Question

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.)

Short answer

After completing the implementations of the correlation algorithms and testing them based on the specific criteria, it was found that different algorithms performed better under different conditions. For example, the phase correlation outperformed the regular correlation in situations with _______. For the Fourier-based masked or windowed correlation and sum of squared differences, the results were confirmed to be the same as the direct implementations.

Step-by-step solution

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.

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.

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.

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.

Key concepts

Sum of Squared Differences

The Sum of Squared Differences (SSD) is one of the simplest methods used in image processing to measure the similarity between two images.
To calculate SSD, you subtract the pixel values of one image from the corresponding pixel values of another image.
You then square these differences and sum them up across all pixels:

• Let's denote two images as 'I' and 'J'.
• For each pixel with coordinates (x, y), find the difference: \( I(x, y) - J(x, y) \).
• Square the difference: \( (I(x, y) - J(x, y))^2 \).
• Sum all the squared differences: \( \sum_{x,y} (I(x, y) - J(x, y))^2 \).

SSD is sensitive to image alignment and exposure differences. It's best used in noise-free conditions where images are naturally aligned.
Using SSD becomes problematic when there are large regions of occlusion or large shifts in the intended area; noise and different lighting conditions can also affect the results greatly.

Normalized Cross-Correlation

Normalized Cross-Correlation (NCC) is used when we want to remove the effects of differing image brightness and contrast.
It normalizes the pixel intensity to provide a correlation value between -1 and 1:

• A value closer to 1 indicates high similarity.
• A value near -1 indicates inverse similarity.
• A value of 0 means no similarity.

To calculate NCC: - You first subtract the mean from each pixel value to consider only intensity variations that mean something across the images. - Then compute the product of these normalized pixel values across images while accounting for their standard deviations. - Finally, sum these up, providing a measure that's insensitive to uniform brightness shifts.
NCC is often favored for its robustness in practical scenarios, especially where images might have differing exposures or areas with uniform brightness.

Phase Correlation

Phase Correlation focuses on detecting translations between two images using frequency domain analysis.
It leverages the Fourier Transform to analyze shifts:

• First, both images are transformed to the frequency domain using the Fast Fourier Transform (FFT).
• The cross-power spectrum is computed by dividing the conjugate of one FFT by the other.
• Inverse FFT is then applied to this spectrum to locate the peak, indicating the translation.

Phase Correlation is powerful due to its ability to withstand noise and exposure variations, providing accurate displacement measurement.
It generally outperforms methods like SSD when dealing with repetitive patterns or frequency-based changes, which can confuse spatial domain methods.

Fourier-based Implementations

Fourier-based Implementations extend various correlation measures into the frequency domain, where computational power and efficiency can be capitalized upon.
These methods take correlation measures like SSD or cross-correlation and apply them through the lens of Fourier analysis:

• This involves computing the Fourier Transform of the images.
• The essential operations, like multiplication or conjugation, are performed in this frequency space.
• Efficient transforms (such as FFT) significantly accelerate the process.

By using Fourier-based approaches, one can perform operations over larger images with less computational time due to FFT's fast operating ability.
This method is suitable for applications requiring high-speed analysis without losing the core information regarding alignment or similarity, important in real-time video processing or large image datasets.