Vector Quantization（学习Free Mind知识整理）

阅读http://blog.pluskid.org/?p=57文章中的一些知识整理：

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矢量量化（Vector Quantization）其实也就是逼近，VQ 是将一个向量空间中的点用其中的一个有限子集来进行编码的过程。Vector Quantization------------>http://www.data-compression.com/vq.html#lbg

因为在很多各种能用的聚类方法都可以用，下面是Free Mind的python实现：

from scipy.cluster.vq import kmeans, vq
from numpy import array, reshape, zeros
from mltk import image

vqclst = [2, 10, 100, 256]

data = image.read('example.jpg')
(height, width, channel) = data.shape

data = reshape(data, (height*width, channel))
for k in vqclst:
    print 'Generating vq-%d...' % k
    (centroids, distor) = kmeans(data, k)
    (code, distor) = vq(data, centroids)
    print 'distor: %.6f' % distor.sum()
    im_vq = centroids[code, :]
    image.write('result-%d.jpg' % k, reshape(im_vq,
        (height, width, channel)))

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Vector Quantization也介绍了LBG算法，LBG-VQ算法是一个迭代算法，它交替地调整P和C（两个优化准则Nearest NeighborCondition 最近邻条件： Vector Quantization（学习Free Mind知识整理）和Centroid Condition质心条件： ），使失真度不断地趋向于它的局部最小值（有点EM的思想哦）。最小的C（码书Codebook）和P（空间划分）。

LBG Design Algorithm

Given . Fixed to be a ``small'' number.
Let and Calculate
Splitting: For , set Set .
Iteration: Let . Set the iteration index.
1. For , find the minimum value of over all . Let be the index which achieves the minimum. Set
2. For , update the codevector
3. Set .
4. Calculate
5. If , go back to Step (i).
6. Set . For, set as the final codevectors.
Repeat Steps 3 and 4 until the desired number of codevectors is obtained.

LBG-VQ的matlab实现：

LBG：

function [codebook index] = LBG_training(ImgPxl, Nc, Vector_Len, er, x, y) 
  Training_idx = (x*y)/Vector_Len ; 
  Training_set = reshape(ImgPxl(:), Training_idx, Vector_Len) ; 

  codebook = zeros(Nc, Vector_Len) ; 

  for i = 1 : Nc 
    % 此为最原始设计之初使化 Codebook 算法，但因效果太差 
    % 更换为下面那一种先做一些平均的动作后取得之算法。 
    %mean(Training_set((i-1)*Training_idx/Nc+1:i*Training_idx/Nc,:)) 
    % 

    Child_num = Training_idx/Nc ; 

    codebook(i,:) = mean( 
        Training_set( 
           ((i-1)*Child_num+1)+Child_num/4:i*Child_num-Child_num/4,: 
        ) 
    ) ; % 因为这一行太长，所以断行成这样..  =.=b 
  end 

  % 给予一个初始的平均改善临界值 ( > 0 就行了.. ) 
  thd = 10 ; 

  % 给予一个极大的上一代 MSE 
  er_o = 1000000000000000 ; 

  % 设定初始代数 
  tol = 0 ; 

  % 最多 training 代数 
  mtol = 1000 ; 

  indxe = zeros(Training_idx,1) ; 
  ver_d = zeros(Training_idx,1) ; 

  % Training 初始化 
  for i = 1 : Training_idx 
    tmp = Training_set(i,:) ; 
    er_ds = sum( 
             ((tmp(ones(size(codebook,1),1),:)-codebook).^2).’ 
            )/Vector_Len ; 

    [ver_d(i) index(i)] = min(er_ds) ; 
  end 

  while( er < thd && tol < mtol) 
    for i = 1 : Nc 
      codeset = find(index == i) ; 
      if ~isempty(codeset) 
        codebook(i,:) = mean([Training_set(codeset,:) ;codebook(i,:)]) ; 
      else 
        % 更新法则，非常精简及没有特别的理论版。  
        if i == 1 
          codebook(i,:) = mean(codebook([2 3],:)) ; 
        elseif i == Nc 
          codebook(i,:) = mean(codebook([Nc-1 Nc-2],:)) ; 
        else 
          codebook(i,:) = mean(codebook([i+1 i-1],:)) ; 
        end 
      end 
    end 

    while (1) 
      %以新的 Codebook 针对每一笔 block ，进行选择 Code 的动作 
      for i = 1 : Training_idx 
        tmp = Training_set(i,:) ; 
        % 计算这一代的误差 
        er_ds = sum( 
            ((tmp(ones(size(codebook,1),1),:)-codebook).^2).’ 
                )/Vector_Len ; 
        [ver_d(i) index(i)] = min(er_ds) ; 
      end 

      if (~(er_o == mean(ver_d)))  
        break ; 
      else 
        for i = 2 : Nc-1 
          codebook(i,:) = mean(codebook([i-1 i i+1],:)) ; 
        end     
      end 
    end 

    thd = abs((er_o – mean(ver_d.^2))/er_o) ; 
    er_o = mean(ver_d.^2) ; 

    tol = tol + 1 ; 
  end 
  tol 
  thd

VQ:

function EzVQ(ImgFile, Nc, Vector_Len, er, x, y) 
    ImgPxl = double(imread(ImgFile)) ; 

    [codebook index] = LBG_training(ImgPxl, Nc, Vector_Len, er, x, y) ; 

    VQImg = ones((x*y)/Vector_Len, Vector_Len) ; 

    % 利用 codebook 来进行译码 
    VQImg = codebook(index,:) ; 

    VQImg = reshape(VQImg(:), x, y) ; 
    MSE = mse(ImgPxl – VQImg) 

    figure(gcf()+1) ; 
    imshow(uint8(ImgPxl)) ; 
    title(‘Source Image’) ; 
    figure(gcf()+1) ; 
    imshow(uint8(VQImg)) ; 
    title(‘VQ Image’) ;

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原文：

http://www.data-compression.com/vq.shtml

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Vector Quantization（学习Free Mind知识整理）

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