WaveletTransformExtensions

小波变换工具

The first DWT was invented by the Hungarian mathematician Alfréd Haar. For an input represented by a list of 2n numbers, the Haar wavelet transform may be considered to simply pair up input values, storing the difference and passing the sum. This process is repeated recursively, pairing up the sums to provide the next scale: finally resulting in 2n-1 differences and one final sum.

Suppose you are given N values

x = (x1, x2, … xN)

where N is even.(X向量的元素的个数必须是偶数)

We take pair-wise average of numbers

sk = (x2k + x2k+1)/2 for k=0, …, N/2 -1

For example,

x = (6, 12, 15, 15, 14, 12, 120, 116) -> s = (9, 15, 13, 118)

We need second list of data d so that the original list x can be recovered from s and d.

For dk (called directed distances), we have:

dk = (x2k - x2k+1)/2 for k=0, …, N/2 -1

The process is invertible since:

sk + dk = (x2k + x2k+1)/2 + (x2k - x2k+1)/2 = x2k

sk - dk = (x2k + x2k+1)/2 - (x2k - x2k+1)/2 = x2k+1

So we map x = (x1, x2, … , xN) to (s | d) = (s1, … , sN/2 | d1, … , dN/2).

Using our example values, we have:

(6, 12, 15, 15, 14, 12, 120, 116) -> (9, 15, 13, 118 | -3, 0, 1, 2)

This process is repeated recursively for s:

(9, 15, 13, 118 | -3, 0, 1, 2) -> (12, 65.5 | -3, -52.5 | -3, 0, 1, 2)

(12, 65.5 | -3, -52.5 | -3, 0, 1, 2) -> (38.75 | -26.75 | -3, -52.5 | -3, 0, 1, 2)

So final result is:

(38.75, -26.75, -3, -52.5, -3, 0, 1, 2)

Why might people prefer the data in this form?

We can identify large changes in the differences portion d of the transform. It is easier to quantize the data in this form. The transform concentrates the information (energy) in the signal in fewer values. And the obvious answer: fewer digits!! In case of images, we need 2D FWT. First, we perform 1D FWT for all rows, and next, for all columns. For color Images, we deal with RGB components of color, and perform Haar Transform for each component separately. Any component (R G B) has values from 0 to 255 to before transformation we scale this values. For displaying image after transformation, we scale back transformed values.