Correlation Implement and compare the performance of the following correlation algorithms: \- sum of squared differences (8.1) \- sum of robust differences (8.2) \- sum of absolute differences (8.3) \- bias-gain compensated squared differences (8.9) \- normalized cross-correlation (8.11) \- windowed versions of the above (8.22-8.23) \- Fourier-based implementations of the above measures (8.18-8.20) \- phase correlation (8.24) \- gradient cross-correlation (Argyriou and Vlachos 2003). Compare a few of your algorithms on different motion sequences with different amounts of noise, exposure variation, occlusion, and frequency variations (e.g., high-frequency textures, such as sand or cloth, and low-frequency images, such as clouds or motion-blurred video). Some datasets with illumination variation and ground truth correspondences (horizontal motion) can be found at http://vision.middlebury.edu/stereo/data/ (the 2005 and 2006 datasets). Some additional ideas, variants, and questions: 1\. When do you think that phase correlation will outperform regular correlation or SSD? Can you show this experimentally or justify it analytically? 2\. For the Fourier-based masked or windowed correlation and sum of squared differences, the results should be the same as the direct implementations. Note that you will have to expand (8.5) into a sum of pairwise correlations, just as in (8.22). (This is part of the exercise.)

Step by step solution

01

Implement Correlation Algorithms

Implement each of the given correlation algorithms. These algorithms include - sum of squared differences, sum of robust differences, sum of absolute differences, bias-gain compensated squared differences, normalized cross-correlation, windowed versions, Fourier-based implementations, phase correlation and gradient cross-correlation. Each algorithm will be implemented separately and tested on example data to ensure it performs correctly.

02

Test and Compare Algorithms

Test each implemented algorithm on different motion sequences. These sequences should vary with respect to factors like noise, exposure variation, occlusions, and frequency variations. Measure and record the performance of each algorithm on these sequences. Analysis will be based on how well they correlate the sequences under these varying conditions.

03

Phase Correlation Analysis

Analyze situations wherein phase correlation might outperform regular correlation or SSD. This will be done through both experimental testing and analytical justification. Experimental testing will involve running the phase correlation algorithm on sequences where it's hypothesized to perform well and noting the results.

04

Fourier-based and SSD Comparison

For the Fourier-based masked or windowed correlation and sum of squared differences, confirm that their results should be the same as the direct implementations. This entails expanding equation (8.5) into a sum of pairwise correlations, similar to equation (8.22), and then using this to check the results.

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