信號(hào)處理原理：第X章 稀疏編碼

1、信號(hào)處理原理Principle of Signal Processing1Sparse Coding第x 章 稀疏編碼稀疏編碼應(yīng)用23Assumption: the signal x was created by x=D0 with a very sparse 0. Missing values in x imply missing rows in this linear system. By removing these rows, we get .Now solveIf 0 was sparse enough, it will be the solution of the above pr

2、oblem! Thus, computing D0 recovers x perfectly. Image Inpainting: Theory=4 Inpainting: The PracticeGiven y, we try to recover the representation of x, by solving We define a diagonal mask operator W representing the lost samples, so thatWe use a dictionary that is the sum of two dictionaries, to get

3、 an effective representation of both texture and cartoon contents. This also leads to image separation Elad, Starck, & Donoho (05)5 Inpainting ResultsSourceOutcomeDictionary: Curvelet (cartoon) + Global DCT (texture)6 Inpainting ResultsSourceOutcomeDictionary: Curvelet (cartoon) + Overlapped DCT (te

4、xture)7 Inpainting Results20%50%80%8 Inpainting Results70% Missing SamplesDCT (RMSE=0.04)Haar (RMSE=0.045)K-SVD (RMSE=0.03)90% Missing SamplesDCT (RMSE=0.085_Haar (RMSE=0.07)K-SVD (RMSE=0.06)9Solution: force shift-invariant sparsity for each NxN patch of the image, including overlaps. Denoising: The

5、ory and PracticeGiven a noisy image y, we can clean it by solving With K-SVD, we cannot train a dictionary for an entire image. How do we go from local treatment of patches to a global prior?Can we use the K-SVD dictionary? 10Our priorExtracts the(i,j)th patch From Local to Global TreatmentFor patch

6、es,our MAP penalty becomes11Option 1:Use a database of images: works quite well (0.5-1dB below the state-of-the-art) Option 2: Use the corrupted image itself ! Simply sweep through all NxN patches(with overlaps) and use them to trainImage of size 1000 x1000 pixels 106 examples to use more than enoug

7、h.This works much better! What Data to Train On?12K-SVDx and D knownx and ij knownCompute D to minimize using SVD, updating one column at a timeD and ij knownCompute x bywhich is a simple averaging of shifted patchesImage Denoising: The AlgorithmCompute ij per patch using matching pursuit13Initial d

8、ictionary (overcomplete DCT) 64256 Denoising ResultsSourceResult 30.829dBObtained dictionaryafter 10 iterationsNoisy image 14 Denoising Results: 3DSource:Vis. Male Head (Slice #137)PSNR=12dB2d-KSVD:PSNR=27.3dB3d-KSVD:PSNR=32.4dB15 Image CompressionProblem: compressing photo-ID images.General purpose

9、 methods (JPEG, JPEG2000) do not take into account the specific family. By adapting to the image-content,better results can be obtained.16 Compression: The AlgorithmTraining set (2500 images)Detect main features and align the images to a common reference (20 parameters) TrainingDivide each image to

10、disjoint 15x15 patches, and for each compute a unique dictionary Divide to disjoint patches, and sparse-code each patchCompressionDetect features and alignQuantize and entropy-code17 Compression Results11.9910.8310.9310.498.928.718.817.898.615.564.825.58Results for 820 bytes per imageOriginalJPEGJPEG 2000PCAK-SVDBottom:RMSE val