This repo contains my homework for the class wavelet analysis at USTC in spring 2024. Each folder contains one PDF that explains the corresponding project, I hope that there are enough comments and good naming so you can just read the code to get a sense.

• Chebyshev interpolation.
• Image compression with wavelets and wavelet packets.

Image compression

Goals and Structure

Please go through the following files in order.

1. basic_wavelet_compression.m: implement a basic wavelet compression and plot different levels.
2. wavelet_packet_compression.m: use wavelet packet to compress images and compare the results with the previous basic one.
3. different_entropy_and_threshold_wavelet_packet.m: run this file to explore how different threshold and entropy types affects the wavelet packet compression performance.

Methods and Theory

Wavelet Image Compression

Wavelet image decompose the image into a set of coefficients representing different frequency components.

1. Wavelet Transform:

   • An image is decomposed into a series of subbands using a wavelet transform (e.g., Discrete Wavelet Transform, DWT).
   • Each subband represents a different frequency component of the image, capturing details at various resolutions.

2. Thresholding:

   • Coefficients below a certain threshold are set to zero, reducing the amount of data to store.
   • Thresholding can be hard (setting coefficients below the threshold to zero) or soft (reducing the magnitude of coefficients towards zero).

3. Quantization:

   • Remaining coefficients are quantized to reduce the number of bits required for storage.

4. Encoding:

   • Quantized coefficients are encoded using techniques such as Huffman coding or run-length encoding to further compress the data.

5. Reconstruction:

   • The compressed image can be reconstructed by applying the inverse wavelet transform to the thresholded coefficients. We focus on 1, 2 and 5 in this project. Adding in 3 and 4 (Quantization and Encoding) will further increase the compression capability.

Wavelet Packet Image Compression

Wavelet packet image compression extends wavelet compression by allowing decomposition of both approximation and detail coefficients.

1. Wavelet Packet Transform:

   • Similar to wavelet transform, but both approximation and detail subbands are further decomposed, allowing for a richer analysis of the image frequencies.

2. Thresholding:

   • Coefficients are thresholded to remove less significant information.

3. Entropy-Based Compression:

   • Various entropy measures (e.g., Shannon entropy, log energy entropy) can be used to adaptively select the best basis for representing the image, optimizing the trade-off between compression and image quality.

4. Quantization and Encoding:

   • Quantization and encoding are applied to the thresholded coefficients, similar to basic wavelet compression.

5. Reconstruction:

   • The compressed image is reconstructed by applying the inverse wavelet packet transform. We focus on 1, 2, 3 and 5 in this project. Adding in 4 (Quantization and Encoding) will further increase the compression capability.

Results

you can either:

• run the programs, since I've added menus, you can play around with different parameters.
• see the "Results" folder in my submission.