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基于8×8 DCT变换的图像压缩MATLAB实现

news 2025/8/12 13:33:24

将图像分割为8×8块，进行DCT变换、量化和压缩，最后重建图像并评估压缩效果。

function dctImageCompression(imagePath, quality)% 读取图像if nargin < 1imagePath = 'cameraman.tif'; % 默认图像endif nargin < 2quality = 50; % 默认压缩质量(0-100)end% 读取图像并转换为双精度originalImage = imread(imagePath);if size(originalImage, 3) == 3originalImage = rgb2gray(originalImage);endoriginalImage = im2double(originalImage);% 显示原始图像figure('Name', 'DCT图像压缩', 'Position', [100, 100, 1000, 500]);subplot(2, 3, 1);imshow(originalImage);title(sprintf('原始图像 (%dx%d)', size(originalImage,2), size(originalImage,1)));% 定义8×8量化矩阵 (基于JPEG标准)quantizationMatrix = [16  11  10  16  24  40  51  61;12  12  14  19  26  58  60  55;14  13  16  24  40  57  69  56;14  17  22  29  51  87  80  62;18  22  37  56  68 109 103  77;24  35  55  64  81 104 113  92;49  64  78  87 103 121 120 101;72  92  95  98 112 100 103  99];% 调整量化质量 (quality: 1-100)if quality <= 0quality = 1;elseif quality > 100quality = 100;endif quality < 50scalingFactor = 5000 / quality;elsescalingFactor = 200 - 2 * quality;endquantizationMatrix = floor((quantizationMatrix * scalingFactor + 50) / 100);quantizationMatrix(quantizationMatrix < 1) = 1;% 填充图像使其尺寸为8的倍数[rows, cols] = size(originalImage);paddedRows = ceil(rows / 8) * 8;paddedCols = ceil(cols / 8) * 8;paddedImage = padarray(originalImage, [paddedRows-rows, paddedCols-cols], 'replicate', 'post');% 初始化变量dctCoeffs = zeros(size(paddedImage));quantizedCoeffs = zeros(size(paddedImage));reconstructedImage = zeros(size(paddedImage));% 压缩率统计totalCoeffs = numel(paddedImage);nonZeroCoeffs = 0;% 处理每个8×8块for i = 1:8:paddedRowsfor j = 1:8:paddedCols% 提取当前块block = paddedImage(i:i+7, j:j+7);% DCT变换dctBlock = dct2(block);dctCoeffs(i:i+7, j:j+7) = dctBlock;% 量化quantBlock = round(dctBlock ./ quantizationMatrix);quantizedCoeffs(i:i+7, j:j+7) = quantBlock;% 统计非零系数nonZeroCoeffs = nonZeroCoeffs + nnz(quantBlock);% 反量化dequantBlock = quantBlock .* quantizationMatrix;% IDCT重建reconstructedImage(i:i+7, j:j+7) = idct2(dequantBlock);endend% 裁剪回原始尺寸reconstructedImage = reconstructedImage(1:rows, 1:cols);% 计算压缩率和PSNRcompressionRatio = totalCoeffs / nonZeroCoeffs;mse = mean((originalImage(:) - reconstructedImage(:)).^2);psnr = 10 * log10(1 / mse);% 显示结果subplot(2, 3, 2);imshow(paddedImage);title('填充后的图像');subplot(2, 3, 3);imshow(log(abs(dctCoeffs) + 1), []);colormap(jet); colorbar;title('DCT系数 (对数尺度)');subplot(2, 3, 4);imshow(abs(quantizedCoeffs), [0, max(quantizedCoeffs(:))]);colormap(jet); colorbar;title('量化后的系数');subplot(2, 3, 5);imshow(reconstructedImage);title(sprintf('重建图像 (质量: %d%%)', quality));subplot(2, 3, 6);diffImage = abs(originalImage - reconstructedImage) * 5; % 放大误差imshow(diffImage);title(sprintf('误差图 (x5放大)'));% 显示压缩信息fprintf('图像尺寸: %d x %d\n', cols, rows);fprintf('压缩质量: %d%%\n', quality);fprintf('压缩率: %.2f:1\n', compressionRatio);fprintf('非零系数比例: %.2f%%\n', (nonZeroCoeffs/totalCoeffs)*100);fprintf('PSNR: %.2f dB\n', psnr);fprintf('MSE: %.6f\n', mse);% 保存结果saveas(gcf, 'dct_compression_result.png');
end

示例

% 示例1：使用默认图像和默认质量(50%)
dctImageCompression();% 示例2：指定图像和质量
dctImageCompression('peppers.png', 75);% 示例3：高质量压缩
dctImageCompression('mandril.tif', 90);% 示例4：低质量高压缩
dctImageCompression('satellite.jpg', 10);

算法说明

1. DCT变换核心原理

将8×8图像块转换为频域表示
公式： $\frac{1}{4}C(u)C(v)\sum_{x=0}^{7}\sum_{y=0}^{7}f(x,y)\cos\left(\frac{(2x+1)u\pi}{16}\right)\cos\left(\frac{(2y+1)v\pi}{16}\right)$
其中 $\begin{cases} 1/\sqrt{2} & \text{if } u=0 \\1 & \text{otherwise}\end{cases}$