What is being normalized by Keras/TensorFlow BatchNormalization

My question is what is being normalized by BatchNormalization (BN).

I am asking, does BN normalize the channels for each pixel separately or for all the pixels together. And does it do it on a per image basis or on all the channels of the entire batch.

Specifically, BN is operating on X. Say, X.shape = [m,h,w,c]. So with axis=3, it is operating on the "c" dimension which is the number of channels (for rgb) or the number of feature maps.

So lets say the X is an rgb and thus has 3 channels. Does the BN do the following: (this is a simplified version of the BN to discuss the dimensional aspects. I understand that gamma and beta are learned but not concerned with that here.)

For each image=X in m:

1. For each pixel (h,w) take the mean of the associated r, g, & b values.
2. For each pixel (h,w) take the variance of the associated r, g, & b values
3. Do r = (r-mean)/var, g = (g-mean)/var, & b = (b-mean)/var, where r, g, & b are the red, green, & blue channels of X respectively.
4. Then repeat this process for the next image in m,

In keras, the docs for BatchNormalization says:

axis: Integer, the axis that should be normalized (typically the features axis).

For instance, after a Conv2D layer with data_format="channels_first", set axis=1 in BatchNormalization.

But what is it exactly doing along each dimension?