why the integral-image contains extra row and column of zeros?

Question

I am learning how to use the integral-images using opencv with Java API, and i created a test that displays the grayscale image before using the integral-image and after using it. the grayscale image is 10 x 10, and when i converted it to the integral-image i found it 11 x 11 with extra rows of zeros and extra column of zeros as shown below in the output.

please let me know why the integral-image contains extra row and column of zeros?

Code:

    public static void main(String[] args) {
    MatFactory matFactory = new MatFactory();
    FilePathUtils.addInputPath(path_Obj);
    Mat bgrMat = matFactory.newMat(FilePathUtils.getInputFileFullPathList().get(0));
    Mat gsImg = SysUtils.rgbToGrayScaleMat(bgrMat);

    Log.D(TAG, "MainClas", "gsImg.dump(): " + gsImg.dump());
    Mat integralMat = new Mat();
    Imgproc.integral(gsImg, integralMat, CvType.CV_32F);

    Log.D(TAG, "MainClas", "sumMat.dump(): " + integralMat.dump());
}

OutPut:

    1: Debug: MainClass -> MainClas: gsImg.dump(): [2, 1, 7, 5, 1, 11, 2, 7, 9, 11;
        1, 2, 0, 0, 3, 20, 17, 5, 7, 8;
        4, 8, 0, 2, 6, 30, 31, 5, 2, 2;
        39, 43, 47, 44, 38, 62, 60, 37, 37, 39;
        27, 29, 52, 52, 47, 75, 67, 59, 58, 63;
        25, 21, 49, 51, 51, 78, 64, 66, 76, 80;
        40, 36, 50, 46, 41, 56, 42, 45, 47, 49;
        13, 17, 20, 15, 9, 20, 15, 19, 12, 11;
        17, 13, 8, 5, 4, 7, 13, 20, 17, 17;
        2, 4, 7, 9, 8, 6, 6, 7, 7, 8]
    2: Debug: MainClass -> MainClas: sumMat.dump(): [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
        0, 2, 3, 10, 15, 16, 27, 29, 36, 45, 56;
        0, 3, 6, 13, 18, 22, 53, 72, 84, 100, 119;
        0, 7, 18, 25, 32, 42, 103, 153, 170, 188, 209;
        0, 46, 100, 154, 205, 253, 376, 486, 540, 595, 655;
        0, 73, 156, 262, 365, 460, 658, 835, 948, 1061, 1184;
        0, 98, 202, 357, 511, 657, 933, 1174, 1353, 1542, 1745;
        0, 138, 278, 483, 683, 870, 1202, 1485, 1709, 1945, 2197;
        0, 151, 308, 533, 748, 944, 1296, 1594, 1837, 2085, 2348;
        0, 168, 338, 571, 791, 991, 1350, 1661, 1924, 2189, 2469;
        0, 170, 344, 584, 813, 1021, 1386, 1703, 1973, 2245, 2533]

Answer 1

That's the intended behavior. Note that the integral image sum(X,Y) in OpenCV is defined as (see documentation here) the sum of pixels in the original image with indices LESS THAN, sum_(x < X, y < Y), not less than or equal to, those of the integral image. Thus sum(0,0) for example will be the sum of zero pixels, defined to be 0. This is also why the resulting sum image has one more row and column than the original.

The reason for this is that it makes it easier to compute sums etc. over blocks of the image and handle them in a uniform way when they include the top and/or left borders.