Advances in Computer Vision - Vol. 2

By Christopher Brown | Go to book overview
6. Zero-crossings and slopes. A natural question to ask is whether gradient information across scales at the zero-crossings, in addition to their location, can be used to reconstruct the original. Hummel ( 1984, personal communication) has recently shown that this is the case, as one would expect in the light of our results ( Yuille and Poggio, 1983b, 1986). We have been able to simplify and extend the elegant proof by Hummel and obtain the following result: Knowledge of zero-crossing surfaces and magnitude of the x - t gradient over a finite, nonzero interval of the zero-crossing surface is sufficient to determine the image.

Finally we can use the uniqueness results of Curtis ( 1985) in two dimensions to obtain a new proof of uniqueness in one dimension that has implications for inverting the heat equation. This is discussed in Yuille and Poggio ( 1986).


APPENDIX A

If we have a matrix equation
Bx + ̱ = a + ̱ (1)
the necessary and sufficient condition for the existence of a solution is that

(2)

Hence, a necessary and sufficient condition for the nonexistence of a solution is that we can find a vector λ + ̱ = (λ1, . . .,λ), such that
λ1(b11, . . .,b1n) + . . . + λm(bm1, . . .,bmn) = 0 (3)
but for which
λ1a1 + . . . + λmam ≠ 0. (4)


APPENDIX B

Suppose we have a generalized diffusion equation of form

. (1)

-74-

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Advances in Computer Vision - Vol. 2
Table of contents

Table of contents

  • Title Page iii
  • Contents v
  • Contributors vii
  • Introduction ix
  • References xiv
  • 1: The Use of the Facet Model and the Topographic Primal Sketch in Image Analysis 1
  • Introduction 1
  • Summary 45
  • References 45
  • 2: Scaling and Fingerprint Theorems for Zero-Crossings 47
  • Introduction 47
  • Appendix A 74
  • Appendix B 74
  • Appendix B 77
  • Appendix B 77
  • 3: Form Perception Using Transformation Networks: Polyhedra 79
  • 3: Form Perception Using Transformation Networks 79
  • Acknowledgements 116
  • 4: The Parts of Perception 121
  • Summary 139
  • Summary 171
  • References 174
  • Author Index 177
  • Subject Index 180
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