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25
Directionlets: Anisotropic MultiDirectional Representation With Separable Filtering
 Ph.D. dissertation, School Comput. Commun. Sci., Swiss Federal Inst. Technol. Lausanne (EPFL
, 2005
"... Abstract—In spite of the success of the standard wavelet transform (WT) in image processing in recent years, the efficiency of its representation is limited by the spatial isotropy of its basis functions built in the horizontal and vertical directions. Onedimensional (1D) discontinuities in images ..."
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Cited by 57 (7 self)
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Abstract—In spite of the success of the standard wavelet transform (WT) in image processing in recent years, the efficiency of its representation is limited by the spatial isotropy of its basis functions built in the horizontal and vertical directions. Onedimensional (1D) discontinuities in images (edges and contours) that are very important elements in visual perception, intersect too many wavelet basis functions and lead to a nonsparse representation. To efficiently capture these anisotropic geometrical structures characterized by many more than the horizontal and vertical directions, a more complex multidirectional (MDIR) and anisotropic transform is required. We present a new latticebased perfect reconstruction and critically sampled anisotropic MDIR WT. The transform retains the separable filtering and subsampling and the simplicity of computations and filter design from the standard twodimensional WT, unlike in the case of some other directional transform constructions (e.g., curvelets, contourlets, or edgelets). The corresponding anisotropic basis unctions (directionlets) have directional vanishing moments along any two directions with rational slopes. Furthermore, we show that this novel transform provides an efficient tool for nonlinear approximation of images, achieving the approximation power ( 1 55), which, while slower than the optimal rate ( 2), is much better than ( 1) achieved with wavelets, but at similar complexity. Index Terms—Directional vanishing moments, directionlets, filter banks, geometry, multidirection, multiresolution, separable filtering, sparse image representation, wavelets. I.
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 38 (9 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.
Ratedistortion optimized image compression using wedgelets
 in IEEE Int. Conf. on Image Proc. – ICIP ’02
, 2002
"... Most waveletbased image coders fail to model the joint coherent behavior of wavelet coefficients near edges. Wedgelets offer a convenient parameterization for the edges in an image, but they have yet to yield a viable compression algorithm. In this paper, we propose an extension of the zerotreebas ..."
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Cited by 28 (7 self)
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Most waveletbased image coders fail to model the joint coherent behavior of wavelet coefficients near edges. Wedgelets offer a convenient parameterization for the edges in an image, but they have yet to yield a viable compression algorithm. In this paper, we propose an extension of the zerotreebased SpaceFrequency Quantization (SFQ) algorithm by adding a wedgelet symbol to its treepruning optimization. This incorporates wedgelets into a ratedistortion compression framework and allows simple, coherent descriptions of the wavelet coefficients near edges. The resulting method yields improved visual quality and increased compression efficiency over the standard SFQ technique. 1.
Representation and compression of multidimensional piecewise functions using surflets
 IEEE TRANS. INF. THEORY
, 2006
"... We study the representation, approximation, and compression of functions in M dimensions that consist of constant or smooth regions separated by smooth (M − 1)dimensional discontinuities. Examples include images containing edges, video sequences of moving objects, and seismic data containing geolog ..."
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Cited by 11 (2 self)
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We study the representation, approximation, and compression of functions in M dimensions that consist of constant or smooth regions separated by smooth (M − 1)dimensional discontinuities. Examples include images containing edges, video sequences of moving objects, and seismic data containing geological horizons. For both function classes, we derive the optimal asymptotic approximation and compression rates based on Kolmogorov metric entropy. For piecewise constant functions, we develop a multiresolution predictive coder that achieves the optimal ratedistortion performance; for piecewise smooth functions, our coder has nearoptimal ratedistortion performance. Our coder for piecewise constant functions employs surflets, a new multiscale geometric tiling consisting of Mdimensional piecewise constant atoms containing polynomial discontinuities. Our coder for piecewise smooth functions uses surfprints, which wed surflets to wavelets for piecewise smooth approximation. Both of these schemes achieve the optimal asymptotic approximation performance. Key features of our algorithms are that they carefully control the potential growth in surflet parameters at higher smoothness and do not require explicit estimation of the discontinuity. We also extend our results to the corresponding
A vision system for intelligent mission profiles of micro air vehicles
 Vehicular Technology, IEEE Transactions on
, 2004
"... Abstract—Recently, much progress has been made toward the development of smallscale aircraft, known broadly as Micro Air Vehicles (MAVs). Until recently, these platforms were exclusively remotely piloted, with no autonomous or intelligent capabilities, due at least in part to stringent payload rest ..."
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Cited by 10 (0 self)
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Abstract—Recently, much progress has been made toward the development of smallscale aircraft, known broadly as Micro Air Vehicles (MAVs). Until recently, these platforms were exclusively remotely piloted, with no autonomous or intelligent capabilities, due at least in part to stringent payload restrictions that limit onboard sensors. However, the one sensor that is critical to most conceivable MAV missions, such as remote surveillance, is an onboard video camera and transmitter that streams flight video to a nearby ground station. Exploitation of this key sensor is, therefore, desirable, since no additional onboard hardware (and weight) is required. As such, in this paper we develop a general and unified computer vision framework for MAVs that not only addresses basic flight stability and control, but enables more intelligent missions as well. This paper is organized as follows. We first develop a realtime feature extraction method called multiscale linear discriminant
Multiresolution linear discriminant analysis: efficient extraction of geometrical structures in images
 in Proc. IEEE Int’l Conf. Image Processing
, 2003
"... Currently popular feature extraction tools (e.g., Gabor, wavelet analysis) do not economically represent edges in images. As a step towards solving this problem, the wedgelet transform was recently proposed [1]; this transform provides nearly optimal representation of objects in the Horizon model, a ..."
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Cited by 9 (5 self)
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Currently popular feature extraction tools (e.g., Gabor, wavelet analysis) do not economically represent edges in images. As a step towards solving this problem, the wedgelet transform was recently proposed [1]; this transform provides nearly optimal representation of objects in the Horizon model, as measured by the minimax meansquared error. However, there is no reason to assume that the components useful for representing pixel values must also be useful for discriminating between regions in an image. Thus, having the successful extraction of edges as our goal, we propose a novel image analysis method—namely, multiresolution linear discriminant analysis (MLDA). In MLDA, analogously to the wedgelet transform, we seek directions that are efficient for discrimination. The MLDA framework comprises the following components: the MLDA atom, dictionary, tree, graph, and MLDAbased algorithms. In this paper, we explain these components and demonstrate the powerful expressiveness of MLDA, which gives rise to fast geometricalstructureanalysis algorithms. 1.
A Geometric Hidden Markov Tree Wavelet Model
 in Proc. of SPIE Wavelets X
, 2003
"... In the last few years, it has become apparent that traditional waveletbased image processing algorithms and models have significant shortcomings in their treatment of edge contours. The standard modeling paradigm exploits the fact that wavelet coefficients representing smooth regions in images tend ..."
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Cited by 9 (5 self)
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In the last few years, it has become apparent that traditional waveletbased image processing algorithms and models have significant shortcomings in their treatment of edge contours. The standard modeling paradigm exploits the fact that wavelet coefficients representing smooth regions in images tend to have small magnitude, and that the multiscale nature of the wavelet transform implies that these small coefficients will persist across scale (the canonical example is the venerable zerotree coder). The edge contours in the image, however, cause more and more large magnitude wavelet coefficients as we move down through scale to finer resolutions. But if the contours are smooth, they become simple as we zoom in on them, and are well approximated by straight lines at fine scales. Standard wavelet models exploit the grayscale regularity of the smooth regions of the image, but not the geometric regularity of the contours. In this paper, we build a model that accounts for this geometric regularity by capturing the dependencies between complex wavelet coefficients along a contour. The Geometric Hidden Markov Tree (GHMT) assigns each wavelet coefficient (or spatial cluster of wavelet coefficients) a hidden state corresponding to a linear approximation of the local contour structure. The shift and rotationalinvariance properties of the complex wavelet transform allow the GHMT to model the behavior of each coefficient given the presence of a linear edge at a specified orientation — the behavior of the wavelet coefficient given the state. By connecting the states together in a quadtree, the GHMT ties together wavelet coefficients along a contour, and also models how the contour itself behaves across scale. We demonstrate the effectiveness of the model by applying it to feature extraction.
Representation and compression of multidimensional piecewise functions using surflets
 IEEE Trans. Inform. Theory
, 2009
"... Abstract—We study the representation, approximation, and compression of functions in M dimensions that consist of constant or smooth regions separated by smooth (M 0 1)dimensional discontinuities. Examples include images containing edges, video sequences of moving objects, and seismic data containi ..."
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Cited by 5 (2 self)
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Abstract—We study the representation, approximation, and compression of functions in M dimensions that consist of constant or smooth regions separated by smooth (M 0 1)dimensional discontinuities. Examples include images containing edges, video sequences of moving objects, and seismic data containing geological horizons. For both function classes, we derive the optimal asymptotic approximation and compression rates based on Kolmogorov metric entropy. For piecewise constant functions, we develop a multiresolution predictive coder that achieves the optimal rate–distortion performance; for piecewise smooth functions, our coder has nearoptimal rate–distortion performance. Our coder for piecewise constant functions employs surflets, a new multiscale geometric tiling consisting of Mdimensional piecewise constant atoms containing polynomial discontinuities. Our coder for piecewise smooth functions uses surfprints, which
Image compression using multiscale geometric edge models
, 2002
"... Edges are of particular interest for image compression, as they communicate important information, contribute large amounts of highfrequency energy, and can generally be described with few parameters. Many of today's most competitive coders rely on wavelets to transform and compress the image, ..."
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Cited by 5 (1 self)
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Edges are of particular interest for image compression, as they communicate important information, contribute large amounts of highfrequency energy, and can generally be described with few parameters. Many of today's most competitive coders rely on wavelets to transform and compress the image, but modeling the joint behavior of wavelet coefficients along an edge presents a distinct challenge. In this thesis, we examine techniques for exploiting the simple geometric structure which captures edge information. Using a multiscale wedgelet decomposition, we present methods for extracting and compressing a cartoon sketch containing the significant edge information, and we discuss practical issues associated with coding the residual textures. Extending these techniques, we propose a ratedistortion optimal framework (based on the SpaceFrequency Quantization algorithm) using wedgelets to capture geometric information and wavelets to describe the rest. At low bitrates, this method yields compressed images with sharper edges and lower meansquare error.