Quadtree-guided wavelet image coding

II. Discrete Wavelet Transform Coding
A discrete wavelet transform (DWT) (c.f. Mallat [15]) decomposes an image into a collection of 3N+1 subband images, where N is the number of scales. Specifically, a one-dimensional low pass filter and a one-dimensional high pass filter, each based on a mother wavelet, are applied horizontally and vertically to the original image, and the outputs are subsampled by a factor of 2, to create 4 subband images, designated LowLow, Low-High, High-Low and High-High. The three subband images with a High component are considered to be at scale 1. The filtering and subsampling is repeated on the Low-Low subband, creating three subband images at scale 2, and a new Low-Low subband image. Such filtering and subsampling of Low-Low subband images continues until 3N+1 subband images are created, three at each scale and one, the baseband image, that is a low-pass filtered version of the original image. For more detailed discussions of wavelet decompositions, we refer the reader to [15-18]. If the original image is M×M, then the subband images at scale i are M2-i×M2-i, and the baseband image is M2-N×M2-N. For future reference, we mention that each pixel of the baseband is considered to have 22(N-i) descendants in each of the outer subbands at scale i. These pixels correspond to the same locations in the original image. A wavelet image coder is created by quantizing and encoding the pixels (often called coefficients) in the various subband images. A desirable feature of wavelet transforms for image coding is their tendency to create large coefficients in the baseband and small coefficients in the other, we will say outer, bands. Though important details such as edges are represented by outer band coefficients, relatively few of them need to be accurately represented. A variety of methods have been proposed for quantizing and encoding the
Abstract
A hybrid of wavelet and quadtree image coding is introduced. It uses a recently developed quadtree predictive coding technique, having good rate/fidelity performance and very low complexity, to encode the baseband wavelet coefficients. The resulting quadtree decomposition of the baseband is then used to guide the encoding of higher frequency subbands, by indicating which of their coefficients require encoding and which do not. The result is a coding method with performance comparable to those ofith less complexity.
I. Introduction
Subband image coders based on discrete wavelet transforms have produced some of the best, if not the best, image coding results (c.f. [1-7]) in the sense of good rate vs. fidelity characteristics (both PSNR and perceptual) with reasonable complexity. Aside from producing good subband decompositions, recent wavelet coders have also benefited substantially from clever ways of exploiting the similarity of subband images at one scale with those of another [2,4,5,7,8]. Though quadtree image coding has a longer history (c.f. [9-14]), generally speaking it has not produced coders that are as effective as the best wavelet coders. In recent work [14], the present authors demonstrated a predictive version of quadtree coding with significantly less complexity and better rate/fidelity performance than previous quadtree techniques or JPEG. Due to its predictive nature, it avoids the kind of blocking artifacts that have prevented other quadtree methods (and block based methods in general) from achieving good fidelity at low rates. However, at very low rates the predictive nature of the method tends to cause some blurring. So while the new quadtree method is less complex than wavelet coding, its perceptual quality is not as good at low rates, for example below .25 bpp.
QUADTREE-GUIDED WAVELET IMAGE CODING1
Chia-Yuan Teng David L. Neuhoff Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, MI 48109
1Supported in part by NSF Grant NCR 9415754
In this paper, a new image coding technique that marries wavelet and quadtree predictive coding is proposed. From the point of view of wavelet coding, the quadtree predictive approach is an excellent way to encode the baseband. Moreover, the quadtree decomposition of the baseband provides a measure of local activity that can be used to guide the coding of the higher subbands, thereby exploiting the similarity between subbands at different scales in a simple and effective way. In fact, the new technique works well with only one- or two-scale wavelet decomposition. Thus, it is simpler than other wavelet coding approaches. From the point of view of quadtree coding, the blurring caused by low rate coding is avoided because the quadtree is applied to only the baseband -- the details are coded in a different fashion. Though a wavelet decomposition must be performed, the method is not much more complex than the original quadtree method, because only a one- or two-scale decomposition is needed, the quadtree method is applied only to the baseband, and simple methods are used to encode the remaining subbands. The paper is organized as follows. A few facts about discrete wavelet transforms are given in Section II. Quadtree coding and the quadtree predictive method are briefly summarized in Section III. Section IV describes the new hybrid of wavelet and quadtree methods. Experimental results, including comparisons with wavelet coding, quadtree predictive coding and JPEG, are given in Section V. Concluding remarks are contained in Section VI.