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Shape from shading through photometric motion

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Abstract

We present a new shape from shading algorithm, extending to the single-input case, a recently introduced approach to the photometric motion process. As proposed by Pentland, photometric motion is based on the intensity variation, due to the motion, at a given point on a rotating surface. Recently, an alternative formulation has also appeared, based on the intensity change at a fixed image location. Expressing this as a function of reflectance-map and motion-field parameters, a constraint on the shape of the imaged surface can be obtained. Coupled with an affine matching constraint, this has been shown to yield a closed-form expression for the surface function. Here, we extend such formulation to the single-input case, by using the Green’s function of an affine matching equation to generate an artificial pair to the input image, corresponding to an approximate rendition of the imaged surface under a rotated view. Using this, we are able to obtain high quality shape-from-shading estimates, even under conditions of unknown reflectance map and light source direction, as demonstrated here by an extensive experimental study.

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Authors and Affiliations

  1. Instituto de Computação, Universidade Federal Fluminense, Niterói, RJ, 24210-240, Brazil

    João L. Fernandes & José R. A. Torreão

Authors
  1. João L. Fernandes
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  2. José R. A. Torreão
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Correspondence to João L. Fernandes.

Appendix

Appendix

Here, we derive the Green’s function of the differential equation (21). Identifying k ≡ a/σ 2, let us rewrite that equation as

$$ \frac{1}{2}I_{2}^{\prime \prime } + \left( {\frac{x + a}{{\sigma^{2} }}} \right)I_{2}^{\prime } + \frac{1}{2}\left[ {k^{2} + \frac{{(x + a)^{2} + \sigma^{2} }}{{\sigma^{4} }}} \right]I_{2} = k^{2} I_{1} .$$
(31)

The Green’s function, G(x,x 0), will then satisfy

$$ \frac{1}{2}G^{\prime\prime}(x,x_{0} ) + \left( {\frac{x + a}{{\sigma^{2} }}} \right)G^{\prime}(x,x_{0} ) + \frac{1}{2}\left[ {k^{2} + \frac{{(x + a)^{2} + \sigma^{2} }}{{\sigma^{4} }}} \right]G(x,x_{0} ) = k^{2} \delta (x - x_{0} ) .$$
(32)

The impulse function vanishes identically for x > x 0 or x < x 0, and thus G(x,x 0) will be the homogeneous solution to (32), as long as x ≠ x 0. This is easily found to be the Gabor function

$$ W(x) = e^{ikx} e^{{ - \frac{{(x + a)^{2} }}{{2\sigma^{2} }}}} .$$
(33)

On the other hand, at x = x 0, G(x,x 0) must be such that the left-hand side of (32) yields an impulse discontinuity. Assuming G(x,x 0) to be continuous, and integrating both sides of (32) with respect to x, around x = x 0, we find that this requires a discontinuity in derivative at x 0, namely,

$$ G^{\prime}(x_{0 + } ,x_{0} ) - G^{\prime}(x_{0 - } ,x_{0} ) = 2k^{2} ,$$
(34)

where \( x_{0 + } = x_{0} + \varepsilon , \) and \( x_{0 - } = x_{0} - \varepsilon , \) for an infinitesimal ε.

Bearing such conditions in mind, we find that a bounded G(x, x 0), tending to zero at infinity, can be obtained as the imaginary part of the complex kernel

$$ K(x,x_{0} ) = \left\{ {\begin{array}{*{20}c} {N(x_{0} )W(x)W(x_{0} )} & {x > x_{0} } \\ 0 & {x < x_{0} } \\ \end{array} } \right. $$
(35)

K(x,x 0) will then, by construction, satisfy (32) for x > x 0 or x < x 0, there remaining only the factor N(x 0) to be determined from the discontinuity condition of (34). After some trivial manipulation, this yields

$$ N(x_{0} ) = \frac{{2k^{2} }}{{\text{Im} [W^{\prime}(x_{0} )W^{*}(x_{0} )]}} ,$$
(36)

and we obtain

$$ G(x,x_{0} ) = 2k^{2} \frac{{\text{Im} [W(x)W^{*}(x_{0} )]}}{{\text{Im} [W^{\prime}(x_{0} )W^{*}(x_{0} )]}} .$$
(37)

Substituting W(x) from (33), we finally get

$$ G(x,x_{0} ) = 2k\sin [k(x - x_{0} )]e^{{ - \left[ {\frac{{(x + a)^{2} - (x_{0} + a)^{2} }}{{2\sigma^{2} }}} \right]}} ,$$
(38)

thus completing our proof.

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Fernandes, J.L., Torreão, J.R.A. Shape from shading through photometric motion. Pattern Anal Applic 13, 35–58 (2010). https://doi.org/10.1007/s10044-009-0156-z

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