The Schr&amp;#x00F6;dinger distance transform (SDT) for point-sets and curves

Sethi, Manu; Rangarajan, Anand; Gurumoorthy, Karthik S.

doi:10.1109/cvpr.2012.6247676

Cited by 17 publications

(27 citation statements)

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“…The main thrust of the present work is to provide an answer to the following question: Can we synthesize a single wave function capable of representing both the distance transform and its gradient density? In [16] we hinted at one such representation but did not provide any theoretical underpinnings to substantiate our claim. In contrast, we not only introduce a wave function φ(X) capable of integrating both the distance transform and its gradient density in a single representation but also provide theoretical and experimental support for the new representation.…”

Section: A New Wave Function For Joint Estimationmentioning

confidence: 98%

A Schrödinger formalism for simultaneously computing the Euclidean distance transform and its gradient density

Gurumoorthy

Rangarajan

2014

Proceedings of the 2014 Indian Conference on Computer Vision Graphics and Image Processing

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In this work, we employ the well-known Hamilton-Jacobi to Schröd-inger connection to present a unified framework for computing both the Euclidean distance function and its gradient density in two dimensions. Previous work in this direction considered two different formalisms for independently computing these quantities. While the two formalisms are very closely related, their lack of integration is theoretically troubling and practically cumbersome. We introduce a novel Schrödinger wave function for representing the Euclidean distance transform from a discrete set of points. An approximate distance transform is computed from the magnitude of the wave function while the gradient density is estimated from the Fourier transform of the phase of the wave function. In addition to its simplicity and efficient O(N log N ) computation, we prove that the wave function-based density estimator increasingly, closely approximates the distance transform gradient density (as a free parameter approaches zero) with the added benefit of not requiring the true distance function.

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Section: A New Wave Function For Joint Estimationmentioning

confidence: 98%

A Schrödinger formalism for simultaneously computing the Euclidean distance transform and its gradient density

Gurumoorthy

Rangarajan

2014

Proceedings of the 2014 Indian Conference on Computer Vision Graphics and Image Processing

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Section: Rush Distant Early Warningmentioning

confidence: 99%

“…In the past few years, we have been exploring the notion of uncertainty in the distance transform representation [19,20]. The eikonal equation ∇S = 1…”

Section: Rush Distant Early Warningmentioning

confidence: 99%

“…Finally, instead of computing distance functions, we compute wave functions which by virtue of satisfying a linear differential equation, inherit the properties of linearity and superposition. For more details, please see [22,20].What we gain-the introduction of uncertainty and the ability to add and superpose wave functions-more than makes up for what we lose-certainty in 7 the distance function representation. Having discussed the benefits of a wave function representation over those of a distance function, we now demonstrate one of its advantages-specifically the ability to represent a set of planar curves with uncertainty-relative to shape density functions which were previously used to represent point-sets with uncertainty.…”

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confidence: 99%

“…When one shape contains two curves and another three curves, their scalar distance transform fields cannot be easily combined. Consequently, while this representation has flourished with active contours and level sets seeing wide usage in shape extraction, there has not been much interest in these (more syntactic) representations for computing shape statistics.In the past few years, we have been exploring the notion of uncertainty in the distance transform representation [19,20]. The eikonal equation ∇S = 1 is an example of a static Hamilton-Jacobi equation.…”

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confidence: 99%

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Revisioning the unification of syntax, semantics and statistics in shape analysis

Rangarajan

2014

Pattern Recognition Letters

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Hidden patterns, rendered visible by hindsight, often stand revealed as strong influences. Rama Chellappa's lab in the mid to late '80s molded our character and scholarship in more ways than one. Rama ably handled the transition from image processing to computer vision and established an applied math and computing infrastructure from which we continue to benefit. In particular, the themes important to King-Sun Fu-syntax, semantics and statistics-were all debated in Rama's lab at that time. We argue that this triad remains important. With syntactic representations losing mindshare to statistics, we remain in the hunt for unification. And with the syntax versus semantics debate unresolved, it deserves a hearing as well. We offer two themes-uncertainty and interaction-to aid in the process of unification. First, we show that complex wave functions carry probabilistic location information in their magnitude and syntactic (curve) information in their phase while representing uncertainty at a fundamental level. Next, after reviewing work in analytic philosophy, we connect semantics to intentional, mental content. Analytic philosophy reminds us to take human experience seriously but remaining physicalist if possible. To this end, we introduce a nondualist interactionist model of experience, wherein compositional (physical) subjects are constantly shaping and being shaped by a physical world. We then demonstrate that wave functions can accommodate * Corresponding Author Email address: anand@cise.ufl.edu (Anand Rangarajan) 1 This work is partially supported by NSF IIS RI:1143963. Preprint submitted to Pattern Recognition LettersJuly 22, 2013 interaction, closely tracking previous work in physics on the measurement problem. The linearity and superposition properties of wave functions allow for literal addition of waves created by human interaction with shapes. Finally, we briefly survey the current situation in the human-computer interaction (HCI) field and argue that mathematical models of interaction akin to those in pattern recognition can aid HCI. We close by arguing that we can follow in Fu's footsteps and incorporate the mathematical modeling of human interaction into pattern recognition.Keywords: syntax, semantics, statistics, uncertainty, interaction, wave functions, Schrödinger, Hamilton-Jacobi, phase, topology, level-sets, distance transform, intentionality, qualia, compositionality, HCI InvocationHold the flame 'til the dream ignites A spirit with a vision is a dream with a mission Rush, MissionLuck is often described as "being in the right place at the right time". Looking back, it seems clear that we were very fortunate indeed to inhabit Prof. While we return to these themes frequently in this work, it is to the specifics of Rama's contributions-and the manner in which they illustrate the deployment of applied math in computer vision-that we now turn.Since we're using a signal processing to computer vision crossover perspective, we bypass Rama's considerable work on Markov random fields in image pr...

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Resonant Deformable Matching: Simultaneous Registration and Reconstruction

Corring

Rangarajan

2016

Computer Vision – ECCV 2016

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The Schrödinger distance transform (SDT) for point-sets and curves

Cited by 17 publications

References 17 publications

A Schrödinger formalism for simultaneously computing the Euclidean distance transform and its gradient density

A Schrödinger formalism for simultaneously computing the Euclidean distance transform and its gradient density

Revisioning the unification of syntax, semantics and statistics in shape analysis

Resonant Deformable Matching: Simultaneous Registration and Reconstruction

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