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Benjamin Kunsberg, Daniel Holtmann-Rice, Steven Zucker; Global constraints on integral curves of shaded surfaces. Journal of Vision 2015;15(12):967. doi: https://doi.org/10.1167/15.12.967.
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© ARVO (1962-2015); The Authors (2016-present)
The problem of shape from shading has been studied for many decades. Due to its ill-posed nature, almost every approach has attempted to reduce the ambiguity at the start. By making local assumptions regarding the light sources and surface, the problem can be reduced in complexity. However, due to the many ad hoc constraints, the resulting algorithms are often brittle and uninformative in describing possible biological structure. Rather, we use shape from shading as a test model to understand how the brain resolves geometric ambiguity. Our entire approach (VSS work 2012 - 2015) has been to mathematically represent ambiguity until it is at a scale sufficient to resolve itself. To this end, we have considered increasingly larger-scale features: local orientation flows (2012), critical contours (2013), and ridges (2013). In this work, we consider the global scale. There are global themes in an image that tie together the local patches. The difficulty is quantitatively describing these constraints without prior knowledge of the local patches. What does the boundary shape of a balloon animal tell you about the total air inside the balloon? We derive two theorems on the global geometry of a Lambertian surface. These are applicable regardless of light source direction or local geometry. Thus, they can be applied directly to constrain the local patches. One theorem restricts the total Gaussian curvature of the surface. The second is a relationship between the geodesic curvature of an isophote and its level value. To our knowledge, this is the first time that global constraints on isophotes have been derived. We prove these theorems and show experimental evidence illustrating their accuracy and use.
Meeting abstract presented at VSS 2015
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