ADAPTIVE PRINCIPAL COMPONENTS AND IMAGE DENOISING D. Darian Muresan Digital Multi-Media Design (DMMD) 1633 N. Quinn St., Suite 304 Arlington, VA. 22209 darian@dmmd.net Thomas W. Parks Electrical and Computer Engineering, Cornell University Ithaca, NY 14853 parks@ece.cornell.edu ABSTRACT This paper presents a novel approach to image denoising using adaptive principal components. Our assumptions are that the image is corrupted by additive white Gaussian noise. The new denoising technique performs well in terms of im- age visual fidelity, and in terms of PSNR values, the new technique compares very well against some of the most re- cently published denoising algorithms. 1. INTRODUCTION This paper investigates the problem of image denoising when the image is corrupted by additive white Gaussian noise, which is a valid assumption for images obtained through scanning or other image capturing devices. A lot of work on noise reduction is based on wavelet thresholding [1], a simple and very effective denoising method. The basic idea is to project the noisy signal onto a properly selected or- thogonal set of basis functions, such that the high frequency coefficients are mostly due to noise. Then, the small high frequency coefficients can be safely set to zero, preserving the structure of the original signal, while removing noise. Finding the best signal representation and the proper thresh- old is discussed in detail in [2]. A large percentage of the image denoising algorithms assume an orthogonal basis decomposition of the signal. While this may be an efficient way to decompose the im- age for compression purposes, several authors [3, 4, 5] have shown that an over-complete representation of the signal is superior for image denoising. The main advantage of over-complete expansion is summarized by [3] as a sup- pression of the Gibbs phenomena. In [3] the Translation- Invariant denoising algorithm is achieved by shifting the signal multiple times, denoising each shifted signal sep- arately (using orthogonal decompositions for each shift), shifting back and then averaging the results. When denois- ing shifted versions of the signal, edge artifacts occur at different locations. When the signals are shifted back and This work was supported by NSF MIP9705349, TI and Kodak averaged these edge artifacts are averaged as well. The au- thors of [3] showed that a uniform thresholding in a Trans- lation Invariant denoising does well in eliminating some of the edge artifacts seen in orthogonal wavelet denoising. The authors in [5] extend the idea of [3] by simultane- ously processing all the shifted versions to obtain more ac- curate statistical models for signal components. The work of [4] extends the idea of wavelet thresholding to an adap- tive wavelet thresholding method based on context model- ing. Each wavelet coefficient is modeled as a random vari- able of a generalized Gaussian distribution with an unknown parameter. Experimentally, their adaptive thresholding us- ing shift-invariant non-subsampled wavelet transform (SI- AdaptShrink) is one of the best denoising algorithms. All denoising algorithms reviewed are some form of a low pass filter. The assumption is that noise is captured by the high frequency coefficients and by filtering these co- efficients the unwanted noise is removed. Unfortunately, edges also have high frequency components and by remov- ing noise, high frequency components belonging to edges are also removed. This is accentuated when using separable wavelets, as is the case with most denoising algorithms in literature. By generating 2-D basis sets, which have vec- tors lined up along edges, and not across them, the high fre- quency coefficients caused by edges are much smaller. This in turn improves the denoising algorithm. The selection of 2-D locally adaptive basis sets is the main contribution of this paper. 2. THE ALGORITHM Signal decompositions based on edge direction [6] decom- pose an image, based on both scale and local edge direction, using steerable filters. Our approach uses principal compo- nents (PC) on local image patches to derive a 2-D, locally adaptive basis set. The local principal components provide the best local 1 basis set and the largest eigenvector is in the 1 The basis set minimizes the sum of the squares of the errors between the first basis and a set of training vectors, , representative of the local