Abstract

The relations between parabolic and planar points of a Lambertian surface M and critical points of the corresponding image irradiance E are studied. It is proved that critical points of E, with the exception of nondegenerate global maxima, occur at points on M with zero Gaussian curvature and that critical points of E that are stable with respect to changes of the position of the light source occur at planar points of M. Furthermore, it is shown that at global maxima of E there exists a simple relation between the principal curvatures of M and L, the graph of E. The relations between planar (parabolic) points of L and planar (parabolic) points of M are also analyzed. Finally, some relationships between isophotes of E and lines of curvature of M are investigated.

© 1994 Optical Society of America

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  1. J. J. Koenderink, A. van Doorn, “Photometric invariant related to solid shape,” Opt. Acta 27, 981–996 (1980).
    [CrossRef]
  2. A. L. Yuille, “Zero crossings on lines of curvature,” Comput. Vis. Graphics Image Process. 45, 68–87 (1989).
    [CrossRef]
  3. M. P. Do Carmo, Differential Geometry of Curves and Surfaces (Prentice-Hall, Englewood Cliffs, N.J., 1976), Chap. 3, pp. 134ff.
  4. K. J. Falconer, The Geometry of Fractal Sets (Cambridge U. Press, Cambridge, 1985), Chap. 3, pp. 28–50.
    [CrossRef]
  5. B. K P. Horn, Robot Vision (MIT Press, Cambridge, Mass., 1986), Chaps. 10 and 11, pp. 202ff.
  6. A. P. Pentland, “Local shading analysis,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 170–187 (1984).
    [CrossRef]
  7. L. J. Corwin, R. H. Szczarba, Calculus in Vector Spaces (Dekker, New York, 1979), Chap. 8, pp. 315ff.
  8. J. L. Goldberg, Matrix Theory with Applications (McGraw-Hill, New York, 1991), Chap. 5, pp. 237–238.
  9. J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations (Dover, New York, 1952), pp. 4 and 23ff and p. 40.
  10. R. T. Frankot, R. Chelappa, “A method for enforcing integrability in shape from shading algorithms,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 439–451 (1988).
    [CrossRef]
  11. W. F. Bishof, M. Ferraro, “Curved Mondrians: shading analysis of patterned objects,” Computat. Intell. 5, 121–126 (1989).
    [CrossRef]
  12. M. Berger, B. Gostiaux, Differential Geometry: Manifolds, Curves and Surfaces (Springer-Verlag, Berlin, 1988), Chap. 10, pp. 346ff.

1989 (2)

A. L. Yuille, “Zero crossings on lines of curvature,” Comput. Vis. Graphics Image Process. 45, 68–87 (1989).
[CrossRef]

W. F. Bishof, M. Ferraro, “Curved Mondrians: shading analysis of patterned objects,” Computat. Intell. 5, 121–126 (1989).
[CrossRef]

1988 (1)

R. T. Frankot, R. Chelappa, “A method for enforcing integrability in shape from shading algorithms,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 439–451 (1988).
[CrossRef]

1984 (1)

A. P. Pentland, “Local shading analysis,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 170–187 (1984).
[CrossRef]

1980 (1)

J. J. Koenderink, A. van Doorn, “Photometric invariant related to solid shape,” Opt. Acta 27, 981–996 (1980).
[CrossRef]

Berger, M.

M. Berger, B. Gostiaux, Differential Geometry: Manifolds, Curves and Surfaces (Springer-Verlag, Berlin, 1988), Chap. 10, pp. 346ff.

Bishof, W. F.

W. F. Bishof, M. Ferraro, “Curved Mondrians: shading analysis of patterned objects,” Computat. Intell. 5, 121–126 (1989).
[CrossRef]

Chelappa, R.

R. T. Frankot, R. Chelappa, “A method for enforcing integrability in shape from shading algorithms,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 439–451 (1988).
[CrossRef]

Corwin, L. J.

L. J. Corwin, R. H. Szczarba, Calculus in Vector Spaces (Dekker, New York, 1979), Chap. 8, pp. 315ff.

Do Carmo, M. P.

M. P. Do Carmo, Differential Geometry of Curves and Surfaces (Prentice-Hall, Englewood Cliffs, N.J., 1976), Chap. 3, pp. 134ff.

Falconer, K. J.

K. J. Falconer, The Geometry of Fractal Sets (Cambridge U. Press, Cambridge, 1985), Chap. 3, pp. 28–50.
[CrossRef]

Ferraro, M.

W. F. Bishof, M. Ferraro, “Curved Mondrians: shading analysis of patterned objects,” Computat. Intell. 5, 121–126 (1989).
[CrossRef]

Frankot, R. T.

R. T. Frankot, R. Chelappa, “A method for enforcing integrability in shape from shading algorithms,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 439–451 (1988).
[CrossRef]

Goldberg, J. L.

J. L. Goldberg, Matrix Theory with Applications (McGraw-Hill, New York, 1991), Chap. 5, pp. 237–238.

Gostiaux, B.

M. Berger, B. Gostiaux, Differential Geometry: Manifolds, Curves and Surfaces (Springer-Verlag, Berlin, 1988), Chap. 10, pp. 346ff.

Hadamard, J.

J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations (Dover, New York, 1952), pp. 4 and 23ff and p. 40.

Horn, B. K P.

B. K P. Horn, Robot Vision (MIT Press, Cambridge, Mass., 1986), Chaps. 10 and 11, pp. 202ff.

Koenderink, J. J.

J. J. Koenderink, A. van Doorn, “Photometric invariant related to solid shape,” Opt. Acta 27, 981–996 (1980).
[CrossRef]

Pentland, A. P.

A. P. Pentland, “Local shading analysis,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 170–187 (1984).
[CrossRef]

Szczarba, R. H.

L. J. Corwin, R. H. Szczarba, Calculus in Vector Spaces (Dekker, New York, 1979), Chap. 8, pp. 315ff.

van Doorn, A.

J. J. Koenderink, A. van Doorn, “Photometric invariant related to solid shape,” Opt. Acta 27, 981–996 (1980).
[CrossRef]

Yuille, A. L.

A. L. Yuille, “Zero crossings on lines of curvature,” Comput. Vis. Graphics Image Process. 45, 68–87 (1989).
[CrossRef]

Comput. Vis. Graphics Image Process. (1)

A. L. Yuille, “Zero crossings on lines of curvature,” Comput. Vis. Graphics Image Process. 45, 68–87 (1989).
[CrossRef]

Computat. Intell. (1)

W. F. Bishof, M. Ferraro, “Curved Mondrians: shading analysis of patterned objects,” Computat. Intell. 5, 121–126 (1989).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (2)

R. T. Frankot, R. Chelappa, “A method for enforcing integrability in shape from shading algorithms,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 439–451 (1988).
[CrossRef]

A. P. Pentland, “Local shading analysis,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 170–187 (1984).
[CrossRef]

Opt. Acta (1)

J. J. Koenderink, A. van Doorn, “Photometric invariant related to solid shape,” Opt. Acta 27, 981–996 (1980).
[CrossRef]

Other (7)

M. Berger, B. Gostiaux, Differential Geometry: Manifolds, Curves and Surfaces (Springer-Verlag, Berlin, 1988), Chap. 10, pp. 346ff.

L. J. Corwin, R. H. Szczarba, Calculus in Vector Spaces (Dekker, New York, 1979), Chap. 8, pp. 315ff.

J. L. Goldberg, Matrix Theory with Applications (McGraw-Hill, New York, 1991), Chap. 5, pp. 237–238.

J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations (Dover, New York, 1952), pp. 4 and 23ff and p. 40.

M. P. Do Carmo, Differential Geometry of Curves and Surfaces (Prentice-Hall, Englewood Cliffs, N.J., 1976), Chap. 3, pp. 134ff.

K. J. Falconer, The Geometry of Fractal Sets (Cambridge U. Press, Cambridge, 1985), Chap. 3, pp. 28–50.
[CrossRef]

B. K P. Horn, Robot Vision (MIT Press, Cambridge, Mass., 1986), Chaps. 10 and 11, pp. 202ff.

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Equations (25)

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E ( x , y ) = λ ρ ( x , y ) N ( x , y ) · s if N ( x , y ) · s 0 , E ( x , y ) = 0 if N ( x , y ) · s < 0 .
E ( x , y ) = λ ρ ( x , y ) × [ f x ( x , y ) s 1 + f y ( x , y ) s 2 + 1 ] [ 1 + f x 2 ( x , y ) + f y 2 ( x , y ) ] 1 / 2 ( 1 + s 1 2 + s 2 2 ) 1 / 2 ,
E / x | ( x 0 , y 0 ) = N / x | ( x 0 , y 0 ) · s = 0 ,
E / x | ( x 0 , y 0 ) = N / y | ( x 0 , y 0 ) · s = 0 .
E x x | ( x 0 , y 0 ) = ( f x x 2 + f x y 2 ) | ( x 0 , y 0 ) ,
E y y | ( x 0 , y 0 ) = ( f y y 2 + f x y 2 ) | ( x 0 , y 0 ) ,
E x y | ( x 0 , y 0 ) = ( f x x f x y + f y y f x y ) | ( x 0 , y 0 ) .
H E ( x 0 , y 0 ) = [ ( f x x 2 + f x y 2 ) | ( x 0 , y 0 ) ( f x x f x y + f y y f x y ) | ( x 0 , y 0 ) ( f x x f x y + f y y f x y ) | ( x 0 , y 0 ) ( f y y 2 + f x y 2 ) | ( x 0 , y 0 ) ] .
H f ( x 0 , y 0 ) = [ f x x | ( x 0 , y 0 ) f x y | ( x 0 , y 0 ) f x y | ( x 0 , y 0 ) f y y | ( x 0 , y 0 ) ] ,
H E ( x 0 , y 0 ) = H f ( x 0 , y 0 ) H f ( x 0 , y 0 ) .
λ 1 ( q ) = β 1 2 ( p ) , λ 2 ( q ) = β 2 2 ( p ) .
K L ( q ) = K M 2 ( p ) ,
/ x ( N / x · N ) = 2 N / x 2 · N + N / x · N / x = 0 ,
2 N / x 2 · N = N / x · N / x .
H E ( x 0 , y 0 ) = [ 2 E / x 2 | ( x 0 , y 0 ) 0 0 2 E / y 2 | ( x 0 , y 0 ) ] .
d E / d t = d N / d t · s = 0 ,
r : ( u , υ ) { x ( u , υ ) , y ( u , υ ) , z ( u , υ ) } , r : ( u , υ ) = [ x ( u , υ ) , y ( u , υ ) , z ( u , υ ) ] S . 3,12
α [ a , b ] = { x [ u ( t ) , υ ( t ) ] , y [ u ( t ) , υ ( t ) ] , z [ u ( t ) , υ ( t ) ] } t [ a , b ] .
N = ( f / x , f / y , 1 ) [ 1 + ( f / x ) 2 + ( f / x ) 2 ] 1 / 2 .
I = g 11 ( d x ) 2 + 2 g 12 d x d y + g 22 ( d y ) 2 ,
[ g i j ] = [ 1 + ( f / x ) 2 f / x f / y f / x f / y 1 + ( f / x ) 2 ] .
I I = b 11 ( d x ) 2 + 2 b 12 d x d y + b 22 ( d y ) 2 ,
[ b i j ] = [ 1 + ( f / x ) 2 + ( f / y ) 2 ] 1 / 2 × [ 2 f / 2 x 2 f / x y 2 f / x y 2 f / 2 y ] .
K = | H f | [ 1 + ( f / x ) 2 + ( f / y ) 2 ] 1 / 2 ,
k i = λ i [ 1 + ( f / x ) 2 + ( f / y ) 2 ] 1 / 2 ,

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