Abstract
Our goal as described in the overview is to provide a complete analysis of the views of natural scenes involving geometric features, shade/shadow, and apparent contours resulting from viewer movement. Our approach to this will involve progressively adding more detailed structure to simpler situations. The starting point for this is the case where we have a single object whose boundary is a smooth surface \(M \subset \mathbb{R}^{3}\) without geometric features. Hence, for the remainder of this chapter we always assume M is a compact smooth surface without boundary.
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Notes
- 1.
In general a smooth function f whose 4-jet at (0, 0) is f 4(x, y) = L 2 + B + C, where L, B, C are linear, cubic and quartic forms in x, y respectively, has type at least A 3 at x = y = 0 if and only if B = LQ for a quadratic form Q and exactly A 3 provided in addition L does not divide Q 2 − 4C.
- 2.
We are indebted to Jan Koenderink for this example and for the insight that with some care the general case can be deduced from this very special one.
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Damon, J., Giblin, P., Haslinger, G. (2016). Apparent Contours for Projections of Smooth Surfaces. In: Local Features in Natural Images via Singularity Theory. Lecture Notes in Mathematics, vol 2165. Springer, Cham. https://doi.org/10.1007/978-3-319-41471-3_3
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