Final Project: Pyramid-Based Texture Analysis and Synthesis

Implementing Oriented Filters

Objective: Create and verify oriented filters, and demonstrate that the sum of all oriented filters for a given band equals a standard Laplacian band-pass filter.

The oriented filters are used to extract direction-specific information from images, which is a critical component of the steerable pyramid. These filters operate in the spatial domain to extract oriented features at various frequency bands. The implementation involves validating the filters by summing them and showing that their combination is equivalent to a Laplacian band-pass filter. To demonstrate their utility, a set of sample images is convolved with these filters, and the resulting direction-specific features are visualized.

Results

Each oriented filter corresponds to high-frequency waves in a specific direction. Adding up the band-pass results of all oriented filters gives the high-frequency waves across all directions, which is equivalent to filtering directly with a Laplacian filter. Although there may be slight differences—caused by an insufficient number of directional filters—if the number of directional filters is sufficiently large, the results should be consistent.

Constructing the Oriented Laplacian Pyramid

Objective: Decompose an image into multiple scales and orientations using the steerable pyramid.

The pyramid construction begins by applying a high-pass filter to compute the high-frequency residual. Next, a low-pass filter is used to compute the low-frequency band. For each scale, the low-frequency band is further decomposed into multiple directional subbands using oriented filters. This process repeats for each scale, progressively downsampling the low-frequency band. The final low-frequency residual is stored as the base of the pyramid.

Results

The first direction corresponds to the vertical (up and down) direction, the second direction corresponds to the bottom-left to top-right direction, the third direction corresponds to the horizontal (left and right) direction, and the fourth direction corresponds to the top-left to bottom-right direction. It can be observed that the filters for each direction highlight edges perpendicular to that direction.

Histogram Matching

Objective: Match the histogram of a noisy image to the source image in both the spatial domain and the pyramid domain.

Histogram matching ensures that the intensity or color distribution of an image matches the source texture. By iteratively matching the frequency space and pixel space of the image, the frequency and pixel distributions of the noise gradually approach those of the texture distribution, achieving the process of texture synthesis.

Results

I also performed histogram matching on each channel of the three-channel image, resulting in the following output. Although it still appears noisy, it now reflects the corresponding colors of the target image and has a similar histogram distribution. The images below show the original noise, the target image, and the matched noise, with histograms representing the grayscale distributions of each.

Texture Synthesis

Objective: Generate textures using the match-texture algorithm.

This step uses the histogram-matched pyramid as a basis. The synthesis begins with a noisy image and progressively adjusts its pyramid representation to match that of the source texture. The process involves reconstructing the image from the adjusted pyramid while maintaining histogram alignment at each level.

To get better representation of the image, we use PCA for image processing and feature extraction. And final results are shown below.

Results