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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 needle-shap ..."
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Cited by 432 (21 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 needle-shaped 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 n-term 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 n-term 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
, 2000
"... 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 JPEC2000. It is not only intended to provide rate-distortion ..."
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Cited by 258 (2 self)
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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 JPEC2000. It is not only intended to provide rate-distortion 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 bit-errors and region-of-interest 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 E-commerce.
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 179 (11 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 154447ITU-T Recommendation T.800) that is being issued in six pa ..."
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Cited by 111 (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 154447ITU-T 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
An overview of JPEG 2000
- in Proc. IEEE Data Compression Conf., Snowbird, UT
, 2000
"... JPEG-2000 is an emerging standard for still image compression. This paper provides a brief history of the JPEG-2000 standardization process, an overview of the standard, and some description of the capabilities provided by the standard. Part I of the JPEG-2000 standard specifies the minimum complian ..."
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Cited by 95 (2 self)
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JPEG-2000 is an emerging standard for still image compression. This paper provides a brief history of the JPEG-2000 standardization process, an overview of the standard, and some description of the capabilities provided by the standard. Part I of the JPEG-2000 standard specifies the minimum compliant decoder, while Part II describes optional, value-added extensions. Although the standard specifies only the decoder and bitstream syntax, in this paper we describe JPEG-2000 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
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 91 (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
Efficient, Low-Complexity Image Coding with a Set-Partitioning Embedded Block Coder
- IEEE Trans. Circuits Systems Video Technology
, 2004
"... We propose an embedded, block-based, image wavelet transform coding algorithm of low complexity. It uses a recursive set-partitioning 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 71 (18 self)
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We propose an embedded, block-based, image wavelet transform coding algorithm of low complexity. It uses a recursive set-partitioning 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 state-of-the-art 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.
Three-Dimensional Embedded Subband Coding with Optimized Truncation (3-D ESCOT)
- 3D ESCOT)”, Applied and Computational Harmonic Analysis10
, 2001
"... This paper presents an efficient video coding algorithm: Three-dimensional embedded subband coding with optimized truncation (3-D ESCOT), in which coefficients in different subbands are independently coded using fractional bit-plane coding and candidate truncation points are formed at the end of ..."
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Cited by 57 (20 self)
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This paper presents an efficient video coding algorithm: Three-dimensional embedded subband coding with optimized truncation (3-D ESCOT), in which coefficients in different subbands are independently coded using fractional bit-plane coding and candidate truncation points are formed at the end of each fractional bit-plane. A rate-distortion 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, memory-constrained temporal wavelet transforms are applied along entire motion threads. Block-based motion threading is implemented in conjunction with 3-D ESCOT in a real video coder. Extension of 3-D ESCOT to object-based coding is also addressed. Experiments demonstrate that 3-D ESCOT outperforms MPEG-4 for most test sequences at the same bit rate. # 2001 Academic Press 1.
Adaptive directional lifting-based wavelet transform for image coding
- IEEE TRANS. IMAGE PROCESS
, 2007
"... We present a novel 2-D wavelet transform scheme of adaptive directional lifting (ADL) in image coding. Instead of alternately applying horizontal and vertical lifting, as in present practice, ADL performs lifting-based prediction in local windows in the direction of high pixel correlation. Hence, it ..."
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Cited by 44 (12 self)
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We present a novel 2-D wavelet transform scheme of adaptive directional lifting (ADL) in image coding. Instead of alternately applying horizontal and vertical lifting, as in present practice, ADL performs lifting-based prediction in local windows in the direction of high pixel correlation. Hence, it adapts far better to the image orientation features in local windows. The ADL transform is achieved by existing 1-D wavelets and is seamlessly integrated into the global wavelet transform. The predicting and updating signals of ADL can be derived even at the fractional pixel precision level to achieve high directional resolution, while still maintaining perfect reconstruction. To enhance the ADL performance, a rate-distortion optimized directional segmentation scheme is also proposed to form and code a hierarchical image partition adapting to local features. Experimental results show that the proposed ADL-based image coding technique outperforms JPEG 2000 in both PSNR and visual quality, with the improvement up to 2.0 dB on images with rich orientation features.
