**La trasformata H (H-transform)**

La tecnica di compressione delle immagini che e' stata usata e' basata sulla H-transform (Fritze et al. 1977; vedi anche Richter 1978 e Capaccioli et al. 1988). La H-transform e' una genaralizzazione bidimensionale della trasformata di Haar (Haar1910). La H-transform e' calcolata per un'immagine di dimensione 2^N x 2^N come segue:

- Dividi l'immagine in blocchi di 2 x 2 pixels. Chiama i 4 pixels di un blocco A00, A10, A01 e A11.
- Per ogni blocco calcola 4 coefficienti:
h0 = (A11 + A10 + A01 + A00) / 2,

hx = (A11 + A10 - A01 - A00) / 2,

hy = (A11 - A10 + A01 - A00) / 2,

hc = (A11 - A10 - A01 + A00) / 2.

- Costruisci un'immagine di 2^(N - 1) x 2^(N - 1) dal valore di h0 per ogni blocco di 2 x 2. Dividi questa immagine in blocchi di 2 x 2 e ripeti il calcolo precedente. Ripeti il processo N volte, riducendo la dimensione dell'immagine di un fattore 2 per ogni passo, finche' rimane un solo valore h0..

If the image is nearly noiseless, the H-transform is somewhat easier to compress than the original image because the differences of adjacent pixels (as computed in the H-transform) tend to be smaller than the original pixel values for smooth images. Consequently, fewer bits are required to store the values of the H-transform coefficients than are required for the original image. For very smooth images the pixel values may be constant over large regions, leading to transform coefficients which are zero over large areas.

Noisy images still do not compress well when transformed, though. Suppose there is noise sigma in each pixel of the original image. Then from simple propagation of errors, the noise in each of the H-transform coefficients is also sigma. To compress noisy images, divide each coefficient by S sigma, where S ~1 is chosen according to how much loss is acceptable. This reduces the noise inthe transform to 0.5 / S, so that large portions of the transform are zero (ornearly zero) and the transform is highlycompressible.

Why is this better than simply thresholding the original image? If we simply divide the image by sigma then we lose all information on objects that are within 1 sigma of sky in a single pixel, but that are detectable by averaging a block of pixels. On the other hand, in dividing the H-transform by sigma, we preserve the information on any object that is detectable by summing a block of pixels! The quantized H-transform preserves the mean of the image forevery block of pixels having a mean significantly different than that of neighboring blocks of pixels.

**Coding the H-transform**

The quantized H-transform has a rather peculiar structure. Not only are large areas of the transform image zero, but the non-zero values are concentrated strongly in the lower-order coefficients. The best approach we have found to code the coefficient values efficiently is quadtree coding of each bitplane of the transform array. Quadtree coding has been used for many purposes (see Samet 1984 for a review); the particular form we are using was suggested by Huang and Bijaoui (1991) for image compression.

- Divide the bitplane up into 4 quadrants. For each quadrant code a `1' if there are any 1-bits in the quadrant, else code a `0'.
- Subdivide each quadrant which is not all zero into 4 more pieces and code them similarly. Continue until one is down to the level of individual pixels.

We follow the quadtree coding with a fixed Huffman coding that uses 3 bits for quadtree values which are common (eg. 0001, 0010, 0100, and 1000) and uses 4 or 5 bits for less common values. This reduces the final compressed file size by about 10% at little computational cost. Slightly better compression can beachieved by following quadtree coding with arithmetic coding (Witten, Neal, andCleary 1987), but the CPU costs of arithmetic coding are not, in our view,justified for 3-4% better compression.

For completely random bitplanes, quadtree coding can actually use more storage than simply writing the bitplane directly; in that case we just dump the bitplane with no coding.

** Astrometric and Photometric Properties of Compressed Image**

We have conducted experiments to study the degradation of astrometry and photometry on the compressed images compared to the original images (White,Postman, and Lattanzi 1992). Even the most highly compressed images have very good photometric properties for both point sources and extended sources; indeed,photometry of extended objects can be improved by the adaptive filtering of the H-transform (Cappacioli et al. 1988). Astrometry is hardly affected by the compression for modest compression factors (up to about a factor of 20 for our digitized Schmidt plates), but does begin to degrade for images with higher compression factors.