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Subsections

2 Image compression methods

2.1 The principle

Numerical image information is simply coded as an array of intensity values, reproducing the geometry of the detectors used for the observation or the densitometer used for plate digitization. The object signal will be stored with noise, background variations, and so on. The relevant information depends on the application domain, and represents what the astronomer wants to study. The information of relevance reflects the limits of the observing instrument and of the digitization process. Reducing the amount of data to be coded will require that the relevant information be selected in the image and that the coding process be reorganized so that we emphasize the relevant information and drop noise and non-meaningful data. For this, we can focus on the region of interest, filter out noise, and quantize coarsely to take into account the limits of our human visual system if the images are only used for browsing.

Furthermore, the usual pixel array representation associated with images stores a lot of redundant information due to correlation of intensity values between nearby pixels and between different scales for large image structures or slowly varying background. A good compression scheme should aim at concentrating on the meaningful information in relation to the scientific purpose of the imaging (survey) project and code it efficiently, thereby limiting as much as possible the redundancy.

For this study, we examined the major available image compression packages, and compared their strategies with respect to these goals.

2.2 Compression packages

Methods used in astronomy include HCOMPRESS (White et al. 1992), FITSPRESS (Press 1992), and JPEG (Furht 1995). These are all based on linear transforms, which in principle help to reduce the redundancy of pixel values in a block and decorrelate spatial frequencies or scales. Two other methods have also been proposed for astronomical image compression: one using mathematical morphology, and another based on the pyramidal median transform (a nonlinear transform). A specific decompression method has also been developed in Bijaoui et al. (1996) in order to reduce artifacts relative to the HCOMPRESS method. In the signal processing domain, two other recent approaches are worthy of mention. The first is based on fractals, and the second uses a bi-orthogonal wavelet transform.

We first briefly review all of these methods, and then compare them in the framework of astronomical images.

HCOMPRESS

HCOMPRESS (White et al. 1992) was developped at Space Telescope Science Institute (STScI, Baltimore), and is commonly used to distribute archive images from the Digital Sky Survey DSS1 and DSS2. It is based on the Haar wavelet transform. The algorithm consists of
1.
applying a Haar wavelet transform to the data,
2.
quantizing the wavelet coefficients linearly as integer values,
3.
applying a quadtree to the quantified value, and
4.
using a Huffman coder.
Sources are available at
http://www.stsci.edu/software/hcompress.html

HCOMPRESS with iterative decompression

Iterative decompression was proposed in Bijaoui et al. (1996) to decompress files which were compressed using HCOMPRESS. The idea is to consider the decompression problem as a restoration problem, and to add constraints on the solution in order to reduce the artifacts.

FITSPRESS

FITSPRESS (Press 1992) uses a threshold on very bright pixels and applies a linear wavelet transform using the Daubechies-4 filters. The wavelet coefficients are thresholded according to a noise threshold, quantized linearly and runlength encoded. This was developed at the Center for Astrophysics, Harvard. Sources are available at ftp://cfata4.harvard.edu/pub/fitspress08.tar.Z.

JPEG

JPEG is the standard video compression software for single frame images (Furht 1995). It decorrelates pixel coefficients within 8 $\times$ 8 pixel blocks using the discrete cosine transform (DCT) and uniform quantization.

Wavelet

Various wavelet packages exist which support image compression, leading to more sophisticated compression methods. The wavelet transform we used is based on a bi-orthogonal wavelet transform (using Antonini-Daubechies 7/9 coefficients) with non-uniform coding (Taubman & Zakhor 1994), and arithmetic encoding. Source code is available at http://www.cs.dartmouth.edu/$\sim$gdavis

Fractal

The image is decomposed into blocks, and each block is represented by a fractal. See Fisher (1994) for more explanation.

Mathematical morphology

This method (Starck et al. 1998), denoted MathMorph in this work, is based on mathematical morphology (erosion and dilation). It consists of detecting structures above a given level, the level being equal to the background plus three times the noise standard deviation. Then, all structures are compressed by using erosion and dilation, followed by quadtree and Huffman coding. This method relies on a first step of object detection, and leads to high compression ratios if the image does not contain a lot of information, as is often the case in astronomy.

Pyramidal median transform

The principle of this compression method (Starck et al. 1996; Starck et al. 1998), denoted PMT here, is to select the information we want to keep, by using the pyramidal median transform, and to code this information without any loss. Thus the first phase searches for the minimum set of quantized multiresolution coefficients which produce an image of "high quality''. The quality is evidently subjective, and we will define by this term an image such as the following: Lost information cannot be recovered, so if we do not accept any loss, we have to compress what we take as noise too, and the compression ratio will be low (3 or 4 only).

2.3 Remarks on these methods

The pyramidal median transform (PMT) is similar to the mathematical morphology (MathMorph) method in the sense that both try to understand what is represented in the image, and to compress only what is considered as significant. The PMT uses a multiresolution approach, which allows more powerful separation of signal and noise. The latter two methods are both implemented in the MR/1 package (see http://ourworld.compuserve.com/homepages/multires).


  
Table 1: Description and comparison of the different steps in the compression packages tested

\begin{tabular}[t]
{\vert c\vert c\vert c\vert c\vert c\vert c\vert}
\hline
 Sof...
 .... Morph. & Dilation &Noise estimation & -- & bitplanes & \\  \hline\end{tabular}

Each of these methods belongs to a general scheme where the following steps can be distinguished:

1.
Decorrelation of pixel values inside a block, between wavelength, scales or shape, using orthogonal or nonlinear transforms.
2.
Selection and quantization of relevant coefficients.
3.
Coding improvement: geometrical redundancy reduction of the coefficients, using the fact that pixels are contiguous in an array.
4.
Reducing the statistical redundancy of the code.
How each method realizes these different steps is indicated in Table 1.

Clearly these methods combine many strategies to reduce geometrical and statistical redundancy. The best results are obtained if appropriate selection of relevant information has been performed before applying these schemes.

For astronomical images, bright or extended objects are sought, as well as faint structures, all showing good spatial correlation of pixel values and within a wide range of greylevels. Noise background, on the contrary, shows no spatial correlation and fewer greylevels. The removal of noisy background helps in regard to data compression of course. This can be done with filtering, greylevel thresholding, or coarse quantization of background pixels. This is used by FITSPRESS, PMT and MathMorph which divide information into a noise part, estimated as a Gaussian process, and a highly correlated signal part. MathMorph simply thresholds the background noise estimated by a 3-sigma clipping, and quantizes the signal as a multiple of sigma (Huang & Bijaoui 1991). FITSPRESS thresholds background pixels and allows for coarse background reconstruction, but also keeps the highest pixel values in a separate list. PMT uses a multiscale noise filtering and selection approach based on noise standard deviation estimation. JPEG and HCOMPRESS do not carry out noise separation before the transform stage.

Identifying the information loss

Apart from signal-to-noise discrimination, information losses may appear after the transforms at two steps: coefficient selection and coefficient quantization. The interpretable resolution of the decompressed images clearly depends upon these two steps.

If the spectral bandwidth is limited, then the more it is shortened, the better the compression rate. The coefficients generally associated with the high spatial frequencies related to small structures (point objects) may be suppressed and lost. Quantization also introduces information loss, but can be optimized using a Lloyd-Max quantizer for example (Proakis 1995).

All other steps, shown in Table 1, such as reorganizing the quantized coefficients, hierarchical and statistical redundancy coding, and so on, will not compromise data integrity. This statement can be made for all packages. The main improvement clearly comes from an appropriate noise/signal discrimination and the choice of a transform appropriate to the objects' signal properties.

Identifing the needs for compression

Following the kind of images and the application needs, different strategies can be used:
1.
Lossy compression: in this case the compression ratio is relatively low (< 5).
2.
Compression without visual loss. This means that one cannot see the difference between the original image and the decompressed one. Generally, compression ratios between 10 and 20 can be obtained.
3.
Good quality compression: the decompressed image does not contain any artifact, but some information is lost. Compression ratios up to 40 can be obtained in this case.
4.
Fixed compression ratio: for some technical reasons, one may decide to compress all images with a compression ratio higher than a given value, whatever the effect on the decompressed image quality.
5.
Signal-to-noise separation: if noise is present in the data, noise modelling can allow very high compression ratios just by including some type of filtering in the wavelet space during the compression.
Following the image types, and the selected strategy, the optimal compression method may vary. The main interest in using a multiresolution framework is to avail of progressive information transfer and visualization.


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