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The process for using K-means clustering algorithm to convert a grayscale image into a binary image in C++ would involve the following steps:

1. Load the grayscale image into a matrix or array.
2. Define the number of clusters (K) to use in the K-means algorithm.
3. Initialize K centroids randomly.
4. Assign each pixel in the grayscale image to the nearest centroid.
5. Compute the mean of each cluster and move the centroid to the mean.
6. Repeat steps 4 and 5 until the centroids converge.
7. Threshold the pixel values in the image based on the centroid assignments to create a binary image.

Here is some example code in C++ that demonstrates this process using the OpenCV library:

#include <opencv2/opencv.hpp>

using namespace cv;

int main(int argc, char** argv)
{
    // Load grayscale image
    Mat image = imread("grayscale_image.jpg", IMREAD_GRAYSCALE);

    // Set number of clusters
    int K = 2;

    // Initialize K-means algorithm
    Mat labels, centers;
    kmeans(image.reshape(1, image.rows * image.cols), K, labels, TermCriteria(TermCriteria::EPS + TermCriteria::COUNT, 10, 1.0), 3, KMEANS_PP_CENTERS, centers);

    // Create binary image from centroid assignments
    Mat binary_image = Mat::zeros(image.size(), CV_8UC1);
    for (int i = 0; i < image.rows; i++) {
        for (int j = 0; j < image.cols; j++) {
            int label = labels.at<int>(i * image.cols + j);
            if (label == 1) {
                binary_image.at<uchar>(i, j) = 255;
            }
        }
    }

    // Display binary image
    imshow("Binary Image", binary_image);
    waitKey(0);

    return 0;
}

Note that this code assumes the input grayscale image is a JPEG file named "grayscale_image.jpg" in the same directory as the executable. Also, the number of clusters (K) is set to 2 for simplicity, but this value can be adjusted depending on the specific image and application.