1.2 Image characteristics

We now consider statistical characteristics of typical images which can permit compression. If all images comprised dots withuncorrelated random intensities, then each pel would need to be coded independently and we could not achieve any usefulgains. However typical images are very different from random dot patterns and significant compression gains are possible.

Some compression can be achieved even if no additional distortion is permitted ( lossless coding ) but much greater compression is possible if some additional distortion isallowed ( lossy coding ). Lossy coding is the main topic of this course but we try to keep the added distortionsnear or below the human visual sensitivity thresholds discussed previously.

Statistical characteristics of signals can often be most readily appreciated by frequency domain analysis since the powerspectrum is the Fourier transform of the autocorrelation function. The 2-D FFT is a convenient tool for analysingimages. shows the $256\times 256$ pel 'Lenna' image and its Fourier log power spectrum. Zero frequency is at the centre of the spectrum imageand the log scale shows the lower spectral components much more clearly.

The bright region near the centre of the spectrum shows that the main concentration of image energy is at low frequencies, whichimplies strong correlation between nearby pels and is typical of real-world images. The diagonal line of spectral energy at about-30°is due to the strong diagonal edges of the hat normal to this direction. Similarly the near-horizontal spectral linecomes from the strong near-vertical stripe of hair to the right of the face. Any other features are difficult to distinguish inthis global spectrum.

A key property of real-world images is that their statistics arenot stationary over the image. demonstrates this by splitting the 'Lenna' image into 64 blocks of $32\times 32$ pels, and calculating the Fourier log power spectrum of each block. The wide variation in spectra is clearlyseen. Blocks with dominant edge directions produce spectra with lines normal to the edges, and those containing the feathers ofthe hat generate a broad spread of energy at all frequencies. However a bright centre, indicating dominant lowfrequency components, is common to all blocks.

We conclude that in many regions of a typical image, most of the signal energy is contained in a relatively small number ofspectral components, many of which are at low frequencies. However, between regions, the location of the maincomponents changes significantly.

The concentration of spectral energy is the key to compression. If a signal can be reconstructed from its Fouriertransform, and many of the transform coefficients are very small, then a close approximation to the original can bereconstructed from just the larger transform coefficients, so only these coefficients need be transmitted .

In practice, the Fourier transform is not very suitable for compression because it generates complex coefficients and it isbadly affected by discontinuities at block boundaries (half-sine windowing was used in and to reduce boundary effects but this would prevent proper reconstruction of the image). In further discussion , we demonstrate the principles of image compression using the Haartransform, perhaps the simplest of all transforms.