Reconstruction of nonperiodic two-dimensional signals from zero crossings

Curtis, Susan R.; Shitz, Shlomo Shamai; Oppenheim, Alan V.

doi:10.1109/tassp.1987.1165212

Cited by 32 publications

(17 citation statements)

References 18 publications

(24 reference statements)

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“…These results rely on the band-limitedness of the signal to compute its analytic extension [44]- [46], which is a nonstable characterization. The properties of zero-crossings of functions convolved with a LoG have also been studied, more specifically its multiscale version [47], [48].…”

Section: A Marr's Theory Of Vision and The Raw Primal Sketchmentioning

confidence: 99%

“…We are now able to precisely formalize the objective of the reconstruction algorithm: We are searching for an image corresponding to an admissible Marr-like wavelet pyramid such that (44) where is the given wavelet primal sketch. This objective can be reached by alternating between the projectors and , a procedure that is proven to converge to the orthogonal projection on [55].…”

Section: Image Reconstruction From Wavelet Primal Sketchmentioning

confidence: 99%

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Complex Wavelet Bases, Steerability, and the Marr-Like Pyramid

Ville

Unser

2008

IEEE Trans. on Image Process.

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Abstract-Our aim in this paper is to tighten the link between wavelets, some classical image-processing operators, and David Marr's theory of early vision. The cornerstone of our approach is a new complex wavelet basis that behaves like a smoothed version of the Gradient-Laplace operator. Starting from first principles, we show that a single-generator wavelet can be defined analytically and that it yields a semi-orthogonal complex basis of , irrespective of the dilation matrix used. We also provide an efficient FFT-based filterbank implementation. We then propose a slightly redundant version of the transform that is nearly translation-invariant and that is optimized for better steerability (Gaussian-like smoothing kernel). We call it the Marr-like wavelet pyramid because it essentially replicates the processing steps in Marr's theory of early vision. We use it to derive a primal wavelet sketch which is a compact description of the image by a multiscale, subsampled edge map. Finally, we provide an efficient iterative algorithm for the reconstruction of an image from its primal wavelet sketch.Index Terms-Feature extraction, primal sketch, steerable filters, wavelet design. MULTISCALE transforms are powerful tools for signal and image processing, computer vision, and for modeling biological vision. A prominent example is the 1-D wavelet transform, which acts as a multiscale version of an th-order derivative operator, where is the number of vanishing moments of the wavelet [1]. Its extension to multiple dimensions and to 2-D, in particular, is typically achieved by forming tensorproduct basis functions. However, such separable wavelets are not well matched to the singularities occuring in images such as lines and edges which can be arbitrarily oriented and even curved. Consequently, there has been a considerable research effort in developing alternative multiscale transforms that are better tuned to the geometry of natural images. Notable examples of these "geometrical x-lets" include biologically-inspired 2-D Gabor transforms [2] [24]. The derivation of the filters is not based on wavelets, but rather obtained through a numerical optimization process. Special constraints are imposed to ensure that the frequency response of the filters is essentially polar-separable and that the decomposition is simple to invert numerically (approximate tight-frame property).In this paper, we present an alternative approach based on an explicit analytical and spline-based formulation of complex wavelet bases of . Special care is given to design basis functions that best match the properties of the visual system, in accordance with Marr's theory of early vision [25]. To motivate our construction, we specify a number of properties that are highly desirable and that are fulfilled, sometimes implicitly, by a number of classical image-processing algorithms.• Invariance. We aim at invariance with respect to elementary geometric operations such as translation, scaling, and rotation. Traditional wavelet transforms only satisfy these propert...

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Section: A Marr's Theory Of Vision and The Raw Primal Sketchmentioning

confidence: 99%

Section: Image Reconstruction From Wavelet Primal Sketchmentioning

confidence: 99%

Complex Wavelet Bases, Steerability, and the Marr-Like Pyramid

Ville

Unser

2008

IEEE Trans. on Image Process.

View full text Add to dashboard Cite

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“…This is in contrast to conventional uniform sampling schemes based on the assumption that signals are bandlimited (Whittaker-Shannon-Nyquist-Kotel'nikov sampling theorem [4]). Curtis et al extended Logan's formalism to two-dimensional signals that are neither octaveband nor periodic [5]. A more recent development in zero-crossings-based signal reconstruction approaches has been within the framework of compressed sensing, where the underlying signal is assumed to be sparse in some basis [6].…”

Section: Introductionmentioning

confidence: 99%

A sub-Nyquist sampling method for computing the level-crossing-times of an analog signal: Theory and applications

Seelamantula

2010

2010 International Conference on Signal Processing and Communications (SPCOM)

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We address the problem of computing the levelcrossings of an analog signal from samples measured on a uniform grid. Such a problem is important, for example, in multilevel analog-to-digital (A/D) converters. The first operation in such sampling modalities is a comparator, which gives rise to a bilevel waveform. Since bilevel signals are not bandlimited, measuring the level-crossing times exactly becomes impractical within the conventional framework of Shannon sampling. In this paper, we propose a novel sub-Nyquist sampling technique for making measurements on a uniform grid and thereby for exactly computing the level-crossing times from those samples. The computational complexity of the technique is low and comprises simple arithmetic operations. We also present a finite-rate-ofinnovation sampling perspective of the proposed approach and also show how exponential splines fit in naturally into the proposed sampling framework. We also discuss some concrete practical applications of the sampling technique.Index Terms-level-crossing, sub-Nyquist sampling, sparse signals, splines, finite rate of innovation.

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“…(ii) Computations: Research on computations of scale-space deal with such aspects as discretizing LoG [9], computing 2-D LoG in terms of 1-D LoGs [10], fast computations of LoG [11], and computing z.c.s with subpixel accuracy [10]. (iii) Applications: Scale-space has been used as apowerful tool in a number of applications such as image reconstruction [12][13][14][15], image segmentation [16,17], image description [3,[8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], and object detection [4].…”

Section: Introductionmentioning

confidence: 99%

Impact of scale on signed, unsigned, authentic and phantom zero crossings in a noisy environment

Topkar

Sood

1995

J Math Imaging Vis

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Abstract. Scale-space description of an image refers to the descriptions of the same image at different resolutions. One popular scale-space representation, the topic of this research, is obtained by passing an image through a bank of smoothing filters (each tuned to a different scale) and detecting the zero crossings (z.c.s) of the second derivative (Laplacian) of the outputs. Any z.c. based algorithm will be affected by the corruption of the z.c.s due to the input noise. Statistical analysis of the z.c.s is used to determine the effect of scale change on the different types of z.c.s. We also study the computational and performance trade-offs involved in choosing scale. Statistical analysis of the z.c.s is nontrivial. The 2 main reasons are: (i) detection of z.c.s is a nonlinear operation, and (ii) the noise at different scales is correlated. In this paper we identify different types of z.c.s and compute the densities of their occurrence in the presence of white and colored noise. The formulae for computing the z.c. densities are applicable to any smoothing filter. The special case of Gaussian smoothing filters is investigated in depth. It is demonstrated how the a priori knowledge about the input image is reflected in the performance of an algorithm which uses the z.c.s as the primitives. Other applications of the statistical analysis of z.c.s include design of optimum smoothing filter, performance analysis and active multiscaling. It is quantitatively demonstrated how the performance of a scale-space system improves as more and more a priori knowledge about the scene is incorporated.

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Reconstruction of nonperiodic two-dimensional signals from zero crossings

Cited by 32 publications

References 18 publications

Complex Wavelet Bases, Steerability, and the Marr-Like Pyramid

Complex Wavelet Bases, Steerability, and the Marr-Like Pyramid

A sub-Nyquist sampling method for computing the level-crossing-times of an analog signal: Theory and applications

Impact of scale on signed, unsigned, authentic and phantom zero crossings in a noisy environment

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