摘 要：随着数字图像数据的快速增长，高效压缩技术成为解决存储与传输瓶颈的关键问题。本研究以傅里叶变换为核心，探讨其在图像压缩中的效率表现，并与离散余弦变换（DCT）和小波变换进行对比分析。研究旨在评估傅里叶变换在不同场景下的适用性及其性能优势。通过构建基于快速傅里叶变换（FFT）的压缩算法模型，结合多种量化策略和熵编码方法，对标准测试图像集进行了系统实验。结果表明，傅里叶变换在处理周期性特征显著的图像时表现出较高的能量集中度和较低的重构误差，尤其在中低压缩比条件下优势明显。然而，在高压缩比下，其性能略逊于DCT和小波变换。本研究的主要贡献在于揭示了傅里叶变换在特定应用场景中的独特价值，并提出了一种改进的能量优化算法，有效提升了压缩效率与图像质量的平衡。这一发现为选择合适的变换方法提供了理论依据，同时为图像压缩技术的进一步发展奠定了基础。
关键词：傅里叶变换；图像压缩；快速傅里叶变换；离散余弦变换；能量优化算法
A Comparative Study on the Efficiency of Fourier Transform in Image Compression
英文人名
Directive teacher:×××
Abstract：With the rapid growth of digital image data, efficient compression techniques have become a critical solution to address storage and transmission bottlenecks. This study focuses on the Fourier Transform to evaluate its performance in image compression and compares it with the Discrete Cosine Transform (DCT) and Wavelet Transform. The aim is to assess the applicability and performance advantages of the Fourier Transform under various scenarios. By constructing a compression algorithm model based on the Fast Fourier Transform (FFT), combined with multiple quantization strategies and entropy coding methods, systematic experiments were conducted on a standard test image set. The results indicate that the Fourier Transform demonstrates high energy concentration and low reconstruction error when processing images with significant periodic features, particularly showing advantages under medium and low compression ratios. However, under high compression ratios, its performance is slightly inferior to that of DCT and wavelet transform. The primary contribution of this study lies in revealing the unique value of the Fourier Transform in specific application contexts and proposing an improved energy optimization algorithm, which effectively enhances the balance between compression efficiency and image quality. This finding provides a theoretical basis for selecting appropriate transform methods and lays a foundation for further development of image compression technology.
Keywords: Fourier Transform；Image Compression；Fast Fourier Transform；Discrete Cosine Transform；Energy Optimization Algorithm
目 录
引言 1
一、傅里叶变换基础与图像压缩原理 1
(一)傅里叶变换基本概念 1
(二)图像压缩的核心原理 2
(三)变换域与图像特征分析 2
二、傅里叶变换在图像压缩中的实现方法 3
(一)离散傅里叶变换的应用 3
(二)频域系数的选择策略 3
(三)数据量与精度的权衡 4
三、效率比较：傅里叶变换与其他方法 4
(一)与离散余弦变换的对比 5
(二)与小波变换的效率分析 5
(三)不同算法的适用场景 5
四、实验评估与结果分析 6
(一)测试数据集与实验设计 6
(二)压缩比与重建质量评估 6
(三)实验结果讨论与优化建议 7
结论 7
参考文献 9
致谢 9
