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New tight frames of curvelets and optimal representations of objects with piecewise C² singularities
 COMM. ON PURE AND APPL. MATH
, 2002
"... This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along C2 edges. Conceptually, the curvelet transform is a multiscale pyramid with many directions and positions at each length scale, and needleshap ..."
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Cited by 232 (17 self)
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This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along C2 edges. Conceptually, the curvelet transform is a multiscale pyramid with many directions and positions at each length scale, and needleshaped elements at fine scales. These elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets. For instance, curvelets obey a parabolic scaling relation which says that at scale 2−j, each element has an envelope which is aligned along a ‘ridge ’ of length 2−j/2 and width 2−j. We prove that curvelets provide an essentially optimal representation of typical objects f which are C2 except for discontinuities along C2 curves. Such representations are nearly as sparse as if f were not singular and turn out to be far more sparse than the wavelet decomposition of the object. For instance, the nterm partial reconstruction f C n obtained by selecting the n largest terms in the curvelet series obeys ‖f − f C n ‖ 2 L2 ≤ C · n−2 · (log n) 3, n → ∞. This rate of convergence holds uniformly over a class of functions which are C 2 except for discontinuities along C 2 curves and is essentially optimal. In comparison, the squared error of nterm wavelet approximations only converges as n −1 as n → ∞, which is considerably worst than the optimal behavior.
The JPEG2000 Still Image Coding System: An Overview
 IEEE Transactions on Consumer Electronics
, 2000
"... Abstract With the increasing use of multimedia technologies, image compression requires higher performance as well as new features. To address this need in the specific area of still image encoding, a new standard is currently being developed, the JPEG2000. It is not only intended to provide rate ..."
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Cited by 184 (2 self)
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Abstract With the increasing use of multimedia technologies, image compression requires higher performance as well as new features. To address this need in the specific area of still image encoding, a new standard is currently being developed, the JPEG2000. It is not only intended to provide ratedistortion and subjective image quality performance superior to existing standards, but also to provide features and functionalities that current standards can either not address efficiently or in many cases cannot address at all. Lossless and lossy compression, embedded lossy to lossless coding, progressive transmission by pixel accuracy and by resolution, robustness to the presence of biterrors and regionofinterest coding, are some representative features. It is interesting to note that JPEG2000 is being designed to address the requirements of a diversity of applications, e.g. Internet, color facsimile, printing, scanning, digital photography, remote sensing, mobile applications, medical imagery, digital library and Ecommerce.
The JPEG2000 still image compression standard
 IEEE Signal Proc. Mag
, 2001
"... The development of standards (emerging and established) by the International Organization for Standardization (ISO), the International Telecommunications Union (ITU), and the International Electrotechnical Commission (IEC) for audio, image, and video, for both transmission and storage, has led to wo ..."
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Cited by 108 (9 self)
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The development of standards (emerging and established) by the International Organization for Standardization (ISO), the International Telecommunications Union (ITU), and the International Electrotechnical Commission (IEC) for audio, image, and video, for both transmission and storage, has led to worldwide activity in developing hardware and software systems and products applicable to a number of diverse disciplines [7], [22], [23], [55], [56], [73]. Although the standards implicitly address the basic encoding operations, there is freedom and flexibility in the actual design and development of devices. This is because only the syntax and semantics of the bit stream for decoding are specified by standards, their main objective being the compatibility and interoperability among the systems (hardware/software) manufactured by different companies. There is, thus, much room for innovation and ingenuity. Since the mid 1980s, members from both the ITU and the ISO have been working together to establish a joint international standard for the compression of grayscale and color still images. This effort has been known as JPEG, the Joint
An overview of the JPEG2000 still image compression standard
 Signal Processing: Image Communication
, 2002
"... In 1996, the JPEGcommittee began to investigate possibilities for a new still image compression standard to serve current and future applications. This initiative, which was named JPEG2000, has resulted in a comprehensive standard (ISO 154447ITUT Recommendation T.800) that is being issued in six pa ..."
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Cited by 77 (0 self)
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In 1996, the JPEGcommittee began to investigate possibilities for a new still image compression standard to serve current and future applications. This initiative, which was named JPEG2000, has resulted in a comprehensive standard (ISO 154447ITUT Recommendation T.800) that is being issued in six parts. Part 1, in the same vein as the JPEG baseline system, is aimed at minimal complexity and maximal interchange and was issued as an International Standard at the end of 2000. Parts 2–6 define extensions to both the compression technology and the file format and are currently in various stages of development. In this paper, a technical description of Part 1 of the JPEG2000 standard is provided, and the rationale behind the selected technologies is explained. Although the JPEG2000 standard only specifies the decoder and the codesteam syntax, the discussion will span both encoder and decoder issues to provide a better
A Tutorial on Modern Lossy Wavelet Image Compression: Foundations of JPEG 2000
, 2001
"... The JPEG committee has recently released its new image coding standard, JPEG 2000, which will serve as a supplement for the original JPEG standard introduced in 1992. Rather than incrementally improving on the original standard, JPEG 2000 implements an entirely new way of compressing images based o ..."
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Cited by 63 (0 self)
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The JPEG committee has recently released its new image coding standard, JPEG 2000, which will serve as a supplement for the original JPEG standard introduced in 1992. Rather than incrementally improving on the original standard, JPEG 2000 implements an entirely new way of compressing images based on the wavelet transform, in contrast to the discrete cosine transform (DCT) used in the original JPEG standard. The significant change in coding methods between the two standards leads one to ask: What prompted the JPEG committee to adopt such a dramatic change? The answer to this question comes from considering the state of image coding at the time the original JPEG standard was being formed. At that time wavelet analysis and wavelet coding were still
An overview of JPEG 2000
 in Proc. IEEE Data Compression Conf., Snowbird, UT
, 2000
"... JPEG2000 is an emerging standard for still image compression. This paper provides a brief history of the JPEG2000 standardization process, an overview of the standard, and some description of the capabilities provided by the standard. Part I of the JPEG2000 standard specifies the minimum complian ..."
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Cited by 60 (2 self)
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JPEG2000 is an emerging standard for still image compression. This paper provides a brief history of the JPEG2000 standardization process, an overview of the standard, and some description of the capabilities provided by the standard. Part I of the JPEG2000 standard specifies the minimum compliant decoder, while Part II describes optional, valueadded extensions. Although the standard specifies only the decoder and bitstream syntax, in this paper we describe JPEG2000 from the point of view of encoding. We take this approach, as we believe it is more amenable to a compact description more easily understood by most readers. 1
Efficient, LowComplexity Image Coding with a SetPartitioning Embedded Block Coder
 IEEE Trans. Circuits Systems Video Technology
, 2004
"... We propose an embedded, blockbased, image wavelet transform coding algorithm of low complexity. It uses a recursive setpartitioning procedure to sort subsets of wavelet coefficients by maximum magnitude with respect to integer powers of two thresholds. It exploits two fundamental characteristics ..."
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Cited by 50 (18 self)
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We propose an embedded, blockbased, image wavelet transform coding algorithm of low complexity. It uses a recursive setpartitioning procedure to sort subsets of wavelet coefficients by maximum magnitude with respect to integer powers of two thresholds. It exploits two fundamental characteristics of an image transform  the well defined hierarchical structure, and energy clustering in frequency and in space. We describe the use of this coding algorithm in several implementations and show extensive comparisons with other stateoftheart coders, such as SPIHT and JPEG2000. We conclude that this algorithm, in addition to being highly flexible, retains all the desirable features of these algorithms and is entirely competitive to them in compression efficiency.
ThreeDimensional Embedded Subband Coding with Optimized Truncation (3D ESCOT)
 3D ESCOT)”, Applied and Computational Harmonic Analysis10
, 2001
"... This paper presents an efficient video coding algorithm: Threedimensional embedded subband coding with optimized truncation (3D ESCOT), in which coefficients in different subbands are independently coded using fractional bitplane coding and candidate truncation points are formed at the end of ..."
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Cited by 49 (19 self)
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This paper presents an efficient video coding algorithm: Threedimensional embedded subband coding with optimized truncation (3D ESCOT), in which coefficients in different subbands are independently coded using fractional bitplane coding and candidate truncation points are formed at the end of each fractional bitplane. A ratedistortion optimized truncation scheme is used to multiplex all subband bitstreams together into a layered one. A novel motion threading technique is proposed to form threads along the motion trajectories in a scene. For efficient coding of motion threads, memoryconstrained temporal wavelet transforms are applied along entire motion threads. Blockbased motion threading is implemented in conjunction with 3D ESCOT in a real video coder. Extension of 3D ESCOT to objectbased coding is also addressed. Experiments demonstrate that 3D ESCOT outperforms MPEG4 for most test sequences at the same bit rate. # 2001 Academic Press 1.
Image Compression by Linear Splines over Adaptive Triangulations
"... This paper proposes a new method for image compression. The method is based on the approximation of an image, regarded as a function, by a linear spline over an adapted triangulation, D(Y ), which is the Delaunay triangulation of a small set Y of significant pixels. The linear spline minimizes the d ..."
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Cited by 35 (7 self)
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This paper proposes a new method for image compression. The method is based on the approximation of an image, regarded as a function, by a linear spline over an adapted triangulation, D(Y ), which is the Delaunay triangulation of a small set Y of significant pixels. The linear spline minimizes the distance to the image, measured by the mean square error, among all linear splines over D(Y ). The significant pixels in Y are selected by an adaptive thinning algorithm, which recursively removes less significant pixels in a greedy way, using a sophisticated criterion for measuring the significance of a pixel. The proposed compression method combines the approximation scheme with a customized scattered data coding scheme. We demonstrate that our compression method outperforms JPEG2000 on two geometric images and performs competitively with JPEG2000 on three popular test cases of real images.