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Patch Match

Some Very Terse Notes on the PatchMatch Randomized Correspondence Algorithm

What is PatchMatch​

At a high level, PatchMatch is an algorithm that efficiently finds approximate nearest neighbor matches between image patches. It leverages randomness to provide fast convergence and has applications which we will conquer in our homeworks, including inpainting, retargeting, and synthesis.

Definitions​

  1. Image Patch: A small, contiguous region of an image, typically square, that captures local structure and texture. Patches are often used to represent and compare local image features.

  2. Nearest Neighbor Field: A mapping from each pixel in an image to its nearest neighbor in another image. In the context of PatchMatch, this mapping is between patches in two images.

  3. Spatial Coherence: The property of natural images where neighboring pixels or patches tend to have similar values. This property is exploited by PatchMatch to guide the search for nearest neighbors. Coherence is defined as:

Coherence=Similarity of neighboring patchesDissimilarity of non-neighboring patches\text{Coherence} = \frac{\text{Similarity of neighboring patches}}{\text{Dissimilarity of non-neighboring patches}}

Theoretical Foundations​

  1. Basic Premise: PatchMatch iterates over image patches, using a random search to guess initial correspondences, then refines these guesses through propagation and random search steps. The algorithm exploits the coherence in natural images, where similar patches tend to cluster spatially. Viewing an image as a collection of overlapping patches introduces a local perspective to global image analysis. The algorithm's reliance on spatial coherence parallels signal processing concepts such as the Wiener filter (might be my week 4 slides), which also assumes neighboring values in a signal (or image) are correlated.

  2. Search Space Reduction: Traditional nearest neighbor search in high-dimensional spaces, as required for comparing image patches, suffers from the curse of dimensionality. PatchMatch mitigates this by focusing the search on likely candidates through a guided random search, leveraging spatial coherence.

  3. Propagation Step: Exploits the assumption that if a patch at location (x,y)(x, y) matches well with a patch at (x′,y′)(x', y'), then neighboring patches are likely to find good matches in the vicinity of (x′,y′)(x', y'). This local smoothness assumption allows for efficient propagation of good matches.

  4. Random Search: Complements the propagation step by randomly sampling the search space around the current best match, exponentially decreasing the search window's size. This step helps escape local minima and improves the chances of finding a global optimum.

  5. Iterative Refinement: The algorithm iteratively applies propagation and random search steps, rapidly converging to a good approximation of the nearest neighbor field. Each iteration refines the match quality, with significant improvements often seen within just a few iterations.

Frequency Domain Analysis

While PatchMatch operates in the spatial domain, its principles resonate with frequency domain methods like the Fourier Transform. Both approaches dissect the image into components (patches or frequency bands) to facilitate analysis and processing.

Motivations, Applications, and Advantages​

  1. Image Inpainting: PatchMatch's ability to find matching patches is crucial for inpainting, where missing or damaged parts of images are filled by borrowing information from similar patches elsewhere in the image.

  2. Content-Aware Resizing: By identifying and manipulating patches, PatchMatch facilitates content-aware resizing, allowing images to be resized without distorting key features.

  3. Computational Efficiency: The introduction of randomness for initial guess improvement, combined with local propagation, enables PatchMatch to achieve results quickly, making it suitable for interactive applications.

PatchMatch Algorithm​

Pseudocode for the PatchMatch Algorithm​

Here's an outline of its core steps, followed by optimization strategies including vectorization and dynamic programming, and a brief analysis of its time and space complexity.

function PatchMatch(imageA, imageB, patchSize)
    Initialize nearestNeighborField (NNF) with random values
    for iteration = 1 to maxIterations do
        // Propagation: Improve guess by comparing with adjacent patches
        if iteration is odd then
            for x = 1 to imageWidth do
                for y = 1 to imageHeight do
                    Propagate(NNF, x, y, patchSize, isOddIteration=true)
                    RandomSearch(NNF, x, y, imageA, imageB, patchSize)
        else
            for x = imageWidth down to 1 do
                for y = imageHeight down to 1 do
                    Propagate(NNF, x, y, patchSize, isOddIteration=false)
                    RandomSearch(NNF, x, y, imageA, imageB, patchSize)
    return NNF
function Propagate(NNF, x, y, patchSize, isOddIteration)
    currentBest = NNF[x, y]
    if isOddIteration then
        left = NNF[x-1, y]
        up = NNF[x, y-1]
        CheckAndUpdateBest(NNF, x, y, left, patchSize)
        CheckAndUpdateBest(NNF, x, y, up, patchSize)
    else
        right = NNF[x+1, y]
        down = NNF[x, y+1]
        CheckAndUpdateBest(NNF, x, y, right, patchSize)
        CheckAndUpdateBest(NNF, x, y, down, patchSize)

function RandomSearch(NNF, x, y, imageA, imageB, patchSize)
    currentBest = NNF[x, y]
    searchRadius = max(imageWidth, imageHeight)
    while searchRadius > 1 do
        randomPatch = ChooseRandomPatch(imageB, searchRadius, currentBest)
        CheckAndUpdateBest(NNF, x, y, randomPatch, patchSize)
        searchRadius = searchRadius / 2

Thought Process and Optimization Opportunities​

  1. Brute-force Approach: This code iterates over every pixel in the target image and compares its patch with every possible patch in the source image. It's a direct but slow method due to the nested loops.

  2. Optimization - Vectorization and Avoid Repetition: The patch_distance computation is ripe for optimization. By using NumPy operations over loops, we can reduce computation time. However, due to the need to compare all pairs of patches, complete vectorization is challenging without significantly increasing memory usage. The brute-force method recalculates distances for overlapping patches. Caching results for previously calculated patches or using integral images to quickly calculate patch sums could reduce redundant calculations.

  3. Dynamic Programming: Can you use Dynamic Programming to speed up the computations? It could be used to propagate good matches to adjacent pixels, which is more efficient. What do you think?

  4. Space Complexity: The NNF array has a size proportional to the target image (O(targeth∗targetw\mathcal{O}(\text{target}_h * \text{target}_w)), storing two integers (coordinates) for each pixel.

  5. Time Complexity: O(targeth∗targetw∗(sourceh−k+1)∗(sourcew−k+1)∗k2)\mathcal{O}(\text{target}_h * \text{target}_w * (\text{source}_h-k+1) * (\text{source}_w-k+1) * k^2) due to the exhaustive search. This highlights the need for the PatchMatch algorithm, which significantly improves the runtime.

Problems​

Problem 1​

What is the time complexity of the brute-force implementation of the PatchMatch algorithm?

  • O(N∗M∗iter∗S)\mathcal{O}(N * M * iter * S)
  • O(N∗M∗S)\mathcal{O}(N * M * S)
  • O(N∗M∗iter)\mathcal{O}(N * M * iter)
  • O(N∗M)\mathcal{O}(N * M)

Problem 2​

Explain how the PatchMatch algorithm works. What are the key differences between the Brute Force approach and the PatchMatch algorithm?

Problem 3​

What is the space complexity of the PatchMatch algorithm?

  • O(N∗M∗iter∗S)\mathcal{O}(N * M * iter * S)
  • O(N∗M∗S)\mathcal{O}(N * M * S)
  • O(N∗M∗iter)\mathcal{O}(N * M * iter)
  • O(N∗M)\mathcal{O}(N * M)

Problem 4​

In the context of the Patch Match algorithm, "coherence" refers to a specific property that significantly influences the algorithm's performance and accuracy. Which of the following best describes the role of coherence in Patch Match?

  • Coherence ensures that the algorithm runs at a constant time complexity regardless of the image size, by enforcing a strict order in which patches are compared.

  • It refers to the likelihood that neighboring pixels in the image space will have similar or closely related best matching patches, which the algorithm exploits to propagate good matches across the image.

  • Coherence is the process by which the algorithm removes all noise from an image before computing the nearest-neighbor field (NNF), ensuring that only coherent (noise-free) patches are analyzed.

  • The term describes the algorithm's ability to maintain temporal consistency in video sequences, ensuring that patch matches do not flicker or change abruptly from one frame to the next.

Problem 5​

Which of the following is a potential optimization strategy for the PatchMatch algorithm?

  • Vectorization
  • Dynamic Programming
  • Parallel Processing
  • All of the above