Haar wavelet (2x2)

Original picture

Haar transformation

The two-dimensional Haar wavelet is useful for transforming data into 4 bands where b0 contains the DC level of each block while b1-b3 contains higher frequency components.

By recursively applying the transformation on b0 until b0 reaches a size of 1 we get frequency information about different areas in the picture at different scales.

Input block of 2x2 pixels
p0 p1
p2 p3

Forward transformation
b0 = (p0 + p1) + (p2 + p3)
b1 = (p0 - p1) + (p2 - p3)
b2 = (p0 + p1) - (p2 + p3)
b3 = (p0 - p1) - (p2 - p3)

Reverse transformation
p0 = ((b0 + b1) + (b2 + b3)) / 4
p1 = ((b0 - b1) + (b2 - b3)) / 4
p2 = ((b0 + b1) - (b2 + b3)) / 4
p3 = ((b0 - b1) - (b2 - b3)) / 4