Advances in Astronomy

Volume 2015 (2015), Article ID 484379, 8 pages

http://dx.doi.org/10.1155/2015/484379

## The Impact of KLT Coder on the Image Distortion in Astronomy

Faculty of Electrical Engineering, Czech Technical University in Prague, 166 27 Prague, Czech Republic

Received 3 August 2015; Accepted 4 November 2015

Academic Editor: Josep M. Trigo-Rodríguez

Copyright © 2015 Petr Pata. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Presented paper is devoted to the application of Karhunen-Loève transform (KLT) for compression and to study of KLT impact on the image distortion in astronomy. This transform is an optimal fit for images with Gaussian probability density function in order to minimize the root mean square error (RMSE). The main part of the encoder is proposed in relation to statistical image properties. Selected astronomical image processing algorithms are used for the encoder testing. The astrometry and point spread function distortion are selected as the most important criteria. The results are compared with JPEG2000 standard. The KLT encoder provides better results from the RMSE point of view. These results are promising and show the novel approach to the design of lossy image compression algorithms and also suitability for algorithms of image data structuring for retrieving, transfer, and distribution.

#### 1. Introduction

Novel robotic imaging systems acquire huge amount of data during their service [1]. This data is necessary for archiving and distribution among servers and users. Therefore, searching for suitable compression standard is still rather important task [2, 3]. There are many compression standards matched to the human visual system (HVS) for multimedia purposes [4]. These methods are optimized for subjective image quality. The most popular standards are JPEG2000 (based on the wavelet filtering) and classical JPEG with discrete cosine transforms. Multimedia compression algorithms have different quality criteria. The HVS is fundamental for these standards [5]. In particular, they are human visual perception properties such as spatial and temporal resolution, sensitivity to brightness difference, and spectral response. Astronomical images are devoted to completely different purpose and processing tasks. Therefore, this novel standard may not be suitable for an application aimed at human observer. This is typical for astronomical image compression in astronomy, medicine, and other scientific applications [6].

The lossless standards are often preferred for compression in astronomy. The popular algorithm is FITSPRESS developed at the Center for Astrophysics, Harvard. The FITSPRESS algorithm uses Daubechies-4 filters of the wavelet transform and application of the run length encoding and Huffman code [7]. The second standard HCOMPRESS has been developed at the Space Telescope Science Institute (STScI, Baltimore). The HCOMPRESS is used for distributing archive images from Digital Sky Survey DSS1 and DSS2. The Haar transform with blocks of pixels is used for this standard and it is extremely fast [8]. The multiscale approach is also suitable for lossless image compression. This principle is included in the astronomical context coder (ACC) [9]. Lossless algorithms have limited capability of achievable compression ratio (up to 5 : 1) according to noise level in image data and limited redundancy.

The lossy algorithms offer higher compression ratios than lossless approach. The removal of irrelevancy from image can bring an irreversible distortion of compressed image. It has a direct influence on distortion of data and therefore it is necessary to study the distortion nature. The PHOTZIP is a lossy method based on the image function modeling. The compression ratio can be driven by the setting of acceptable astronomical measurement error [10]. Popular modern image compression standards (JPEG2000, JPEG-LS, or JPEGXR) offer also high compression ratios for wide range of image classes including astronomical applications [11, p. 200] [12]. These standards are designed with very good performance for common images [13]. They can take advantage from special properties of selected special image types [14]. These classes include biomedicine, security, and satellite applications. The special effort has been spent for robotic telescope and other systems with large amount of acquired image data.

The Karhunen-Loève transform (KLT) is the integral transform with optimal data reduction properties. The KLT offers the best data fit with root mean square error minimization for image data with Gaussian probability distribution. The KLT has also close relation to PCA or Hotelling transform used in image processing and pattern recognition [15]. The main advantage of this transform is a signal decomposition into optimal decorrelated components in KLT vector space. The KLT could be extended for applications in image processing [16] and components decorrelation could be used in image compression. The optimal KLT bases are signal dependent and it is necessary to construct base system for each image separately for the optimal Gaussian decorrelation. These characteristics give high potential of the KLT for signal and image processing based on the decorrelation. Compression is one of these tasks. The optimal decorrelation is highly efficient for hyperspectral imaging especially [17]. The lossy compression algorithms can be used for large image archives and intelligent management of image data [3, 18].

#### 2. Karhunen-Loève Transform

The Karhunen-Loève transform (KLT) is well-known integral transform for Gaussian signals processing [15]. Let us assume that there is a set of images with Gaussian probability density distribution: where and are column and row indices (i.e., pixel elements). These matrices are elements of vector space with defined scalar productover complex numbers. Then is a mutual energy of both elements. The symbol is used for elements from dual vector space . The relation between and is defined by relation (2). Orthogonality in defined vector space could be written as conditionWhen we can observe for orthonormal elements, then we can find a set of independent elementswhich satisfy condition of the orthonormalityThere, is well-known Kronecker delta. When the set is full, it can cover whole vector space . Then the set is the orthonormal base of the vector space. The projection of the image to this orthonormal base is And inverse transform for the image reconstruction isThe mean value could be obtained aswhere is probability density function (pdf) of the distribution random image. The covariance matrix has elementsWhen the has Gaussian characteristic, the eigenmatrices of the covariance matrices are an optimal base in the vector space in terms of mean square error criterion. The characteristic equation of the covariance matrix is There, are eigenvalues of the covariance matrix and are equal to energy contained in relevant spectral components. The set of base images can be sorted according to size of the eigenvalues . The size of eigenvalues is shown in Figure 1 for a wide field image example. M7-300ff.dat is the image from the BOOTES system (see Figure 2). These images have size 1536 × 1024 pixels with pixel size of *μ*m and 16-bit quantization depth.