Cameraman image

Let ${\bf x}$ be a $n^2\times 1$ vector created from a square $n\times n$ image. The matrix ${\bf A}$ is constructed in order to produce the 2-dimensional DCT from the input image ${\bf x}$ which is of dimension $N=n^2$. For the input image, we used the “cameraman" image, down-sampled to 128$\times$128, shown in Figure 7.3.

Figure 7.3: Original cameraman image.

For the 128$\times$128 image, matrix ${\bf A}$ and the orthogonal complement matrix ${\bf B}$ are huge, 16384$\times$16384. Luckily, they do not need to be explicitly constructed. Rather, products such as ${\bf A}^\prime {\bf x}$ or ${\bf B}^\prime {\bf x}$ and the derivatives (5.15) may be computed using the 2D DCT and the optimization (5.15) can be accomplished without second derivatives.

Figure 7.4 shows the same image reproduced by inverse DCT of the lower $48\times 48$ DCT coefficients. This image is the pseudo-inverse spectral solution (5.11) and has negative values and values greater than 1, so is not a valid image.

Figure: Cameraman image reconstructed using $48\times 48$ DCT coefficients.

Figure 7.5 shows the MaxEnt solution $\hat{\mbox{\boldmath $\lambda$}}({\bf z}^*)$, to maximizing (7.6) subject to (5.12). Like the image in Figure 7.4, it is feature reproducing, but has values in $(0,1)$. Note the better image characteristics at edges and lines. See software/image_ex2.m.

Figure: Maximum entropy image for doubly-bounded data, for the given $48\times 48$ DCT coefficients.

For comparison purposes, we re-ran the maximization using the positive assumption of section 5, obtaining the solution to the maximization of (5.14) subject to (5.12). This is essentially a maximum entropy 2-D spectrum because the 2-D DCT of the image can be considered the auto-correlation function. It is the formulation used by a significant amount of the literature in image reconstruction [42]. Figure 7.6 shows the resulting image. Although still showing improvement in sharpness over Figure 7.4, we see immediately the effect of having no upper bound on the pixel intensity and increased Gibbs-effect. The result is less pleasing to the eye than Figure 7.5.

Figure: Maximum entropy image for positive (singly-bounded) data, for the given $48\times 48$ DCT coefficients. Same as traditional maximum entropy spectral estimate.