| BZs >gnuplot.ps Acta Polytechnica Vol. 51 No. 2/2011 Application of Wavelet Transform for Image Denoising of Spatially and Time Variable Astronomical Imaging Systems M. Blažek, E. Anisimova, P. Páta Abstract We report on our efforts to formulate algorithms for image signal processing with the spatially and time variant Point- Spread Function (PSF) and inhomogeneous noise of real imaging systems. In this paper we focus on application of the wavelet transform for denoising of the astronomical images with complicated conditions. They influence above all accuracy of themeasurements and thenew source detection ability. Ouraim is to test theusefulness ofWavelet transform (as the standard image processing technique) for astronomical purposes. Keywords: Point-Spread Function, image processing, IRAF, Daophot, wide-field, noise reduction, wavelet, astronomy. 1 Introduction Our research reflects the needs for effective data pro- cessing from systems with complicated PSF or im- portant role of noise like wide-field cameras or pre- cise astronomical telescopes (e.g. MAIA [1], BART, D50 [2] and BOOTES [3]). A large amount of data is displaced because of big errors during data reduction (e.g. information found on the edges of the image from fish-eye type cameras). Standard photometrical packages (e.g. IRAF [4]) include algorithms for PSF extraction dependent lin- early or quadratically on the position in the image [5]. However, this does not change the analytic part of PSF — only the fitting residua occurring in the look- up table (the experimental part of the constructed PSF) depending on the image cursor, where the con- volution proceeds. This model however does not re- shape the stars at the edges of wide-field images (an example of such an asymmetric PSF is shown in Fig- ure 1), although it is useful in the crowded fields as shown in Figure 2. Fig. 1: Example of the asymmetric stellar Point-Spread Function from the edge of a wide-field camera The formulation of new noise space dependent models (other than simple darkframe) and automatic variant PSF construction is above all useful for wide- field detectors. With the new algorithms, higher pre- cision (sensitivity) and lower data littering can be obtained. All-sky cameras for Gamma-Ray Bursts or meteor detections require optoelectronic analysis specific for each system because of the different de- pendence of the parameters. Automatic data reduction usually does not in- clude time variant PSF because of the slow changes of the external conditions (camera cooling, atmosphere, direction of observation), which become important during long and precise measurement. The parame- ters of PSF photometry (in comparison to aperture photometry) have to be acquired manually. The al- gorithms for evaluating them automatically on series of images will enable faster and more precise pho- tometry if we assume similar and slowly changing conditions of the observations. Typically one-by-one image analysis does not expect the dependence of the PSF parameters on the previous images. Aberration modelling using Zernike polynomi- als [6] can provide better results for space-variant systems, while the Fourier or Wavelet transform can provide guidance for time-variant analysis of PSF. 2 Wavelet transform for astronomic image processing The discrete wavelet transform [7] is widely used in image processing to get rid of noise, while it is not commonly used in astronomy as higher uncertain- ties can occur. We tested the application of this method on stellar field data taken by the robotic tele- 11 Acta Polytechnica Vol. 51 No. 2/2011 Fig. 2: Methods of PSF construction from an over-crowded stellar field (NGC 6791) — significant coma present. The second column represents look-up tables — additive corrections from the analytic function to the observed empirical PSF [5] I. row – analytic function only II. row – constant PSF over the image containing both the analytic function and corrections III. row – look-up tables depending linearly on position IV. row – look-up tables depending quadratically on position scope ‘D50’ at the Ondrejov observatory (CZ). The wavelet transform was proceeded with Matlab1 soft- ware, while IRAF software was used for astronomic processing with the DAOPHOT package2 [5]. We focused on 3 specific targets: • denoising • stellar magnitude changes • the influence of the wavelet transform on the ef- ficiency of the new object detection algorithms Five specific wavelet families were used for the tests: Daubechies, Biorthogonal, Reverse Biorthog- onal, Symlets and Coiflets, with the decomposition level up to 3 [8]. Those families were chosen to cover both orthogonal and biorthogonal wavelet shapes and the most used wavelet families in standard image sig- nal processing. Only soft thresholding [9] was ap- plied, due to its better peak signal to noise ratio for image denoising, and no thresholding was done to 1http://www.mathworks.com/ 2http://iraf.noao.edu/ 12 Acta Polytechnica Vol. 51 No. 2/2011 the approximation coefficients of the wavelet dyadic decomposition. The test image is shown in Fig- ure 3. The efficiency of the wavelet transform on the test image for each wavelet was checked using the DAOPHOT routines [5] under IRAF software package and with standard aperture photometry al- gorithms. Fig. 3: Test image for wavelet transform 2.1 Results of the denoising We observed the changes of the standard deviation of the sky (background) value depending strongly on the decomposition level and the intensities of the nearby stars. The first level of decomposition shows only small variations among the wavelets that were used, lowering the noise dispersion by 40 %–50 %. Even greater efficiency can be gained around faint stars for the second decomposition level with higher differences in the influence between the wavelets. For the third decomposition level, the influence of the denoising on the astronomical image was even nega- tive for several wavelets (e.g. most of the Daubechies and Reversed Biorthogonal wavelets) because of the strong artifacts of the transform (Figure 4). All wavelet denoising results can be seen in Table 1. The first and second column of the table represent the wavelet that was used. The first column for each de- composition level shows the mean percentage lower- ing (effectivity E) of the background noise deviation σ around stars according to the equation E = 1 − σA σB , (1) where the subscripts B and A represent background noise deviation before the wavelet transform applica- tion and after the transform respectively. The second and third column for each decomposition level in Ta- ble 1 show the maximum Δmmax and mean Δm er- rors of the photometric magnitudes after the wavelet transform. The Signal-To-Noise Ratio of the stars selected for the statistics is from 150 to 3 100. 2.2 Magnitude changes Most of the photometrical magnitudes were un- changed or changed only slightly during the wavelet transform, and all of those changes were hidden in- side the 3-σ band of the counted instrumental magni- tudes. From this point of view, the wavelet transform can therefore be considered as a low-loss modification of scientific images with low influence on the photo- metrical information. Detailed errors of the magni- tude changes can be seen in Table 1. 2.3 Detection of new sources The same detection algorithms for transformed im- ages as for the original images were used for the search of new object candidates and for the pair- ing with their counterparts in the astronomical cat- alogues. However, no new sources were found using the DAOPHOT routines [5]. The influence of the wavelets on the Point-Spread Function of the image was unfortunately too strong to get any new infor- mation from the noise level. No advantages for the new source detection algorithms were therefore ob- tained from the wavelet denoising. Future research is therefore suggested. (a) (b) (c) Fig. 4: Effect of the wavelet transform on (a) the original stellar field, (b) the Daubechies 3 wavelet third decomposition level, (c) the Reverse Biorthogonal 3.3 wavelet third decomposition level 13 Acta Polytechnica Vol. 51 No. 2/2011 Table 1: Efficiency E of the denoising of the background deviation of the stellar surroundings (1). Maximum Δmmax and mean Δm error of photometry after application of given Wavelet transform Wavelet Dec. Level 1 Dec. Level 2 Dec. Level 3 E Δmmax Δm E Δmmax Δm E Δmmax Δm [%] [mag] [mag] [%] [mag] [mag] [%] [mag] [mag] Daubechies 1 47 0.026 0.010 71 0.052 0.020 79 0.098 0.031 2 47 0.014 0.004 73 0.030 0.012 55 0.067 0.026 3 47 0.005 0.002 69 0.046 0.014 35 0.102 0.030 4 48 0.008 0.002 60 0.037 0.010 24 0.092 0.029 5 48 0.012 0.002 57 0.020 0.006 22 0.061 0.026 6 48 0.012 0.003 43 0.021 0.008 1 0.094 0.034 7 46 0.011 0.003 39 0.039 0.007 4 0.114 0.032 8 44 0.008 0.002 49 0.013 0.005 10 0.090 0.034 9 45 0.008 0.002 34 0.021 0.007 −7 0.118 0.037 10 45 0.014 0.002 28 0.036 0.008 −10 0.122 0.039 Biorthogonal 1.1 47 0.026 0.010 71 0.052 0.020 79 0.098 0.031 1.3 45 0.007 0.002 72 0.017 0.006 58 0.061 0.016 1.5 43 0.004 0.001 70 0.012 0.002 51 0.044 0.012 2.2 48 0.010 0.004 69 0.060 0.016 71 0.110 0.023 2.4 48 0.006 0.001 70 0.025 0.009 66 0.051 0.017 2.6 48 0.008 0.001 69 0.032 0.007 54 0.055 0.016 2.8 48 0.009 0.002 71 0.014 0.004 56 0.040 0.012 3.1 34 0.012 0.005 45 0.076 0.016 38 0.102 0.018 3.3 45 0.008 0.002 67 0.025 0.007 58 0.028 0.008 3.5 48 0.010 0.002 67 0.037 0.008 53 0.070 0.012 3.7 48 0.011 0.002 68 0.015 0.005 – – – 3.9 48 0.012 0.002 61 0.018 0.005 47 0.033 0.009 4.4 48 0.012 0.002 68 0.025 0.009 37 0.077 0.026 5.5 44 0.009 0.001 64 0.018 0.006 −15 0.080 0.026 6.8 47 0.014 0.003 65 0.018 0.006 11 0.064 0.024 Coiflets 1 48 0.010 0.003 71 0.053 0.014 61 0.099 0.026 2 48 0.010 0.002 66 0.043 0.009 26 0.083 0.027 3 47 0.013 0.003 55 0.039 0.007 15 0.099 0.025 4 47 0.014 0.003 42 0.041 0.007 1 0.090 0.028 5 47 0.014 0.003 30 0.041 0.007 −10 0.105 0.027 Symlets 2 47 0.014 0.004 73 0.030 0.012 55 0.067 0.026 3 47 0.005 0.002 69 0.046 0.014 35 0.102 0.030 4 47 0.008 0.002 66 0.046 0.011 31 0.086 0.026 5 45 0.007 0.002 62 0.045 0.010 23 0.119 0.028 6 47 0.011 0.003 61 0.015 0.005 18 0.060 0.023 7 46 0.015 0.002 51 0.165 0.009 – 0.068 0.028 8 47 0.012 0.003 43 0.045 0.008 2 0.099 0.029 Reverse 1.1 47 0.026 0.010 71 0.052 0.020 79 0.098 0.031 Biorthogonal 1.3 44 0.015 0.004 68 0.054 0.016 45 0.118 0.031 1.5 43 0.006 0.002 65 0.034 0.007 23 0.106 0.021 2.2 51 0.009 0.002 66 0.044 0.010 28 0.087 0.020 2.4 49 0.007 0.002 67 0.018 0.007 11 0.057 0.020 2.6 47 0.010 0.002 57 0.035 0.008 14 0.067 0.022 2.8 47 0.011 0.002 47 0.017 0.005 8 0.063 0.019 3.1 34 0.006 0.002 11 0.019 0.004 −115 0.039 0.009 3.3 45 0.004 0.002 32 0.015 0.004 −85 0.030 0.010 3.5 47 0.006 0.002 36 0.020 0.006 −53 0.057 0.014 3.7 48 0.009 0.002 33 0.015 0.005 −9 0.043 0.011 3.9 48 0.010 0.002 28 0.019 0.006 −9 0.055 0.016 4.4 48 0.009 0.002 67 0.015 0.006 24 0.046 0.019 5.5 48 0.013 0.003 64 0.010 0.003 12 0.063 0.023 14 Acta Polytechnica Vol. 51 No. 2/2011 3 Conclusion Space dependent PSF and noise models are necessary for precise and complete data reduction from imag- ing systems with wide-field lenses. We therefore plan to measure the optoelectronic characteristics and de- scribe the geometric distortion of all-sky detectors to test and formulate new algorithms of variant noise and PSF. The advantage of the denoising algorithms should be namely new discoveries of objects hidden in the noise. Although the background noise dis- tribution was positively modified with most of the wavelet transforms, no advantage for the new source detection algorithms was found yet. In our future effort we’d like to focus as well on the detection al- gorithms of the transformed astronomical images to obtain photometric methods possibly useful in prac- tice. Acknowledgement This project was supported by the Czech Techni- cal University in Prague grant SGS CVUT 2010 – OHK3-066/10. References [1] Vítek, S., Koten, P., Páta, P., Fliegel, K.: Double-station automatic video observation of the meteors, Advances in Astronomy 2010, Ar- ticle ID 943145, 4 pages (2010). [2] Nekola, M. et al: Robotic telescopes for high en- ergy astrophysics in Ondřejov, Experimental As- tronomy, Vol. 28, 2010, Issue 1, p. 79–85. [3] Castro-Tirado, A. J. et al.: The Burst Ob- server and Optical Transient Exploring Sys- tem (BOOTES), A&A Suppl., Vol. 138, 1999, p. 583–585. [4] Massey, P.: A User’s Guide to CCD Reductions with IRAF, NOAO (1997). [5] Stetson, Peter B.: DAOPHOT – A computer pro- gram for crowded-field stellar photometry, As- tronomical Society of the Pacific, Publications, Vol. 99, 1987, p. 191–222. [6] Řeřábek, M., Páta, P., Koten, P.: Processing of the astronomical image data obtained from uwfc optical systems, Image Reconstruction from Incomplete Data V, Proc. SPIE 7076, 70760L (2008). [7] Mallat, S.: A theory for multiresolution signal decomposition: the wavelet representation, IEEE Pattern Anal. and Machine Intell., Vol. 11, no. 7, 1989, pp. 674–693. [8] Daubechies, I.: Ten lectures on wavelets, CBMS- NSF regional conference series in applied math- ematics, Lectures delivered at the CBMS con- ference on wavelets, University of Lowell, Mass., June 1990, Philadelphia: Society for Industrial and Applied Mathematics – SIAM (1992). [9] Donoho, D. L.: De-noising by soft-thresholding, IEEE, Trans. on Inf. Theory, Vol. 41, 3, 1995, pp. 613–627. Martin Blažek E-mail: blazem10@fel.cvut.cz Elena Anisimova Petr Páta Czech Technical University in Prague Faculty of Electrical Engineering Technická 2, Prague 6 15