K-Means Color Segmentation
Given a list of pixel intensities and K clusters, perform K-means clustering for exactly 10 iterations.
Initialization: Pick K evenly spaced values from the sorted unique intensities. If there are N unique values, select indices at positions round(i×(N−1)/(K−1)) for i=0,1,...,K−1. If N≤K, use all unique values (pad with the last value if needed). For K=1, use the middle unique value.
Algorithm (repeat 10 times):
- Assign each pixel to the nearest center (lowest index breaks ties)
- Update each center to the mean of its assigned pixels (keep old center if no pixels assigned)
Return the list of cluster assignments (0-indexed).
Example:
pixels = [1, 2, 3, 100, 101, 102] K = 2
[0, 0, 0, 1, 1, 1]
- The unique intensities are sorted to get [1, 2, 3, 100, 101, 102]. With K=2, we select two evenly spaced values. The indices are calculated as round(0×(5)/(2−1))=0 and round(1×(5)/(2−1))=5, so the initial centers are [1, 102].
- We repeat the K-means algorithm for 10 iterations. In each iteration, pixels are assigned to the nearest center. For the first iteration, pixels [1, 2, 3] are assigned to center 1 (index 0) and pixels [100, 101, 102] are assigned to center 2 (index 1).
- After the first iteration, the centers are updated to the mean of their assigned pixels. The new centers become ((1+2+3)/3,(100+101+102)/3)=(2,101).
- The algorithm continues for 9 more iterations, but the assignments and centers will not change significantly, resulting in the final cluster assignments: [0, 0, 0, 1, 1, 1]
Constraints:
- pixels is a list of numeric intensities
- K >= 1
- Run exactly 10 iterations
- Return list of integer assignments (0 to K-1)
Background Knowledge
K-Means clustering is an unsupervised machine learning algorithm used to group similar data points into clusters based on their features. In the context of image segmentation, it can be applied to pixel intensities to segment an image into regions of similar color. The algorithm iteratively updates the centroids (or centers) of the clusters and reassigns the data points to the nearest centroid. The goal is to minimize the sum of the squared distances between each data point and its assigned centroid.
The initialization step is crucial in K-Means clustering, as it can affect the final result. The problem description provides a specific initialization method, which involves selecting K evenly spaced values from the sorted unique intensities. This approach helps to distribute the initial centroids across the range of intensities, increasing the chances of converging to a good solution. Understanding the concept of centroids, distance metrics (e.g., Euclidean distance), and the importance of initialization is essential for tackling this problem.
In addition to K-Means clustering, it's helpful to have a basic understanding of image representation, where each pixel is associated with an intensity value. The problem focuses on grayscale images, where each pixel has a single intensity value. However, the concepts learned here can be extended to color images, where each pixel has multiple color channel values (e.g., RGB).
Algorithm/Approach
The K-Means clustering algorithm follows a pattern of:
- Initialization: Set the initial centroids.
- Assignment: Assign each data point to the nearest centroid.
- Update: Update the centroids based on the assigned data points. This pattern is repeated for a specified number of iterations or until convergence.
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