Comparison with Other Orthogonal Transforms

Here we compare KLT with other orthogonal transforms such as identity transform I (no transform), Walsh-Hadamard transform WHT, discrete cosine transform DCT and discrete Fourier transform DFT. Each row of an image clouds (left panel in figure below) is used as a 1D signal $X$ which is repeated as many times as the number of rows in the image. Different orthogonal transforms $Y=A^TX$ are applied to the signal $X$ and the corresponding covariance matrix $\Sigma_Y$ are obtained and compared to see how well they decorrelate the signal and compact the energy. The figure shows the image and three covariance matrices under transforms I (identify), DCT (DFT and WHT have very similar results to DCT) and PCT:

This figure shows the energy distribution among all elements. The flat curve is the original energy distribution (no transform), the most compact (high on the left, low on the right) is the distribution after PCT. The intermediate ones are by DCT and WHT with similar effects. Note that a conversion of $y=x^{0.3}$ has applied to the intensity of the covariance matrices for the weak values to be still visible.

This table shows the number of elements needed to keep certain percentage of the total dynamic energy (information). It is seen that DCT's performance is very close to that of the optimal PCT.

Percentage:	90	95	99	100
I:	209	230	250	256 (all)
DCT:	10	22	97	256 (all)
PCT:	7	13	55	256 (all)

The orthogonal transforms' properties of signal decorrelation and energy compaction are based on the assumption that the signals are in general continuous due to the nature of most of the physical processes. Given the current value of a certain physical variable, its next value can be predicted with reasonable confidence to be close to the current one. However, in some situations where this assumption is not true, orthogonal transforms such as DCT may increase the signal correlation. The image below (left) is the texture of sand, whose pixels are not correlated. The covariance matrix before (2nd to the left) and after DCT (2rd to the right) and PCT (right) are shown here:

In this case, DCT has increased the signal correlation and made no improvement to energy compaction (same distribution as the original), while the PCT can still completely decorrelate the signal and optimally compact the energy distribution.