLE I in the new metric).duced with our proposed technique (a bias effect in the color from the 7 in our proposed new metric). Both images have been denoised with the same global dictionary.TH 256 ATOMS OF SIZE castle 7 in3 FOR of the water. What is more, the color of the sky is .piecewise CASE IS DIVIDED IN FOUR and some part AND 6 6 3 FOR EACH constant when f= xch MCAULEY AND approach corrected. (a)HEIR “3(b) Original algorithm, HE TOP-RIGHT RESULTS ARE THOSE OBTAINED BY Y is another artifact our AL [28] WITH T Original. 3 MODEL.” T dB. (c) Proposed algorithm, dB. 8 ATOMS. THE BOTTOM-LEFT ARE OUR RESULTS2] ON EACH CHANNEL SEPARATELY WITH 8 8 ATOMS. THE BOTTOM-LEFT ARE OUR RESULTS OBTAINED AND 6OTTOM-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 Coe cients x 6 3 FOR 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. . EACH CASE IS DIVID
  • 57. RESULTS ARE THOSE GIVEN BY MCAULEY AND AL [28] WITH THEIR “3 3 MODEL.” THE TOP-RIGHT RESULTS ARE THOSE O RAINED DICTIONARY. THE BOTTOM-RIGHT ARE THE IMPROVEMENTS OBTAINED WITH THE ADAPTIVE APPROACH WITH 20 IT CALE K-SVD ALGORITHM [2] ON EACH CHANNEL SEPARATELY WITH 8 EPRESENTATION FOR COLOR IMAGE RESTORATION DENOISING ALGORITHM WITH 256 ATOMS OF SIZE 7 7 3 FOR color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( uced with our proposed technique ( in the new metric). in our proposed new metric). Both images have been denoised with the same global dictionary. Some Hot Topics 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 Image f =ch is another artifact our approach corrected. (a) Original. (b) Original algorithm, dB. (c) Proposed algorithm, dB. Dictionary learning:with 256 atoms learned on a generic database of natural images, with two differental.: SPARSE REPRESENTATION FOR COLOR IMAGE RESTORATIONhave negative values, the vectors are presented scaled and shifted to the [0,255] range per channel: (a) 5 MAIRAL et sizes of patches. Note the large number of color-less 5 3 patches; (b) 8 8 atoms. 3 patches. 57 xOR IMAGE RESTORATION 61 Fig. 7. Data set used for evaluating denoising experiments. learning D ing Image; (b) resulting dictionary; (b) is the dictionary learned in the image in (a). The dictionary is more colored than the global one. TABLE I Analysis vs. synthesis:g. 7. Data set used for evaluating denoising experiments. 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. Fig. 2. Dictionaries 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. Js (f ) = min ||x||1 color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( TABLE I in the new metric).duced with our proposed technique (a bias effect in the color from the 7 in our proposed new metric). Both images have been denoised with the same global dictionary.TH 256 ATOMS OF SIZE castle 7 in3 FOR of the water. What is more, the color of the sky is .piecewise CASE IS DIVIDED IN FOUR and some part AND 6 6 3 FOR EACH constant when f= x J (f ) = ||D f ||ch MCAULEY AND approach corrected. (a)HEIR “3(b) Original algorithm, HE TOP-RIGHT RESULTS ARE THOSE OBTAINED BY Y is another artifact our AL [28] WITH T Original. 3 MODEL.” T dB. (c) Proposed algorithm, dB. 8 ATOMS. THE BOTTOM-LEFT ARE OUR RESULTS2] ON EACH CHANNEL SEPARATELY WITH 8 8 ATOMS. THE BOTTOM-LEFT ARE OUR RESULTS OBTAINED a 1 AND 6OTTOM-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 Coe cients x c=D f 6 3 FOR 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. . EACH CASE IS DIVID
  • 58. RESULTS ARE THOSE GIVEN BY MCAULEY AND AL [28] WITH THEIR “3 3 MODEL.” THE TOP-RIGHT RESULTS ARE THOSE O RAINED DICTIONARY. THE BOTTOM-RIGHT ARE THE IMPROVEMENTS OBTAINED WITH THE ADAPTIVE APPROACH WITH 20 IT CALE K-SVD ALGORITHM [2] ON EACH CHANNEL SEPARATELY WITH 8 EPRESENTATION FOR COLOR IMAGE RESTORATION DENOISING ALGORITHM WITH 256 ATOMS OF SIZE 7 7 3 FOR color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( uced with our proposed technique ( in the new metric). in our proposed new metric). Both images have been denoised with the same global dictionary. Some Hot Topics 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 Image f =ch is another artifact our approach corrected. (a) Original. (b) Original algorithm, dB. (c) Proposed algorithm, dB. Dictionary learning:with 256 atoms learned on a generic database of natural images, with two differental.: SPARSE REPRESENTATION FOR COLOR IMAGE RESTORATIONhave negative values, the vectors are presented scaled and shifted to the [0,255] range per channel: (a) 5 MAIRAL et sizes of patches. Note the large number of color-less 5 3 patches; (b) 8 8 atoms. 3 patches. 57 xOR IMAGE RESTORATION 61 Fig. 7. Data set used for evaluating denoising experiments. learning D ing Image; (b) resulting dictionary; (b) is the dictionary learned in the image in (a). The dictionary is more colored than the global one. TABLE I Analysis vs. synthesis:g. 7. Data set used for evaluating denoising experiments. 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. Fig. 2. Dictionaries 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. Js (f ) = min ||x||1 color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( TABLE I in the new metric).duced with our proposed technique (a bias effect in the color from the 7 in our proposed new metric). Both images have been denoised with the same global dictionary.TH 256 ATOMS OF SIZE castle 7 in3 FOR of the water. What is more, the color of the sky is .piecewise CASE IS DIVIDED IN FOUR and some part AND 6 6 3 FOR EACH constant when f= x J (f ) = ||D f ||ch MCAULEY AND approach corrected. (a)HEIR “3(b) Original algorithm, HE TOP-RIGHT RESULTS ARE THOSE OBTAINED BY Y is another artifact our AL [28] WITH T Original. 3 MODEL.” T dB. (c) Proposed algorithm, dB. 8 ATOMS. THE BOTTOM-LEFT ARE OUR RESULTS2] ON EACH CHANNEL SEPARATELY WITH 8 8 ATOMS. THE BOTTOM-LEFT ARE OUR RESULTS OBTAINED a 1 AND 6OTTOM-RIGHT ARE THE IMPROVEMENTS OBTAINED WITH THE ADAPTIVE APPROACH WITH 20 ITERATIONS. Other sparse priors:H GROUP. AS CAN BE SEEN, OUR PROPOSED TECHNIQUE CONSISTENTLY PRODUCES THE BEST RESULTS Coe cients x c=D f 6 3 FOR 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. . EACH CASE IS DIVID |x1 | + |x2 | max(|x1 |, |x2 |)
  • 59. RESULTS ARE THOSE GIVEN BY MCAULEY AND AL [28] WITH THEIR “3 3 MODEL.” THE TOP-RIGHT RESULTS ARE THOSE O RAINED DICTIONARY. THE BOTTOM-RIGHT ARE THE IMPROVEMENTS OBTAINED WITH THE ADAPTIVE APPROACH WITH 20 IT CALE K-SVD ALGORITHM [2] ON EACH CHANNEL SEPARATELY WITH 8 EPRESENTATION FOR COLOR IMAGE RESTORATION DENOISING ALGORITHM WITH 256 ATOMS OF SIZE 7 7 3 FOR color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( uced with our proposed technique ( in the new metric). in our proposed new metric). Both images have been denoised with the same global dictionary. Some Hot Topics 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 Image f =ch is another artifact our approach corrected. (a) Original. (b) Original algorithm, dB. (c) Proposed algorithm, dB. Dictionary learning:with 256 atoms learned on a generic database of natural images, with two differental.: SPARSE REPRESENTATION FOR COLOR IMAGE RESTORATIONhave negative values, the vectors are presented scaled and shifted to the [0,255] range per channel: (a) 5 MAIRAL et sizes of patches. Note the large number of color-less 5 3 patches; (b) 8 8 atoms. 3 patches. 57 xOR IMAGE RESTORATION 61 Fig. 7. Data set used for evaluating denoising experiments. learning D ing Image; (b) resulting dictionary; (b) is the dictionary learned in the image in (a). The dictionary is more colored than the global one. TABLE I Analysis vs. synthesis:g. 7. Data set used for evaluating denoising experiments. 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. Fig. 2. Dictionaries 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. Js (f ) = min ||x||1 color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( TABLE I in the new metric).duced with our proposed technique (a bias effect in the color from the 7 in our proposed new metric). Both images have been denoised with the same global dictionary.TH 256 ATOMS OF SIZE castle 7 in3 FOR of the water. What is more, the color of the sky is .piecewise CASE IS DIVIDED IN FOUR and some part AND 6 6 3 FOR EACH constant when f= x J (f ) = ||D f ||ch MCAULEY AND approach corrected. (a)HEIR “3(b) Original algorithm, HE TOP-RIGHT RESULTS ARE THOSE OBTAINED BY Y is another artifact our AL [28] WITH T Original. 3 MODEL.” T dB. (c) Proposed algorithm, dB. 8 ATOMS. THE BOTTOM-LEFT ARE OUR RESULTS2] ON EACH CHANNEL SEPARATELY WITH 8 8 ATOMS. THE BOTTOM-LEFT ARE OUR RESULTS OBTAINED a 1 AND 6OTTOM-RIGHT ARE THE IMPROVEMENTS OBTAINED WITH THE ADAPTIVE APPROACH WITH 20 ITERATIONS. Other sparse priors:H GROUP. AS CAN BE SEEN, OUR PROPOSED TECHNIQUE CONSISTENTLY PRODUCES THE BEST RESULTS Coe cients x c=D f 6 3 FOR 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. . EACH CASE IS DIVID 2 1 |x1 | + |x2 | max(|x1 |, |x2 |) |x1 | + (x2 2 + x3 ) 2
  • 60. RESULTS ARE THOSE GIVEN BY MCAULEY AND AL [28] WITH THEIR “3 3 MODEL.” THE TOP-RIGHT RESULTS ARE THOSE O RAINED DICTIONARY. THE BOTTOM-RIGHT ARE THE IMPROVEMENTS OBTAINED WITH THE ADAPTIVE APPROACH WITH 20 IT CALE K-SVD ALGORITHM [2] ON EACH CHANNEL SEPARATELY WITH 8 EPRESENTATION FOR COLOR IMAGE RESTORATION DENOISING ALGORITHM WITH 256 ATOMS OF SIZE 7 7 3 FOR color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( uced with our proposed technique ( in the new metric). in our proposed new metric). Both images have been denoised with the same global dictionary. Some Hot Topics 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 Image f =ch is another artifact our approach corrected. (a) Original. (b) Original algorithm, dB. (c) Proposed algorithm, dB. Dictionary learning:with 256 atoms learned on a generic database of natural images, with two differental.: SPARSE REPRESENTATION FOR COLOR IMAGE RESTORATIONhave negative values, the vectors are presented scaled and shifted to the [0,255] range per channel: (a) 5 MAIRAL et sizes of patches. Note the large number of color-less 5 3 patches; (b) 8 8 atoms. 3 patches. 57 xOR IMAGE RESTORATION 61 Fig. 7. Data set used for evaluating denoising experiments. learning D ing Image; (b) resulting dictionary; (b) is the dictionary learned in the image in (a). The dictionary is more colored than the global one. TABLE I Analysis vs. synthesis:g. 7. Data set used for evaluating denoising experiments. 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. Fig. 2. Dictionaries 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. Js (f ) = min ||x||1 color artifacts while reconstructing a damaged version of the image (a) without the improvement here proposed ( TABLE I in the new metric).duced with our proposed technique (a bias effect in the color from the 7 in our proposed new metric). Both images have been denoised with the same global dictionary.TH 256 ATOMS OF SIZE castle 7 in3 FOR of the water. What is more, the color of the sky is .piecewise CASE IS DIVIDED IN FOUR and some part AND 6 6 3 FOR EACH constant when f= x J (f ) = ||D f ||ch MCAULEY AND approach corrected. (a)HEIR “3(b) Original algorithm, HE TOP-RIGHT RESULTS ARE THOSE OBTAINED BY Y is another artifact our AL [28] WITH T Original. 3 MODEL.” T dB. (c) Proposed algorithm, dB. 8 ATOMS. THE BOTTOM-LEFT ARE OUR RESULTS2] ON EACH CHANNEL SEPARATELY WITH 8 8 ATOMS. THE BOTTOM-LEFT ARE OUR RESULTS OBTAINED a 1 AND 6OTTOM-RIGHT ARE THE IMPROVEMENTS OBTAINED WITH THE ADAPTIVE APPROACH WITH 20 ITERATIONS. Other sparse priors:H GROUP. AS CAN BE SEEN, OUR PROPOSED TECHNIQUE CONSISTENTLY PRODUCES THE BEST RESULTS Coe cients x c=D f 6 3 FOR 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. . EACH CASE IS DIVID 2 1 |x1 | + |x2 | max(|x1 |, |x2 |) |x1 | + (x2 2 + x3 ) 2 Nuclear

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