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S.R.Subramanya1 Outline of Vector Quantization of Images.

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1 S.R.Subramanya1 Outline of Vector Quantization of Images

2 S.R.Subramanya2 VQ Coding Outline Divide data (signal) into non-overlapping vectors (Each vector contains ‘n’ elements (pixels/samples)) For each image vector : Find closest vector in codebook Get its index in codebook Encode indices

3 S.R.Subramanya3 VQ Decoding Outline Entropy decode the indices Lookup codebook for each index and retrieve vector Combine vectors to reconstruct image (data)

4 S.R.Subramanya4 VQ Terminology Vector : Group of ‘n’ elements (pixels/samples) Codebook : (Final) set of codevectors Training set : (Initial) set of codevectors Codevector : Vector derived from image/group of images

5 S.R.Subramanya5 Why VQ is better than scalar Quantization Scalar Quantizer :  Treats each pixel independently  Does not use correlation between neighboring pixels Vector Quantizer: Image (data) divided into vectors (blocks) Correlation among pixels in vectors is exploited Block size should be appropriate: Too large block : correlation is lost Too small block : More code vectors If no interpixel correlation, then no gain

6 S.R.Subramanya6 VQ Bitrate Vector size : n (say, n = p x p) pixels Codebook size : L vectors Codebook index size : bits Bit Rate of VQ : bits/pixel

7 S.R.Subramanya7 Distortion Measures xkxk Image closest matching code vector Image vectors : X j Codebook codevectors : V i, V1V1 V2V2 VkVk VLVL Mean Square Error (MSE) (Euclidean Distance) Weighted MSE

8 S.R.Subramanya8 Two Basic Kinds of Codebook Local Codebook  One codebook for each image  Codebook derived from vectors of one image  Good performance (Quality of reconstruction)  More overhead : (1) computation (2) Transmission of CB to decoder

9 S.R.Subramanya9 Two Basic Kinds of Codebook Global Codebook  One codebook for a class of images  Codebook derived from vectors of all images in the class  Less overhead (compared to local codebook)  Lower performance

10 S.R.Subramanya10 Major Issues in VQ Generation (construction) of codebook  concerns what needs to be included in the codebook Design of codebook  concerns structuring codebook entries to minimize search time

11 S.R.Subramanya11 Codebook Generation Generate codebook from a Training set Training Set: Set of vectors derived from image vectors Codevectors should minimize distortion Most commonly used algorithm : LBG algorithm LBG : Linde–Buzo–Gray algorithm

12 S.R.Subramanya12 LBG Algorithm Outline

13 S.R.Subramanya13 Codebook Initialization Three basic schemes: Random Perturb and Split (Bottoming) Pairwise Nearest Neighbor (PNN) clustering (PNN)

14 S.R.Subramanya14 Lloyd’s algorithm 1. Choice of an initial codebook. 2. All points x i are encoded; E VQ is evaluated. 3. If E VQ is small enough, then stop. 4. All centroids y j are replaced by the center- of-gravity of the data x i associated to y j in step 2. 5. Back to step 2.

15 S.R.Subramanya15 Lloyd: example 1. Initialization of the code boook. X i : data. y j :centroids

16 S.R.Subramanya16 Lloyd: example 2. Encoding (nearest-neighbor). x i : data. y j :centroids

17 S.R.Subramanya17 Lloyd: example 4. Decoding (center-of-gravity)

18 S.R.Subramanya18 Lloyd: example 2. Encoding (nearest-neighbor) → new borders

19 S.R.Subramanya19 Lloyd: example 4. Decoding (center-of-gravity) → new positions of centroids

20 S.R.Subramanya20 Lloyd: example 2. Encoding (nearest-neighbor) → new borders

21 S.R.Subramanya21 Lloyd: example 4. Decoding (center-of-gravity) → new positions of centroids

22 S.R.Subramanya22 Lloyd: example 2. Encoding (nearest-neighbor) → new borders

23 S.R.Subramanya23 Lloyd’s algorithm: the names Lloyd’s algorithm Generalized Lloyd’s algorithm Linde- Buzzo-Gray (LBG) algorithm K-means ISODATA … All based on the same principle.

24 S.R.Subramanya24 Lloyd’s algorithm: properties The codebook is modified only after the presentation of the whole dataset. The mean square error (E VQ ) decreases at each iteration. The risk of getting trapped in local minima is high. The final quantizer depends on the initial one

25 S.R.Subramanya25 How to initialize Lloyd’s algorithm ? 1. Randomly in the input space 2. The Q first data points x i Use the first 2 R vectors (code book vectors) in the training sequence as the initial codebook. 3. Q randomly chosen data points x i select several widely spaced words from the training sequence. This approach is sometimes called random code generation.

26 S.R.Subramanya26 4. ‘Product codes’: the product of scalar quantizers Collection of codebooks C i, each consisting of M i vectors. Then the product codebook C is defined as the collection of all M possible concatenations of m words drawn successively from the m codebooks C i. 5. Growing initial set: – a first centroid y 1 is randomly chosen (in the data set) – a second centroid y 2 is randomly chosen (in the data set); if d(y 1, y 2 ) > threshold, y 2 is kept – a third centroid y 3 is randomly chosen (in the data set); if d(y 1, y 3 ) > threshold and d(y 2, y 3 ) > threshold y 3 is kept – … How to initialize Lloyd’s algorithm ? (Cont.)

27 S.R.Subramanya27 How to initialize Lloyd’s algorithm ? (Cont.) 6. ‘pairwise nearest neighbor’: – a first codebook is built with all data points x i – the two centroids nearest one from another are merged (center-of gravity) – … Variant: – the increase of distortion ( E VQ ) is evaluated for the merge of each pair of centroids y j ; the pair giving the lowest increase is merged

28 S.R.Subramanya28 How to initialize Lloyd’s algorithm ? 7. Splitting – a first centroid y 1 is randomly chosen – a second centroid y 1 +ε is created; Lloyd algorithm is applied to the new codebook – two new centroids are created by perturbing the two existing ones; Lloyd algorithm is applied to this 4-centroids codebook – …

29 S.R.Subramanya29 Splitting

30 S.R.Subramanya30 Codebook Design Basic objective : Minimize search time for codevector Full (Exhaustive) Search : very expensive Design emphasis : Organization of codebook

31 S.R.Subramanya31 Tree-structured codebook Product codebook Mean/Residual VQ Interpolative/Residual VQ Gain/Shape VQ Classified VQ Finite State VQ Codebook Organizations

32 S.R.Subramanya32 Tree – Structured Codebook Codebook organized as M–ary tree Number of Levels : Code vectors stored at the leaves Intermediate node : ‘average’ of ‘codevectors’ of children Improved search time Increased Storage cost Performance inferior to full search

33 S.R.Subramanya33 Variations of Tree – Structured VQ Tapered trees:  Non – Uniform number of branches at nodes  Branches per node increases going down the tree ( Ex: 2 branches at level 1, 3 at level 2 etc.) Pruned Trees  Start with (full) large initial tree  Remove code vectors that do not reduce distortion

34 S.R.Subramanya34 Product Codebook Codebook : Cartesian product of many smaller codebooks Vector characterized by many independent features:f 1,f 2...,f N Separate codebook for each feature Smaller codebbok sizes : L 1,L 2...,L N Effective codebook size : L 1,L 2...,L N Actual storage + search complexity O(L 1 +L 2 +...+L N )

35 S.R.Subramanya35 Prediction/Residual Class VQ Predict Original Image Derive Residual Image Data used in prediction: Scalar Quantization + Encoding Residual Image : Vector Quantization Major Types:  Mean/Residual VQ  Interpolative/Residual VQ

36 S.R.Subramanya36 Mean/Residual VQ Image Vectors have similar variations about different mean/ends Remove mean from vectors fewer code vectors

37 S.R.Subramanya37 Mean/Residual VQ Scheme : Compute ‘mean’ of image vectors : M={m 1,m 2...,m N } Quantize M using scalar quantization : (Apply DPCM before Quantization further bitrate reduction) Subtract Quantized means from vector elements Residual vectors Quantize Residual vectors using VQ

38 S.R.Subramanya38 Interpolative/Residual VQ Subsample original image by l in each dimension (typically, l=8) Quantize subsampled value Upsample using Quantized subsampled values (typically, bilinear interpolation is used) Form Residual : (Original – Upsampled) Segment Residual to form 4 x 4 vectors Quantize Residual vectors using VQ

39 S.R.Subramanya39 Classified VQ Several codebooks Each codebook for a specific feature  ex. edges, smooth areas, etc. Codebooks could be of different sizes

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