ABSTRACT

Directional filter banks (DFBs), which were introduced by Bamberger in 1989 [1-3], are digital analysissynthesis filter banks that allow an image to be represented as a collection of subbands. In the analysis section, filters succeeded by downsampling matrix operators decompose an image into a set of subbands. In the synthesis section, these subbands are upsampled via a complementary set of matrix operators, then filtered and merged to reconstruct the original image. Such a description is strikingly reminiscent of a conventional 2-D analysis-synthesis filter bank. The distinguishing feature is that the DFB subbands embody angular information (a different angle for each subband), as opposed to the traditional low-, mid-, and high-frequency information. The directional subbands in the DFB are maximally decimated so that the total number of pixels in the full set of subbands is equal to the number of pixels in the original image. Thus, the subbands are nonredundant or, equivalently stated, they form a critically sampled representation. The DFB has the property that it can achieve exact reconstruction. That is, if the constituent filters are

properly designed, the output of the synthesis section can reconstruct the original image from the

decimated subbands exactly. This property is important in applications where modifications are performed in the subband domain for the purpose of enhancing image quality in synthesis. In addition, DFBs are very efficient from a computational perspective and are typically designed for implementation in a separable two-hand filter bank tree structure. This is a noteworthy characteristic as it enables the DFB to have arithmetic complexity comparable to (though a little higher than) the popular 2-D transforms, like discrete Fourier transforms (DFTs) and discrete cosine transforms (DCTs). DFBs are attractive for many image-processing applications that can benefit from directional analysis

such as noise reduction, edge sharpening, feature enhancement, compression, object recognition, and texture synthesis, to mention a few. Furthermore, DFBs are also attractive from a visual informationprocessing perspective. Research has shown that the first layer of cells in the human visual cortex responds to different orientations and scales. Consequently, DFBs can model in an approximate way the human visual system. All of these reasons have motivated much of the study and interest in directional analysis-synthesis systems in recent years. Since the introduction of the DFB in 1989, there have been a number of extensions and variations that

have been considered, such as relaxation of the maximum decimation condition (leading to directional pyramids), multidimensional extensions of the DFB, nonseparable implementations, and mixed low-, mid-, high-frequency angular decompositions [4-7]. To avoid confusion among these variations, we will refer to the original DFB as the Bamberger DFB (BDFB) in our subsequent discussion and use the term DFB to refer more generically to the broad class of directional decompositions.