The geometric mean filter is an image filtering process meant to smooth and reduce noise of an image.[1] It is based on the mathematic geometric mean. The output image G(x,y) of a geometric mean is given by

Where S(x,y) is the original image, and the filter mask is m by n pixels.

Each pixel of the output image at point (x,y) is given by the product of the pixels within the geometric mean mask raised to the power of 1/mn. For example, using a mask size of 3 by 3, pixel (x,y) in the output image will be the product of S(x,y) and all 8 of its surrounding pixels raised to the 1/9th power.

Using the following original image with pixel (x,y) at the center:

Gives the result of: (5*16*22*6*3*18*12*3*15)^(1/9) = 8.77.

Application

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The geometric mean filter is most widely used to filter out Gaussian noise. In general it will help smooth the image with less data loss than an arithmetic mean filter.[1]

Code example

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The following code shows the application of a geometric mean filter to an image using MATLAB.

% Applies geometric mean filter to image input_noise that has added gaussian noise

[m, n] = size(input_noise);                  
output = zeros(m, n);                       % output image set with placeholder values of all zeros
val = 1;                                    % variable to hold new pixel value

for i = 2:m-2                               % loop through each pixel in original image
    for j = 2:n-2                           % compute geometric mean of 3x3 window around pixel
        p = input_noise(i - 1, j - 1);
        q = input_noise(i - 1, j);
        r = input_noise(i - 1, j + 1);
        s = input_noise(i, j - 1);
        t = input_noise(i, j);
        u = input_noise(i, j + 1);
        v = input_noise(i + 1, j - 1);
        w = input_noise(i + 1, j);
        x = input_noise(i + 1, j + 1);
        
        val = (p * q * r * s * t * u * v * w * x) ^ (1 / 9);
        output(i, j) = val;                 % set output pixel to computed geometric mean
        val = 1;                            % reset val for next pixel
    end
end
 
Input image with added gaussian noise
 
Output image after filter

References

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  1. ^ a b Gonzalez, Rafael (2002). Digital Image Processing 3nd Edition. Prentice Hall. pp. 232–233. ISBN 0201180758.