1378 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 47, NO. 12, DECEMBER 2000 IAVQ—Interval-Arithmetic Vector Quantization for Image Compression Sandro Ridella, Member, IEEE, Stefano Rovetta, Member, IEEE, and Rodolfo Zunino, Member, IEEE Abstract—Interval arithmetic (IA) can enhance vector quantiza- tion (VQ) in image-compression applications. In the interval arith- metic vector quantization (IAVQ) reformulation of classical VQ, prototypes assume ranges of admissible locations instead of being clamped to specific space positions. This provides the VQ-recon- struction process with some degrees of freedom, which do not af- fect the overall compression ratio, but help make up for coarse dis- cretization effects. In image compression, IA attenuates artifacts (such as blockiness) brought about by the VQ schema. This paper describes the algorithms for both the training and the run-time use of IAVQ. Data-driven training endows the methodology with the adaptiveness of standard VQ methods, as confirmed by exper- imental results on real images. Index Terms—Author, please supply index terms. E-mail key- words@ieee.org for more info. I. INTRODUCTION V ECTOR QUANTIZATION (VQ) [1] encodes information by means of a set of prototypes (codewords) in the ob- served domain. Each point in the data space is represented by the codeword that maximizes a similarity criterion. Compres- sion stems from using a codebook whose (log) cardinality is smaller than the number of bits describing a datum. The fact that entire space partitions are encoded by the associated codewords causes VQ to be an efficient compression method. The heavy computational load represents a crucial issue of VQ schemata. It can be tackled by either dedicated hardware circuitry [2]–[6] or accelerated algorithms [7]–[9]. The index-based coding schema allows VQ to attain consid- erable compression ratios in high-dimensional domains. This makes the method suitable for image compression [10]–[13]. Thanks to the possibility of locally representing scene contents, VQ coding schemata may perform effectively in specific appli- cations that involve image understanding [14] and multimedia data processing [15]. When very low bit-rate compression must be attained, VQ techniques, as well as other nonconventional approaches (e.g., wavelets), may represent a valid alternative to standard compression algorithms such as Joint Photographers Expert Group (JPEG) [14]–[16] in specific applications. When considering the quality of reconstructed images, it is well-known that quantizing the space into a few partitions may Manuscript received September 1999; revised August 2000. This work was supported in part by the Italian Ministry for University and Scientific and Tech- nological Research (MURST). This paper was recommended by Associate Ed- itor A. Skodras. The authors are with the Deptartment of Biophysical and Electronic Engineering (DIBE), University of Genoa, 16145 Genova, Italy (e-mail: ridella@dibe.unige.it; rovetta@dibe.unige.it; zunino@dibe.unige.it). Publisher Item Identifier S 1057-7130(00)11029-8. lead to an excessive discretization of represented data. This may give rise to undesired effects, such as blockiness. Removing ar- tifacts is still an open problem [17]–[21]. Any additional in- formation transmitted to remove image defects may tend to in- crease the required bandwidth, hence a myriad of deblurring ap- proaches have been proposed to improve image quality at the decoder end without increasing the number of bits sent. This paper shows that interval arithmetic (IA) [22] can be profitably integrated within the VQ-based paradigm. The major advantage is that reconstruction quality is enhanced without affecting compression performance. Interval arithmetic vector quantization (IAVQ) redefines VQ prototypes and lets them be placed in ranges of admissible locations rather than specific space positions. Thus, a VQ prototype becomes an “interval pro- totype.” Nevertheless, the number of bits and the time for en- coding a datum are the same as those of classical VQ coding. Interval prototypes at the receiver end provide the pixel-re- construction process with degrees of freedom, thus permitting a qualitative improvement in image rendering. Evaluating the method quantitatively is complicated by the lack of a valid model of visual perception, hence the current research adopts mean square error as a standard distortion measure. An approach using the quantization interval has been proposed for JPEG-compressed images [18], [23], where projections onto convex sets (POCS) constrain the reconstruction to be consistent with the encoded bitstream subject to a smoothness constraint. From a regularization perspective, a frequency-do- main method to remove blocking from JPEG-encoded pictures led to a gradient-based quadratic-programming problem [17]. Likewise, IAVQ formulates the image-reconstruction process as a constrained quadratic-programming problem [24], where interval codewords impose bounds to the solution space. The final optimization process implies a local-smoothness assump- tion. In place of the gradient-based method used in [17], a cel- lular neural network (CNN) [25], [26] drives the regularization task and supports the actual image rendering. The major ad- vantage of adopting a CNN lies in the notable efficiency of the eventual circuit implementation. The reconstruction method at the decoder end differs significantly from classical low-pass fil- tering [19], [27], as interval quantities strictly control (as much as slopes in [17]) the filtering action and ultimately prevent gen- eralized blurring effects. In IAVQ, the bounds are domain-adap- tive, as the used codebooks can be trained empirically by simple and fast algorithms. Thus IAVQ ensures the example-driven ability of VQ schemata and keeps their flexibility. The augmented model does not affect either the compression ratio or the speed performance. The circuitry supporting data-coding uses standard VQ hardware implementations [6]. 1057–7130/00$10.00 © 2000 IEEE