题目如下:读取一幅512×512×8比特的单色Lena图像,完成以下步骤。
- 统计该图像的概率直方图,并画出直方图
- 计算该图像的熵
- 对其进行霍夫曼编码
- 分别计算压缩率和冗余度
将要进行处理的图片命名为lena512.jpg
。
接下来运行以下代码:
clc,clf,clear all,close all;
I = imread('lena512.jpg'); % 读取图像
imshow(I,[]); % 显示图像
L = 256;
%获取各符号的概率
[M,N] = size(I);
I0 = I(:);
P = zeros(1,L);
for i = 0:L-1
P(i+1) = length(find(I0 == i))/(M*N);
end
%统计概率直方图
figure;bar(P);
% 计算该图像的熵
H = entropy(I);
% 进行霍夫曼编码
code = huffman(P);
% 计算压缩率和冗余度
Lc = 0;
for i = 1:L
Lc = Lc+P(i)*length(code{i,1});%平均码长
end
m = log2(L);
C = m/Lc; %压缩比
R = 1-H/Lc; %冗余度
R = strcat(num2str(R*100),'%');%冗余度的百分比表示
函数entropy.m
代码如下:
function h = entropy(x, n)
%ENTROPY Computes a first-order estimate of the entropy of a matrix.
% H = ENTROPY(X, N) returns the first-order estimate of matrix X
% with N symbols (N = 256 if omitted) in bits/symbol. The estimate
% assumes a statistically independent source characterized by the
% relative frequency of occurrence of the elements in X.
% Copyright 2002-2004 R. C. Gonzalez, R. E. Woods, & S. L. Eddins
% Digital Image Processing Using MATLAB, Prentice-Hall, 2004
% $Revision: 1.4 $ $Date: 2003/10/26 18:35:35 $
error(nargchk(1, 2, nargin)); % Check input arguments
if nargin < 2
n = 256; % Default for n.
end
x = double(x); % Make input double
xh = hist(x(:), n); % Compute N-bin histogram
xh = xh / sum(xh(:)); % Compute probabilities
% Make mask to eliminate 0's since log2(0) = -inf.
i = find(xh);
h = -sum(xh(i) .* log2(xh(i))); % Compute entropy
函数huffman.m
代码如下:
function CODE = huffman(p)
% HUFFMAN Builds a variable-length Huffman code for a symbol source.
% CODE = HUFFMAN(P) returns a Huffman code as binary strings in
% cell array CODE for input symbol probability vector P. Each word
% in CODE corresponds to a symbol whose probability is at the
% corresponding index of P.
%
% Based on huffman5 by Sean Danaher, University of Northumbria,
% Newcastle UK. Available at the MATLAB Central File Exchange:
% Category General DSP in Signal Processing and Communications.
% Copyright 2002-2004 R. C. Gonzalez, R. E. Woods, & S. L. Eddins
% Digital Image Processing Using MATLAB, Prentice-Hall, 2004
% $Revision: 1.5 $ $Date: 2003/10/26 18:37:16 $
% Check the input arguments for reasonableness.
error(nargchk(1, 1, nargin));
if (ndims(p) ~= 2) | (min(size(p)) > 1) | ~isreal(p) | ~isnumeric(p)
error('P must be a real numeric vector.');
end
% Global variable surviving all recursions of function 'makecode'
global CODE
CODE = cell(length(p), 1); % Init the global cell array
if length(p) > 1 % When more than one symbol ...
p = p / sum(p); % Normalize the input probabilities
s = reduce(p); % Do Huffman source symbol reductions
makecode(s, []); % Recursively generate the code
else
CODE = {'1'}; % Else, trivial one symbol case!
end;
%-------------------------------------------------------------------%
function s = reduce(p);
% Create a Huffman source reduction tree in a MATLAB cell structure
% by performing source symbol reductions until there are only two
% reduced symbols remaining
s = cell(length(p), 1);
% Generate a starting tree with symbol nodes 1, 2, 3, ... to
% reference the symbol probabilities.
for i = 1:length(p)
s{i} = i;
end
while numel(s) > 2
[p, i] = sort(p); % Sort the symbol probabilities
p(2) = p(1) + p(2); % Merge the 2 lowest probabilities
p(1) = []; % and prune the lowest one
s = s(i); % Reorder tree for new probabilities
s{2} = {s{1}, s{2}}; % and merge & prune its nodes
s(1) = []; % to match the probabilities
end
%-------------------------------------------------------------------%
function makecode(sc, codeword)
% Scan the nodes of a Huffman source reduction tree recursively to
% generate the indicated variable length code words.
% Global variable surviving all recursive calls
global CODE
if isa(sc, 'cell') % For cell array nodes,
makecode(sc{1}, [codeword 0]); % add a 0 if the 1st element
makecode(sc{2}, [codeword 1]); % or a 1 if the 2nd
else % For leaf (numeric) nodes,
CODE{sc} = char('0' + codeword); % create a char code string
end
原图:
概率直方图:
算得:
- 熵:H = 7.4471
- 压缩比:C = 1.0709
- 冗余度:R = 0.31214%