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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 13, NO. 7, JULY 2004 867Holographic Image Representation WithReduced Aliasing and Noise EffectsRoman DovgardAbstract—This paper presents a novel holographic imagerepresentation which has a better reconstruction quality aftercropping the hologram than any known representation today.Reconstructed images are very close to ideally low-pass filteredversions of the original image, almost without aliasing. We usethe same test images which were used by Bruckstein et al. in [1],and show empirically that our method has smaller reconstructionerrors, on all of them, than the errors produced by subsamplingand Fourier transform methods presented in [1].Index Terms—Aliasing, fast Fourier transform (FFT), holog-raphy, holographic image representation, inverse indexing,prefiltering, staircase low-pass filter.I. I NTRODUCTIONHOLOGRAPHY is a physical process of “lensless photog-raphy,” where an interference pattern of two laser beamstargeted onto a scene is captured at a film. Every region in thehologram encodes information about the whole scene, makingit possible to reconstruct the scene even from a small portionsof the hologram placed at arbitrary locations in the hologram.Quality of the reconstruction is independent from the locationof a region, depending only on its size.Holographic phenomenon has some impact on image pro-cessing (e.g., see [1]–[5]). We concentrate in this paper onholographic image representations, which were introduced byBruckstein et al. in [1]. Given an image of size ,we attempt to find its holographic representation , s.t. givenany cropped region of we can reconstruct some distortedversion of the whole image . This typically means that everyregion in [segment if is a one-dimensional (1-D) signalor rectangle if is a two-dimensional (2-D) signal] containsinformation about the whole image . Reconstruction qualityshould depend solely on region’s size, but not on its location.This can be accomplished by some smart sampling of or byusing some convolutional and Fourier transform methods. Inthis paper, we combine these two strategies and produce 2-Dhologram s.t. reconstruction of from cropped regions ofhas a better quality than in the previous approaches [1].Holographic image representations were shown, by Bruck-stein, et al. [2] to be useful for compression of images ina holographic way that enables decompression from multiplecompressed versions available in a distributed environment,with decompression quality better than in any one of the indi-Manuscript received May 6, 2002; revised October 31, 2003. The associateeditor coordinating the review of this manuscript and approving it for publica-tion was Dr. Charles D. Creusere.The author is with the Department of Applied Mathematics and ComputerScience, Weizmann Institute of Science (WIS), Rehovot, Israel (e-mail:romad@netvision.net.il).Digital Object Identifier 10.1109/TIP.2004.827228vidual decompressions. Also, they were shown by Brucksteinet al. [3] to be good for holographic image watermarking,where authors multiply spectrum of an image by a real mul-tiplicative mask in such a way that, given the original image,any region in a watermarked image enables them to recoverthe whole watermark.The rest of the paper is organized as follows. In Section II,we introduce inverse indexing approach in which we constructa 2-D array of the same size as the original image in sucha way that any rectangle of yields a sampled version of .In Section III, we introduce staircase low-pass filtering whichenables us to construct a hologram in such a way that recon-structed images come out to be aliasing free, very close to theideally low-passed versions of and of much better quality thanreconstructions from . In Section IV, we provide experimentalevidence for the fact that our algorithm has smaller reconstruc-tion errors than the two algorithms presented in [1].II. I NVERSE I NDEXINGDenote by (here, standsfor a set of natural numbers which satisfy andstandsforasetofnaturalnumbers whichsatisfyfor some natural number )(1)(2)a square hologram region and a sampled image grid, respec-tively.In otherwords, represents a arrayof pixelswhose upper left corner is at row and column ,and represents a sampling grid of pixels, withspacing in both horizontal and vertical directions, whoseupper left corner is at row and column . Also denote by (here)(3)the ideal low-pass mask in the spectral domain, which can beseen as a array of pixels, centered at the center of thespectrum.Every pixel coordinate in the image can be repre-sented in binary encoding as1057-7149/04$20.00 © 2004 IEEEZhejiang University of Media and Communications (218.75.124.130) - 2014/10/8 Download