An Animated Introduction to the Discrete Wavelet Transform

Arne Jensen
Aalborg University

A signal with 8 samples:

56, 40, 8, 24, 48, 48, 40, 16

56		40		8		24		48		48		40		16
	48				16				48				28					8	-8	0	12
			32								38					16	10
							35								-3

• 
  The first row is the original signal
• The next row is composed of
• the mean of samples pairwise

(56 + 40)/2 = 48

• the difference between the the first member of the pair and the computed mean

(56 - 48) = 8

The transform is invertible.

left = mean + difference
left + right = 2 * mean
left - mean = difference
right = mean - difference

We can start from the bottom row: add and subtract the difference to the mean, and repeat the process up to the first row.

Now, we consider to do the threshold filtering about the difference values calculated then re-construct the signal:

• threshold = 4

59		43		11		27		45		45		37		13
	51				19				45				25					8	-8	0	12
			35								35					16	10
							35								0

• threshold = 9

51		51		19		19		45		45		37		13
	51				19				45				25					0	0	0	12
			35								35					16	10
							35								0

Here is now a graphical representation of the results of the thresholding.
Solid line: original signal
Dashed line for thresholding

threshold = 4	threshold = 9