What's the theory behind computing variance of an image?

Question

I am trying to compute the blurriness of an image by using LaplacianFilter.

According to this article: https://www.pyimagesearch.com/2015/09/07/blur-detection-with-opencv/ I have to compute the variance of the output image. The problem is I don't understand conceptually how do I compute variance of an image.

Every pixel has 4 values for every color channel, therefore I can compute the variance of every channel, but then I get 4 values, or even 16 by computing variance-covariance matrix, but according to the OpenCV example, they have only 1 number.

After computing that number, they just play with the threshold in order to make a binary decision, whether the image is blurry or not.

PS. by no means I am an expert on this topic, therefore my statements can make no sense. If so, please be nice to edit the question.

Answer 1

On sentence description:

The blured image's edge is smoothed, so the variance is small.

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1. How variance is calculated.

The core function of the post is:

def variance_of_laplacian(image):
    # compute the Laplacian of the image and then return the focus
    # measure, which is simply the variance of the Laplacian
    return cv2.Laplacian(image, cv2.CV_64F).var()

As Opencv-Python use numpy.ndarray to represent the image, then we have a look on the numpy.var:

Help on function var in module numpy.core.fromnumeric:

var(a, axis=None, dtype=None, out=None, ddof=0, keepdims=<class 'numpy._globals$
    Compute the variance along the specified axis.

    Returns the variance of the array elements, a measure of the spread of a distribution.
    The variance is computed for the flattened array by default, otherwise over the specified axis.

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2. Using for picture

This to say, the var is calculated on the flatten laplacian image, or the flatted 1-D array.

To calculate variance of array x, it is:

var = mean(abs(x - x.mean())**2)

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For example:

>>> x = np.array([[1, 2], [3, 4]])
>>> x.var()
1.25
>>> np.mean(np.abs(x - x.mean())**2)
1.25

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For the laplacian image, it is edged image. Make images using GaussianBlur with different r, then do laplacian filter on them, and calculate the vars:

The blured image's edge is smoothed, so the variance is little.