Gitit Ruckenstein

Efficient decoding of Reed-Solomon codes beyond half the minimum distance

A list decoding algorithm is presented for [n,k] Reed-Solomon (RS) codes over GF(q), which is capable of correcting more than [(n-k)/2] errors. Based on a previous work of Sudan (see J. Compl., vol.13, p.180-93, 1997), an extended key equation (EKE) is derived for RS codes, which reduces to the classical key equation when the number of errors is limited to [(n-k)/2]. Generalizing Masseys (1969) algorithm that finds the shortest recurrence that generates a given sequence, an algorithm is obtained for solving the EKE in time complexity O(l/spl middot/(n-k)/sup 2/), where l is a design parameter, typically a small constant, which s an upper bound on the size of the list of decoded codewords. (The case l=1 corresponds to classical decoding of up to [(n-k)/2] errors where the decoding ends with at most one codeword.) This improves on the time complexity O(n/sup 3/) needed for solving the equations of Sudans algorithm by a naive Gaussian elimination. The polynomials found by solving the EKE are then used for reconstructing the codewords in time complexity O((llog/sup 2/l)k(n+llogq)) using root-finders of degree-l univariate polynomials.

Comprehensive solutions for automatic removal of dust and scratches from images

Dust, scratches, or hair on originals (prints, slides, or negatives) distinctly appear as light or dark artifacts on a scan. These unsightly artifacts have become a major consumer concern. There are several scenarios for removal of dust and scratch artifacts. One scenario is during acquisition, e.g., while scanning photographic media. Another is artifact removal from a digital image in an image editor. For each scenario, a different solution is suitable, with different performance requirements and differing levels of user interaction. This work describes a comprehensive set of algorithms for automatically removing dust and scratches from images. Our algorithms solve a wide range of use scenarios. A dust and scratch removal solution has two steps: a detection step and a reconstruction step. Very good detection of dust and scratches is possible using side information, such as provided by dedicated hardware. Without hardware assistance, dust and scratch removal algorithms generally resort to blurring, thereby losing image detail. We present algorithmic alternatives for dust and scratch detection. In addition, we present reconstruction algorithms that preserve image detail better than previously available alternatives. These algorithms consistently produce visually pleasing images in extensive testing.

Photo Repair and 3D Structure from Flatbed Scanners Using 4- and 2-Source Photometric Stereo

We recently introduced a technique that allows 3D information to be captured from a conventional flatbed scanner [22]. The technique requires no hardware modification and allows untrained users to easily capture 3D datasets. Once captured, these datasets can be used for interactive relighting and enhancement of surface detail on physical objects. We have also found that the method can be used to scan and repair damaged photographs. Since only the 3D structure on these photographs will typically be surface tears and creases, our method provides an accurate procedure for automatically detecting these flaws without any user intervention. Once detected, automatic techniques, such as infilling and texture synthesis, can be leveraged to seamlessly repair such damaged areas. We here provide a more thorough exposition and significant new material. We first present a method that is able to repair damaged photographs with minimal user interaction and then show how we can achieve similar results using a fully automatic process.

Detection of textured areas in natural images using an indicator based on component counts

An algorithm is presented for the detection of textured areas in natural images. Texture detection has potential application to image enhancement, tone correction, defect detection, content classification, and image segmentation. For example, texture detection may be useful for object detection when combined with color models and other descriptors. Sky, e.g., is generally smooth, and foliage is textured. The texture detector presented here is based on the intuition that texture in a natural image is comprised of many components. The measure we develop examines the structure of local regions of the image. This structural approach enables us to detect both structured and unstructured textures at many scales. Furthermore, it distinguishes between edges and texture, and also between texture and noise. Automatic detection results are shown to match human classification of corresponding image areas.

Bounds on the List-Decoding Radius of Reed-Solomon Codes

Techniques are presented for computing upper and lower bounds on the number of errors that can be corrected by list decoders for general block codes and, specifically, for Reed--Solomon (RS) codes. The list decoder of Guruswami and Sudan implies such a lower bound (referred to here as the GS bound) for RS codes. It is shown that this lower bound, given by means of the codes length, the minimum Hamming distance, and the maximal allowed list size, in fact applies to all block codes. Ranges of code parameters are identified where the GS bound is tight for worst-case RS codes, in which case the list decoder of Guruswami and Sudan provably corrects the largest possible number of errors. On the other hand, ranges of parameters are provided for which the GS lower bound can be strictly improved. In some cases the improvement applies to all block codes with a given minimum Hamming distance, while in others it applies only to RS codes.

Lower bounds on the anticipation of encoders for input-constrained channels

An input-constrained channel S is defined as the set of words generated by a finite labeled directed graph. It is shown that every finite-state encoder with finite anticipation (i.e., with finite decoding delay) for S can be obtained through state-splitting rounds applied to some deterministic graph presentation of S, followed by a reduction of equivalent states. Furthermore, each splitting round can be restricted to follow a certain prescribed structure. This result, in turn, provides a necessary and sufficient condition on the existence of finite-state encoders for S with a given rate p:q and a given anticipation a. A second condition is derived on the existence of such encoders; this condition is only necessary, but it applies to every deterministic graph presentation of S. Based on these two conditions, lower bounds are derived on the anticipation of finite-state encoders. Those lower bounds improve on previously known bounds and, in particular, they are shown to be tight for the common rates used for the (1,7)-runlength-limited (RLL) and (2,7)-RLL constraints.

Upper bounds on the list-decoding radius of Reed-Solomon codes

Upper bounds are presented on the number of errors that can be corrected by a list decoder of Reed-Solomon codes. A range of code parameters is identified where the list decoding algorithm of Guruswami and Sudan (see IEEE Trans. Inform. Theory, vol.45, p.1757-67, 1999) attains those bounds.

Nested input-constrained codes

An input-constrained channel, or simply a constraint, is a set S of words that is generated by a finite labeled directed graph. An encoder for S maps, in a lossless manner, sequences of unconstrained input blocks into sequences of channel blocks, the latter sequences being words of S. In most applications, the encoders are finite-state machines and, thus, presented by state diagrams. In the special case where the state diagram of the encoder is (output) deterministic, only the current encoder state and the current channel block are needed for the decoding of the current input block. In this work, the problem of designing coding schemes that can serve two constraints simultaneously is considered. Specifically, given two constraints S/sub 1/ and S/sub 2/ such that S/sub 1//spl sube/S/sub 2/ and two described rates, conditions are provided for the existence of respective deterministic finite-state encoders /spl epsi//sub 1/ and /spl epsi//sub 2/, at the given rates, such that (the state diagram of) /spl epsi//sub 1/ is a subgraph of /spl epsi//sub 2/ Such encoders are referred to as nested encoders. The provided conditions are also constructive in that they imply an algorithm for finding such encoders when they exist. The nesting structure allows to decode /spl epsi//sub 1/ while using the decoder of /spl epsi//sub 2/. Developments in optical recording suggest a potential application that can take a significant advantage of nested encoders.

Nested block decodable runlength-limited codes

Consider a (d/sub 1/, k/sub 1/)-runlength-limited (RLL) constraint that is contained in a (d/sub 2/, k/sub 2/)-RLL constraint, where k/sub 1//spl ges/2d/sub 1/ and d/sub 2/>0, and fix a codeword length q>k/sub 2/. It is shown that whenever there exist block-decodable encoders with codeword length q for those two constraints, there exist such encoders where one is a subgraph of the other: furthermore, both encoders can be decoded by essentially the same decoder. Specifically, a (d/sub 1/, k/sub 1/)-RLL constrained word is decoded by first using a block decoder of the (d/sub 2/, k/sub 2/)-RLL encoder, and then applying a certain function to the output of that decoder.