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On Differential Photometric Reconstruction for Unknown, Isotropic BRDFs
Dec. 2013 (vol. 35 no. 12)
pp. 2941-2955
Manmohan Chandraker, NEC Labs. America, Inc., Cupertino, CA, USA
Jiamin Bai, Electr. Eng. & Comput. Sci. Dept., Univ. of California, Berkeley, Berkeley, CA, USA
Ravi Ramamoorthi, Electr. Eng. & Comput. Sci. Dept., Univ. of California, Berkeley, Berkeley, CA, USA
This paper presents a comprehensive theory of photometric surface reconstruction from image derivatives in the presence of a general, unknown isotropic BRDF. We derive precise topological classes up to which the surface may be determined and specify exact priors for a full geometric reconstruction. These results are the culmination of a series of fundamental observations. First, we exploit the linearity of chain rule differentiation to discover photometric invariants that relate image derivatives to the surface geometry, regardless of the form of isotropic BRDF. For the problem of shape-from-shading, we show that a reconstruction may be performed up to isocontours of constant magnitude of the gradient. For the problem of photometric stereo, we show that just two measurements of spatial and temporal image derivatives, from unknown light directions on a circle, suffice to recover surface information from the photometric invariant. Surprisingly, the form of the invariant bears a striking resemblance to optical flow; however, it does not suffer from the aperture problem. This photometric flow is shown to determine the surface up to isocontours of constant magnitude of the surface gradient, as well as isocontours of constant depth. Further, we prove that specification of the surface normal at a single point completely determines the surface depth from these isocontours. In addition, we propose practical algorithms that require additional initial or boundary information, but recover depth from lower order derivatives. Our theoretical results are illustrated with several examples on synthetic and real data.
Index Terms:
surface reconstruction,computational geometry,gradient methods,image sequences,photometry,stereo image processing,differential photometric reconstruction,lower order derivatives,surface depth,surface normal,constant depth,surface gradient,photometric flow,optical flow,surface information,temporal image derivative,spatial image derivative,photometric stereo,shape-from-shading,surface geometry,photometric invariants,chain rule differentiation,geometric reconstruction,image derivatives,photometric surface reconstruction,comprehensive theory,isotropic BRDF,Light sources,Differential theory,Image reconstruction,Surface reconstruction,Lighting,Photometric measurements,differential theory,Surface reconstruction,general BRDF,photometric invariants
Citation:
Manmohan Chandraker, Jiamin Bai, Ravi Ramamoorthi, "On Differential Photometric Reconstruction for Unknown, Isotropic BRDFs," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 35, no. 12, pp. 2941-2955, Dec. 2013, doi:10.1109/TPAMI.2012.217
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