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The Minimal Structure and Motion Problems with Missing Data for 1D Retina Vision

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Abstract

In this paper we investigate the structure and motion problem for calibrated one-dimensional projections of a two-dimensional environment. The theory of one-dimensional cameras are useful in several areas, e.g. within robotics, autonomous guided vehicles, projection of lines in ordinary vision and vision of vehicles undergoing so called planar motion. In a previous paper the structure and motion problem for all cases with non-missing data was classified and solved. Our aim is here to classify all structure and motion problems, even those with missing data, and to solve them. In the classification we introduce the notion of a prime problem. A prime problem is a minimal problem that does not contain a minimal problem as a sub-problem. We further show that there are infinitely many such prime problems. We give solutions to four prime problems, and using the duality of Carlsson these can be extended to solutions of seven prime problems. Finally we give some experimental results based on synthetic data.

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Authors and Affiliations

  1. Centre for Mathematical Sciences, Lund University, P O Box 118, SE-221 00, Lund, Sweden

    Magnus Oskarsson & Kalle Åström

  2. School of Technology and Society, Malmö University, SE-205 06, Malmö, Sweden

    Niels Chr. Overgaard

Authors
  1. Magnus Oskarsson
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  2. Kalle Åström
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  3. Niels Chr. Overgaard
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Correspondence to Magnus Oskarsson.

Additional information

Magnus Oskarsson received his M.Sc. degree in Engineering Physics in 1997, and Ph.D. in Mathematics in 2002 from the University of Lund, Sweden. His thesis work was devoted to computer vision with applications for autonomous vehicles. He is currently an assistant professor at the Centre for Mathematical Sciences, Lund University, where his teachings include undergraduate and graduate courses in mathematics and image analysis. His main research interests are in algebra and optimization with applications in computer vision, cognitive vision, and image enhancement.

Kalle Åström received his B.Sc. in Mathematics in 1990, M.Sc. degree in Engineering Physics in 1991 and Ph.D. in Mathematics in 1996 from Lund University, Sweden. His thesis was awarded Best Nordic Ph.D. Thesis in pattern recognition and image analysis 1995-1996 at the Scandinavian Conference in Image Analysis, 1997. He has been a postdoctoral research fellow, associate professor and is now professor at the Centre for Mathematical Sciences, Lund University. His teachings include undergraduate and graduate courses in mathematics, image processing and computer vision. His current research interests include stochastic analysis of low level vision, computer vision for autonomous guided vehicles, geometry and algebra of multiple views of points, curves and surfaces, cognitive vision, handwriting recognition and medical image analysis.

Niels Chr. Overgaard completed his PhD degree in the area of mathematical analysis in 2002 at Lund University, Sweden. He received his MSc in mathematics and physics from Aalborg University, Denmark, in 1992. He is currently a Post Doc in the Applied Mathematics Group at Malmö University, Sweden, where he is involved in research on image analysis and mathematical biology. He has contributed 18 papers to international journals and conferences. His main research interests are PDEs, distribution theory, and calculus of variations, and their applications to medical image segmentation, active contours, computer vision problems, and biofilm modelling.

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Oskarsson, M., Åström, K. & Overgaard, N.C. The Minimal Structure and Motion Problems with Missing Data for 1D Retina Vision. J Math Imaging Vis 26, 327–343 (2006). https://doi.org/10.1007/s10851-006-9003-7

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