Quasi-Circular Rotation Invariance in Image Denoising

This paper studies a new method for wavelet-based image denoising which is translation invariant (TI) and rotation invariant (RI). These invariances are crucial in image denoising and, more generally, may play important roles in image modeling. In contrast to other approximately RI methods, like the steerable pyramid, our new method employs standard separable wavelet bases in conjunction with a pseudo-circular image rotation. This scheme does not involve interpolation and hence the observation model (likelihood function) is invariant under this rotation. The superiority of our new method with respect to existing TI (non-RI) techniques is supported by experiments. 1 Introduction A major diiculty in wavelet-based image denoising (and wavelet-based image modeling in general) is the lack of translation invariance (TI) and rotation invari-ance (RI) associated with separable wavelet bases. TI can be obtained fairly easily using the frame formed by all possible translations of the wavelet basis, but RI is more diicult to achieve simply because images are usually sampled on a rectangular lattice. It is well-known that TI denoising methods reduce blocky artifacts and in general produce better image denois-ing results than non-TI methods 1, 2]. However, TI methods still tend to favor certain features that align well with the wavelet basis functions. In the case of Haar wavelets, for example, these features are approximately vertical or horizontal edges. For a slant edge with slope of, say, 30 degrees, or a circular disk, it is easy to see that their shape and orientation will remain unchanged for any translation operation. Hence, TI methods are not well-matched to the above mentioned situations which are very common in natural images. The geometry of natural images can be dealt with more eeectively by designing more complex, often redundant, image representations. The shiftable multiscale transform developed in 3, 4] represents one eeort in this direction. The shiftable multiscale transform (here shiftable refers to both translation and rotation) essentially involves a standard image rotation. Unfortunately, standard rotation is not compatible with rectilinear image sampling schemes, and involves pixel interpolation. Here we propose a new rotation scheme that is better suited to square image lattices and ooers certain advantages. In this paper we develop a new translation and rotation invariant (TRI) denoising method which uses a quasi-circular image rotation scheme. It makes use of the l 1 distance norm, rather than the Euclidean norm underlying standard image rotation, and hence we call this an l 1-rotation. …