## Riemann tensor of motion vision revisited

Optics Express, Vol. 9, Issue 1, pp. 7-8 (2001)

http://dx.doi.org/10.1364/OE.9.000007

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### Abstract

This note shows that the Riemann-space interpretation of motion vision developed by Barth and Watson is neither necessary for their results, nor sufficient to handle an intrinsic coordinate problem. Recasting the Barth-Watson framework as a classical velocity-solver (as in computer vision) solves these problems.

© Optical Society of America

1. E. Barth and A. B. Watson, “A geometric framework for nonlinear visual coding,” Opt. Express **7**, 155–165 (2000), http://www.opticsexpress.org/oearchive/source/23045.htm [CrossRef] [PubMed]

**R**. There is a problem with this interpretation: The native units of x, y, t, and f are quite disparate (x and y having length units, t having time units, and image intensity f having luminance units such as foot-lamberts), and the components of

**R**depend on how these units are chosen. Furthermore, a Riemann tensor implies a metric space (complete with distances and angles), and justifying this structure would require an intrinsic connection among the units of x, y, t, and f. Such a connection was made between x and t in Relativity theory through the vacuum speed of light c (so that ct and x have the same units). However, connecting the units of x, y, t, and f in such a fundamental way seems to pose a deeper challenge, rendering the Riemann constructs of distances and angles more difficult to justify.

1. E. Barth and A. B. Watson, “A geometric framework for nonlinear visual coding,” Opt. Express **7**, 155–165 (2000), http://www.opticsexpress.org/oearchive/source/23045.htm [CrossRef] [PubMed]

**R**component into numerator and denominator (and using only the numerator) may have been a sensible thing to do, given the units problem I have noted. The denominators of the

**R**components (Eq. 2 in [1

1. E. Barth and A. B. Watson, “A geometric framework for nonlinear visual coding,” Opt. Express **7**, 155–165 (2000), http://www.opticsexpress.org/oearchive/source/23045.htm [CrossRef] [PubMed]

**R**elements. Furthermore, ratios of

**R**elements have no dependence on the units of image intensity, and all the numerators are homogeneous functions of each of the coordinates (x,y,t, and f). Therefore, making the theory out of

*ratios*of

**R**components helps solve the coordinate problem.

**R**components are all 2-by-2 determinants, and some ratios of these determinants comprise a Cramer’s-rule solution to two linear equations. The linear system in question is clear from the Appendix of [1

**7**, 155–165 (2000), http://www.opticsexpress.org/oearchive/source/23045.htm [CrossRef] [PubMed]

_{x}+b f

_{y}+f

_{t}and set its partial derivatives in x, y, t equal to zero, which is correct, but the original function F is also zero.] Now, one can write the partial derivatives of Eq. 1 above:

_{3221}/R

_{2121}and -R

_{3121}/R

_{2121}in Eq. 2 of [1

**7**, 155–165 (2000), http://www.opticsexpress.org/oearchive/source/23045.htm [CrossRef] [PubMed]

**7**, 155–165 (2000), http://www.opticsexpress.org/oearchive/source/23045.htm [CrossRef] [PubMed]

**7**, 155–165 (2000), http://www.opticsexpress.org/oearchive/source/23045.htm [CrossRef] [PubMed]

**7**, 155–165 (2000), http://www.opticsexpress.org/oearchive/source/23045.htm [CrossRef] [PubMed]

**R**-component ratios that are used as switches for the motion-detection algorithm. In these cases, the numerators (2×2 determinants) of the

**R**-components can be used instead of the

**R**-components or their ratios. Thus the

**R**components never seem to enter distinctly and testably into the motion-vision model in [1

**7**, 155–165 (2000), http://www.opticsexpress.org/oearchive/source/23045.htm [CrossRef] [PubMed]

## References

1. | E. Barth and A. B. Watson, “A geometric framework for nonlinear visual coding,” Opt. Express |

2. | C. Zetsche and E. Barth, “Direct detection of flow discontinuities by 3D-curvature operators,” Pattern Recognition Lett. |

3. | C. Mota and E. Barth, “On the uniqueness of curvature features,” Proc. Artificial Intell. (Dynamische Perzeption). Köln: Infix Verlag , |

**OCIS Codes**

(150.4620) Machine vision : Optical flow

(330.4060) Vision, color, and visual optics : Vision modeling

(330.4150) Vision, color, and visual optics : Motion detection

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 1, 2001

Published: July 2, 2001

**Citation**

Michael Brill, "Riemann tensor of motion vision revisited," Opt. Express **9**, 7-8 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-1-7

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### References

- E. Barth, and A. B. Watson, "A geometric framework for nonlinear visual coding," Opt. Express 7, 155-165 (2000), http://www.opticsexpress.org/oearchive/source/23045.htm [CrossRef] [PubMed]
- C. Zetsche and E. Barth, "Direct detection of flow discontinuities by 3D-curvature operators," Pattern Recognition Lett. 12, 771-779 (1991). [CrossRef]
- C. Mota and E. Barth, "On the uniqueness of curvature features," Proc. Artificial Intell. (Dynamische Perzeption). K�ln: Infix Verlag, v. 9, pp. 175-178 (2000).

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