Making a Panoramic Digital Image of the Entire Northern Sky Anne M. Rajala anne2006@caltech.edu, x1221, MSC #775 Mentors: Ashish Mahabal and S.G. Djorgovski October 3, 2003 Abstract The Digitized Palomar Observatory Sky Survey consists of approximately 2700 images in three filters that comprise the entire northern sky. Overlaps allow one to match the images and mosaic them together to form one smooth image for the northern hemisphere, providing a wealth of scientific and educational opportunities requiring larger areas of sky. However, the images are not uniform; certain gradients exist as a result of observational conditions and imperfections in the photographic plates used. Simply lining up the stars in the overlaps to put the images together will not produce a smooth result. The edges will be visible and the effect will be of many tiles rather than a single homogeneous image. Thus, the backgrounds must be properly fit and removed. A surface can be fit to each image and divided out to flatten the background. Since each image has a different combination of gradients affecting it, the correct type of surface must be determined first, checked, and applied. Automated scripts will then flatten all the images, allowing the creation of a uniform mosaicked panorama. 1 Introduction The Digitized Palomar Observatory Sky Survey 1 plays host to approximately 2700 digital images comprising the northern hemisphere. Each 1GB image is a 6.5 x 6.5 section of the sky, taken in one of three visible wavelengths at one arcsecond resolution. Because each plate shares a 1.5 overlap with its neighboring plates, it is possible to match them up and mosaic them together to form one smooth panoramic image of the northern hemisphere. This would enable larger sections of the sky to be examined at once, providing a wealth of scientific and educational opportunities. Currently available images in the public domain are usually only one-half to one degree in size. Vignetting is an aberration produced when taking an image of a square field from a round eyepiece. This and other standard effects have been corrected 2 but more corrections need to take place. If the images are not all uniform, then placing them next to each other will not produce one smooth image - the edges will be visible and the effect will be that of many small images rather than a single large image. 1
2 Background Subtraction Certain gradients appear in the background of each image. Some result from imperfections in the photographic plates themselves, others from the amount of moonlight on the nights of the original imaging. Since these conditions vary from night to night and thus from plate to plate, this background must be modeled and removed from each image in order to be able to smoothly transition from one plate to another. Creating, fine-tuning, and carrying out this subtraction process is the primary aim of this project. The Image Reduction and Analysis Facility (IRAF) software 3 provides some tasks that can be utilized sequentially to carry out such processes. Once these have been tested and finalized and all the parameters ascertained, it is necessary to write a script that will analyze an original plate and process it completely in an automated fashion, such that 2700 plates need not be done manually. For a program or a script to complete the processing of a plate, several things have to happen. It needs to look at the plate and determine what order and what type of polynomial is best fit to the surface in both the x- and y-directions (clipping the edges affected by vignetting). It should then take that information and fit such a surface to the image, which then needs to be factored out from the original image. 3 Experimental Phase The way to fit a polynomial to a surface is to try a large number of them and then run tests on them all to see which is best. In principle, a script should be able to do this but it has not yet been written. First, statistical criteria must be ascertained to determine which fit is best. The images were averaged down into one single line and one single column and polynomials were fit and removed from the x- and y-directions respectively. Linear spline, cubic spline, legendre and chebyshev polynomials were tested, each with orders from one to five. Each plate then had 20 fitted polynomials for each direction. Statistics were generated from these images to try to match characteristics of the images to statistical traits. The residual image (the difference between the original image and the polynomial fitted surface) with the best fit will have pixel values closest to zero at all points. The mean and median pixel values should show some indication of this, but finding the ones closest to zero aren t always the right fits. Bright objects such as stars or galaxies in the plate offset the mean pixel value more significantly than the median, so the median is more important to look at. However, if a residual plate has a large bright spot and a large dark spot in different corners, it will not affect the median pixel value enough to flag the fit as a poor one. Therefore, it is also important to have a small local standard deviation in the fitted surface. Formulating the specific statistical criteria for a good fit has not been completed and may not be trivial due to individual imperfections in each original photographic plate and the extremely large scale of these images. Note that these plots show the mean and median pixel values are often negative numbers. This is due to a constant being oversubtracted from the images. 2
Figure 1: Plot of mean pixel values versus median pixel values for the residuals of the different polynomial fits for the x-direction of f721. The size of the symbol indicates the order of the polynomial and the symbol indicates the type of polynomial used (asterisk for legendre, circle for chebyshev, x for linear spline, and triangle for cubic spline). The numbers next to the symbols are the standard deviations. Figure 2: Plot of mean pixel values versus median pixel values for the residuals of the different polynomial fits for the y-direction of f721. The size of the symbol indicates the order of the polynomial and the symbol indicates the type of polynomial used (asterisk for legendre, circle for chebyshev, x for linear spline, and triangle for cubic spline). The numbers next to the symbols are the standard deviations. From the plots in figures one and two, equations were chosen to process f721 in two ways: the first with equations that have the mean and median closest to zero (fifth order legendre in the x-direction and fourth order legendre in the y-direction), and the second with equations that have the mean and median farthest from zero (third order legendre in both x- and y-directions). The plates processed in this way showed that the first resultant plate (with the statistics that should 3
indicate a better fit) was flatter than the second. However, it is not a very significant difference and it is possible that other options with statistics at a comparable distance from the origin to the first plate might be better. Besides looking at the numerical statistics, the flatness was also visually confirmed in this phase. These polynomials were fit with the parameter of rejecting the top 1% of the brightest pixels. This way, the fit ignores the stars or other bright objects that might be in the plate, and only fits the surface to the background. This causes greater accuracy and results in a flatter plate. Ignoring the top 5% or 10% does not improve the fit further. There are methods to create masks to meticulously remove all bright objects before fitting surfaces, but that was not attempted during this project. The following are plots of pixel values of five rows averaged together across one horizontal section of the image. Note that the processed image (Fig. 4) has more consistent background pixel values across the entire image than the original plot (Fig. 3). Figure 3: Plot of pixel values across a cross-section of unprocessed image. The slopes on both edges are indicative of a lack of flatness in the image s background. The standard deviation of these pixels is 260.3, with a median of 4845. The discrepancy is clearly significant because it rises more than two sigmas on each side. This process has been run on nine plates (721-3, 793-5, 865-7 in the f filter that form an approximately 250 degree mosaic). However, in principle a script can work the same way for each of the other plates and may be run in batch mode to complete the set. 4
Figure 4: Plot of pixel values across a cross-section of a processed image. Note that the graph stays constant across the image. The standard deviation of these pixels is 156.1 (with a median of -12.88), considerably lower than the unprocessed image. 4 Mosaicking the Images To mosaic the images, each pixel must be mapped to its coordinates in the sky, and then mapped to a pixel value in the panoramic image. This was done as part of the yoursky project 4 already. For this project, however, algorithms need to also be implemented to create a smooth transition between the plates. If a plate s pixels are being mapped on to the final canvas in a place that overlaps with a previously mapped plate, it should not simply overwrite the existing pixels. Methods will need to be established to determine which pixel is of better quality and to place the preferable one in the final image, or perhaps to average the overlapping pixels. The set of nine plates was run through this code to be mosaicked into a 3x3 block. For comparison, the original unprocessed plates were used as well. 5 Future Possibilities Gal 5 have developed an algorithm in the catalog domain to determine the background slopes in four overlap regions with the adjacent plates. Slopes created from paths going through several plates are weighted according to the length of path. These slopes are then calculated and applied to the 5
Figure 5: Nine original plates (f721-3, f793-5, f865-7) tiled together. Note how the edges are clearly visible, indicating that the backgrounds are not flat. The standard deviation of these pixels is 2352, with a median of 4588. Figure 6: Original nine plates tiled together with a background matching scheme which has a constant additive offset added the images. The standard deviation of these pixels is 216.8, with a median of 76.46. The smaller standard deviation indicates that the background matching option is helpful to the flatness of the plates. However, it does not lower the standard deviation as much as the pre-processing of the plates does, and does not remove the entire background. 6
Figure 7: Same nine plates flattened with 1% pixel rejection and tiled together. The white and black spots result from errors in the polynomial fitting tasks, and occur most often on the right and top edges of a plate. A more tailored mosaicking code would remove more pixels from those edges and less from the left and bottom edges of plates to get a smoother image. The standard devation of these pixels is 80.43 (with a median of -6.139), much lower than the unprocessed images mosaic. Figure 8: Same nine plates flattened with the same 1% pixel rejection and tiled together with the background matching scheme. The standard deviation of these pixels is 122.7 (with a median of 120.1), which is comparable to the processed images without background matching (Fig. 6). plate to yield a smooth transition at the edges. Applying this algorithm to the image domain is more challenging but may be useful in creating a smooth panorama. Currently, Montage 6, another code, is being developed that will have the capability to mosaic 7
together images, which could be a good tool to use after the images have been flattened. At the moment it is using 2MASS images, which are significantly smaller than the DPOSS images. References [1] http://www.astro.caltech.edu/ george/dposs/dposs.html [2] Mahabal, A., et al. Serving the Sky, Virtual Observatories of the Future, R.J. Brunner, S.G. Djorgovski, and A.S. Szalay, eds., Astronomical Society of the Pacific Conference Series, Vol. 225, pp. 193-196, 2001. [3] www.iraf.noao.edu [4] Jacob, J.C., et al. yoursky: Rapid Desktop Access to Custom Astronomical Image Mosaics, Virtual Observatories, A.S. Szalay, ed., Proceedings of SPIE, Vol. 4846, pp. 53-64, 2002. [5] Gal, R. R., PhD Thesis. The Northern Sky Optical Cluster Survay: Galaxy Clusters From 5000 Square Degrees of DPOSS, 2001. [6] http://montage-stage.ipac.caltech.edu 6 Acknowledgements Many special thanks to Ashish Mahabal and S. George Djorgovski, who provided countless hours of assistance in putting together and guiding this project. Thanks also to Joe Jacob of the Jet Propulsion Lab who helped mosaic the images. 8