1.4 Good header table

It is found to be appropriate and convenient to model the distribution of many types of transformed image coefficients byLaplacian distributions. It is appropriate because much real data is approximately modeled by the Laplacian probabilitydensity function (PDF), and it is convenient because the mathematical form of the Laplacian PDF is simple enough to allowsome useful analytical results to be derived.

A Laplacian PDF is a back-to-back pair of exponential decays and is given by:

The mean of this PDF is zero and the variance is given by:

Hence the standard deviation is:

First we analyse the probability of a pel being quantised to each step of the quantiser. This is given by the area under $p(x)$ between each adjacent pair of quantiser thresholds.

• Probability of being at step 0, $p_0=(-(\frac{1}{2}Q)< x< \frac{1}{2}Q)=2(0< x< \frac{1}{2}Q)$
• Probability of being at step $k$ , $p_k=((k-\frac{1}{2})Q< x< (k+\frac{1}{2})Q)$

Now we can calculate the entropy of the subimage:

We can show this in theory by considering the case when $\gg (\frac{x_0}{Q}, 1)$ , when we find that: $$\alpha \approx \frac{Q}{2x_0}$$ $$r\approx 1-\frac{Q}{x_0}\approx 1-2\alpha$$ $$\sqrt{r}\approx 1-\alpha$$ Using the approximation $\log_2(1-\epsilon )\approx -\left(\frac{\epsilon }{\ln 2}\right)$ for small $\epsilon$ , it is then fairly straightforward to show that $$H\approx -\log_2\alpha +\frac{1}{\ln 2}\approx \log_2\left(\frac{2e{x}_0}{Q}\right)$$ We denote this approximation as $H_a$ in , which shows how close to $H$ the approximation is, for $x_0> Q$ (i.e. for $\frac{x_0}{Q}> 0$ dB).

We can compare the entropies calculated using with those that were calculated from the bandpass subimage histograms, as given in these figuresdescribing Haar transform energies and entropies; level 1 energies , level 2 energies , level 3 energies , and level 4 energies . (The Lo-Lo subimages have PDFs which are more uniform and do not fit the Laplacian model well.) The values of $x_0$ are calculated from: $$x_0=\frac{\mathrm{std. dev.}}{\sqrt{2}}=\sqrt{\frac{\mathrm{subimage energy}}{\mathrm{2 (no of pels in subimage)}}}$$ The following table shows this comparison:

We see that the entropies calculated from the energy via the Laplacian PDF method (second column from the right) areapproximately 0.5 bit/pel greater than the entropies measured from the Lenna subimage histograms. This is due to the heaviertails of the actual PDFs compared with the Laplacian exponentially decreasing tails. More accurate entropies can beobtained if $x_0$ is obtained from the mean absolute values of the pels in each subimage. For a Laplacian PDF we can show that