IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 15, NO. 11, NOVEMBER 2006 3375 Fast Splitting -Rooting Method of Image Enhancement: Tensor Representation Fatma Turkay Arslan, Member, IEEE, and Artyom M. Grigoryan, Senior Member, IEEE Abstract—In the tensor representation, a two-dimensional (2-D) image is represented uniquely by a set of one-dimensional (1-D) sig- nals, so-called splitting-signals, that carry the spectral information of the image at frequency-points of specific sets that cover the whole domain of frequencies. The image enhancement is thus reduced to processing splitting-signals and such process requires a modifica- tion of only a few spectral components of the image, for each signal. For instance, the -rooting method of image enhancement can be fulfilled through processing separately a maximum of split- ting-signals of an image , where is a power of two. In this paper, we propose a fast implementation of the -rooting method by using one splitting-signal of the tensor representation with respect to the discrete Fourier transform (DFT). The imple- mentation is described in the frequency and spatial domains. As a result, the proposed algorithms for image enhancement use two 1-D -point DFTs instead of two 2-D -point DFTs in the traditional method of -rooting. Index Terms—Fourier transform, image enhancement, splitting- signal, tensor representation. I. INTRODUCTION D IGITAL image enhancement is necessary to expose critical details that are essential but not clearly seen in the image at hand. Image enhancement focuses on the improvement of dig- ital image quality for visual inspection or for machine analysis, without knowledge about the source of degradation. In medical imaging (e.g., computer tomography and magnetic resonance), three-dimensional images (or a stack of two-dimensional im- ages) of different organs and tissues are produced. However, various sources of interference during image acquisition (e.g., movement of a patient, insufficient performance and noise of imaging devices) make the quality of such images too poor to be directly used for diagnostic purposes. In such cases, to dis- cern the concealed but important information in the images, it is deemed necessary to use various image enhancement methods such as enhancing edges, emphasizing the differences, or re- ducing the noise [1]–[8]. We consider the Fourier transform-based image enhance- ment, although the Hartley, Hadamard, cosine, and other transforms are used for image enhancement as well [9]–[11]. Our focus is on the well-known method of -rooting, which Manuscript received August 3, 2005; revised April 26, 2006. The associate editor coordinating the review of this manuscript and approving it for publica- tion was Dr. Tamas Sziranyi. F. T. Arslan and A. M. Grigoryan are with Department of Electrical and Com- puter Engineering, The University of Texas at San Antonio, San Antonio, TX 78249-0669 USA (e-mail: arslan@lodestar.utsa.edu; amgrigoryan@utsa.edu). Color versions of Figs. 1, 2, 3, and 10 are available online at http://ieeexplore. ieee.org. Digital Object Identifier 10.1109/TIP.2006.881927 was modified later on into - -rooting, modified unsharp masking and filtering [12]–[15], as well as methods based on wavelet transforms [16]–[20]. The main disadvantage in using the -rooting method relates to the difficulty of selection of the value of parameter . This value should be chosen in an “optimal” way to enhance all parts of the image very well. The optimality is with respect to some measure of enhance- ment, the universal form of which has not been found yet. The selection of the optimal parameter is accomplished in the frequency domain through the calculation and analysis of the two-dimensional discrete Fourier transforms (2-D DFT) of the original and enhanced images, and this is thus the main steps that need to be efficiently performed. To solve this problem, a novel tensor form of splitting the mathematical structure of the 2-D DFT was proposed in [23]–[28]. In the tensor representation, an image is considered as a certain totality of 1-D signals (so-called splitting-signals, or image-signals) that carry the spectral information of the 2-D DFT of the image at frequency-points of different (but not disjoint) subsets in the frequency domain. The problem of 2-D image enhance- ment is thus reduced to the splitting -rooting method. The splitting-signals are processed separately with their optimal parameters, to achieve high-quality enhanced images, even when processing only one such signal. In this paper, a part of our research is presented, we focus on new effective realizations of the splitting -rooting method of image enhancement. Based on properties of the tensor represen- tation, two algorithms of image enhancement in the frequency and spatial domains are introduced. The proposed algorithms for image enhancement use two 1-D -point DFTs, instead of two 2-D -point DFTs, when comparing with the traditional method of -rooting. We also show how to enhance an image by using only one coefficient of enhancement for processing a splitting-signal rather than such coefficients, as was done in the splitting -rooting method. The rest of the paper is organized in the following way. In Section II, we review briefly some necessary background ma- terial, including the concept of -rooting, quantative measure of image enhancement, splitting of the 2-D DFT, and the tensor representation of the image with respect to the Fourier trans- form. In Section III, the application of splitting-signals for fast enhancement of the image by the -rooting method is described. The experimental results on different types of images, as well as a complexity and brief comparison of the proposed splitting algorithms with the traditional -rooting method and wavelet transforms are also given. Examples of MATLAB codes for per- forming the enhancement by splitting-signals are given in the Appendix. 1057-7149/$20.00 © 2006 IEEE