IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 16, NO. 7, JULY 2007 1761 Space-Frequency Quantization for Image Compression With Directionlets Vladan Velisavljevic ´, Baltasar Beferull-Lozano, Member, IEEE, and Martin Vetterli, Fellow, IEEE Abstract—The standard separable 2-D wavelet transform (WT) has recently achieved a great success in image processing because it provides a sparse representation of smooth images. However, it fails to efficiently capture 1-D discontinuities, like edges or contours. These features, being elongated and characterized by geometrical regularity along different directions, intersect and generate many large magnitude wavelet coefficients. Since contours are very important elements in the visual perception of images, to provide a good visual quality of compressed images, it is fundamental to preserve good reconstruction of these directional features. In our previous work, we proposed a construction of critically sampled perfect reconstruction transforms with direc- tional vanishing moments imposed in the corresponding basis functions along different directions, called directionlets. In this paper, we show how to design and implement a novel efficient space-frequency quantization (SFQ) compression algorithm using directionlets. Our new compression method outperforms the stan- dard SFQ in a rate-distortion sense, both in terms of mean-square error and visual quality, especially in the low-rate compression regime. We also show that our compression method, does not increase the order of computational complexity as compared to the standard SFQ algorithm. Index Terms—Directional transforms, directional vanishing mo- ments (DVMs), image coding, image orientation analysis, image segmentation, multiresolution analysis, nonseparable transforms, wavelet transforms (WTs). I. INTRODUCTION P ROVIDING efficient transform-based representations of images is an important problem in many areas of image processing, like approximation and compression. An efficient representation requires sparsity, that is, most of the information has to be contained in a few large-magnitude coefficients. The standard 2-D wavelet transform (WT) has become very successful in recent years because it provides a sparse multires- olution representation of natural images due to the presence of vanishing moments in the high-pass (HP) filters (enforced by imposing zeros at ) [3]. This method is conceptually Manuscript received September 7, 2006; revised March 28, 2007. The as- sociate editor coordinating the review of this manuscript and approving it for publication was Dr. Giovanni Poggi. V. Velisavljevic ´ is with the Deutsche Telekom Laboratories, 10587 Berlin, Germany (e-mail: vladan.velisavljevic@telekom.de). B. Beferull-Lozano is with the Group of Information and Communication Systems, Instituto de Robótica-Escuela Técnica Superior de Ingenieria, Universidad de Valencia, 46980, Paterna (Valencia), Spain (e-mail: baltasar. beferull@uv.es). M. Vetterli is with the School of Computer and Communication Sciences, EPFL, CH-1015 Lausanne, Switzerland, and also with the Department of Elec- trical Engineering and Computer Science, University of California, Berkeley, CA 94720 USA (e-mail: martin.vetterli@epfl.ch). Digital Object Identifier 10.1109/TIP.2007.899183 simple and has a low computational complexity because of the simple separable 1-D filtering and subsampling operations. For these reasons, the 2-D WT has been adopted in the image com- pression standard JPEG-2000. However, the performance of the 2-D WT is limited by the spatial isotropy of the basis functions and the construction only along the horizontal and vertical directions, which does not provide enough directionality. For this reason, the standard 2-D WT fails to provide a sparse representation of oriented 1-D discontinuities (edges or contours) in images [1]. These features are characterized by a geometrical coherence that is not properly captured by the isotropic wavelet basis functions. Thus, to provide an efficient representation of contours, the basis functions are required to be anisotropic and to have di- rectional vanishing moments (DVMs) along more than the two standard directions. Several previous approaches, like curvelets [4], contourlets [5], bandelets [6], [7], and wedgeprints [8], have already addressed this nontrivial task. However, these methods have higher complexity than the standard 2-D WT and require nonseparable filtering and filter design. Furthermore, these transforms are often oversampled, thus, making it non- trivial to have efficient image compression methods. Another directional method that resides on content-based adaptation of transform directions has already been reported in [9], where image is segmented and the segments are separately resampled and transformed so that the dominant directions are aligned with the horizontal or vertical direction. Similarly, in [10], the WT is applied along curves such that the energy in the HP subband is minimized. Several recently proposed directional approaches use the lifting scheme [11] in image compression algorithms. This scheme is exploited in [12], where transform directions are adapted pixel-wise throughout images. A similar adaptation is used in [13] and [14], but with more (9 and 11, respectively) different directions. In addition, the method in [13] uses the pixel values at fractional coordinates obtained by interpola- tion. Lifting is also implemented in [10] and in [15], where the wavelet packet decomposition is applied. However, even though these methods are computationally efficient and provide good compression results, they show a weaker performance when combined with zerotree-based compression algorithms. In our previous work [2], [16], [17], we designed critically sampled anisotropic basis functions with DVMs across any two directions with rational slopes, which we called directionlets. Our basis construction retains the separable processing and the computational simplicity of the standard 2-D WT. We showed that directionlets outperform the standard 2-D WT in nonlinear approximation (NLA) of images while keeping a similar com- plexity. In [17], we also analyzed the approximation power of 1057-7149/$25.00 © 2007 IEEE Authorized licensed use limited to: EPFL LAUSANNE. Downloaded on February 17, 2009 at 11:19 from IEEE Xplore. Restrictions apply.