Efficient algorithm for online Variance over image batches

Question

I have a large amount of images and want to calculate the variance (of each channel) across all of them. I'm having problem finding an efficient (and even correct) algorithm for that.

I read on of the Welford's online algorithm but it is way to slow as it is does not vectorize across a single image or a batch of images.
So I'm wondering how to improve the speed of it to by using vectorization or making use of inbuilt variance algorithms.

Answer 1

These are the two functions needed to update/combine the mean and variances of two batches. Both functions can be used with vectors (the 3 color channels) and the mean and variance can be acquired from inbuilt methods like batch.var().

Equations taken from: https://notmatthancock.github.io/2017/03/23/simple-batch-stat-updates.html

# m amount of samples (or pixels) over all previous badges
# n amount of samples in new incoming batch
# mu1 previous mean
# mu2 mean of current batch
# v1 previous variance
# v2 variance of current batch

def combine_means(mu1, mu2, m, n):
    """
    Updates old mean mu1 from m samples with mean mu2 of n samples.
    Returns the mean of the m+n samples.
    """
    return (m/(m+n))*mu1 + (n/(m+n))*mu2

def combine_vars(v1, v2, mu1, mu2, m, n):
    """
    Updates old variance v1 from m samples with variance v2 of n samples.
    Returns the variance of the m+n samples.
    """
    return (m/(m+n))*v1 + n/(m+n)*v2 + m*n/(m+n)**2 * (mu1 - mu2)**2
    

As you see one can simplify them a bit by reusing some calculations like m+n but keeping it in this imo better to understand version.