Abstract

We explore the relationships between two classical means of mathematically representing visual patterns that are invariant under geometric transformations. One, based on integral transforms, in particular, Fourier transforms, produces transform magnitudes invariant to pairwise combinations of translations, rotations, and size changes. The second, based on the degree to which the pattern remains invariant to differential operators (which are the infinitesimal generators of the geometric transformations), results in algebraic relations between pattern structures and group theoretical properties of the transforms. Formal relationships are established between these representations, relating the kernel properties of the integral transforms to the associated Lie transformation groups.

© 1988 Optical Society of America

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References

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  1. A. Rosenfeld, A. C. Kak, Digital Picture Processing, 2nd ed. (Academic, Orlando, Fla., 1982).
  2. Y.-N. Hsu, H. H. Arsenault, G. April, “Rotation-invariant digital pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4012–4015 (1982).
    [CrossRef] [PubMed]
  3. N. J. Ferrier, “Invariance coding in pattern recognition,” M.Sc. thesis (University of Alberta, Edmonton, Alberta, Canada, 1987).
  4. H. W. Guggenheimer, Differential Geometry (McGraw-Hill, New York, 1963).
  5. P. J. Olver, Application of Lie Groups to Differential Equations (Springer-Verlag, Berlin, 1986).
    [CrossRef]
  6. W. C. Hoffman, “The Lie algebra of visual perception,” J. Math. Psychol. 3, 65–98 (1966);errata, 4, 348–349 (1967).
    [CrossRef]
  7. W. C. Hoffman, “Higher visual perception as prolongation of the basic Lie transformation group,” Math. Biosci. 6, 437–471 (1970).
    [CrossRef]
  8. J. S. Mondragón, K. B. Wolf, eds., Lie Methods in Optics (Springer-Verlag, Berlin, 1985).
  9. G. W. Bluman, J. D. Cole, Similarity Methods for Differential Equations (Springer-Verlag, Berlin, 1974).
    [CrossRef]
  10. B. Schutz, Geometrical Methods of Mathematical Physics (Cambridge U. Press, Cambridge, 1980).
  11. M. Spivak, A Comprehensive Introduction to Differential Geometry (Publish or Perish, Berkeley, Calif., 1979), Vol. 1.
  12. D. Casasent, D. Psaltis, “Scale invariant optical transform,” Opt. Eng. 15, 258–261 (1976).
    [CrossRef]
  13. L. V. Ovisiannikov, Group Analysis of Differential Equations (Academic, New York, 1982).

1982 (1)

1976 (1)

D. Casasent, D. Psaltis, “Scale invariant optical transform,” Opt. Eng. 15, 258–261 (1976).
[CrossRef]

1970 (1)

W. C. Hoffman, “Higher visual perception as prolongation of the basic Lie transformation group,” Math. Biosci. 6, 437–471 (1970).
[CrossRef]

1966 (1)

W. C. Hoffman, “The Lie algebra of visual perception,” J. Math. Psychol. 3, 65–98 (1966);errata, 4, 348–349 (1967).
[CrossRef]

April, G.

Arsenault, H. H.

Bluman, G. W.

G. W. Bluman, J. D. Cole, Similarity Methods for Differential Equations (Springer-Verlag, Berlin, 1974).
[CrossRef]

Casasent, D.

D. Casasent, D. Psaltis, “Scale invariant optical transform,” Opt. Eng. 15, 258–261 (1976).
[CrossRef]

Cole, J. D.

G. W. Bluman, J. D. Cole, Similarity Methods for Differential Equations (Springer-Verlag, Berlin, 1974).
[CrossRef]

Ferrier, N. J.

N. J. Ferrier, “Invariance coding in pattern recognition,” M.Sc. thesis (University of Alberta, Edmonton, Alberta, Canada, 1987).

Guggenheimer, H. W.

H. W. Guggenheimer, Differential Geometry (McGraw-Hill, New York, 1963).

Hoffman, W. C.

W. C. Hoffman, “Higher visual perception as prolongation of the basic Lie transformation group,” Math. Biosci. 6, 437–471 (1970).
[CrossRef]

W. C. Hoffman, “The Lie algebra of visual perception,” J. Math. Psychol. 3, 65–98 (1966);errata, 4, 348–349 (1967).
[CrossRef]

Hsu, Y.-N.

Kak, A. C.

A. Rosenfeld, A. C. Kak, Digital Picture Processing, 2nd ed. (Academic, Orlando, Fla., 1982).

Olver, P. J.

P. J. Olver, Application of Lie Groups to Differential Equations (Springer-Verlag, Berlin, 1986).
[CrossRef]

Ovisiannikov, L. V.

L. V. Ovisiannikov, Group Analysis of Differential Equations (Academic, New York, 1982).

Psaltis, D.

D. Casasent, D. Psaltis, “Scale invariant optical transform,” Opt. Eng. 15, 258–261 (1976).
[CrossRef]

Rosenfeld, A.

A. Rosenfeld, A. C. Kak, Digital Picture Processing, 2nd ed. (Academic, Orlando, Fla., 1982).

Schutz, B.

B. Schutz, Geometrical Methods of Mathematical Physics (Cambridge U. Press, Cambridge, 1980).

Spivak, M.

M. Spivak, A Comprehensive Introduction to Differential Geometry (Publish or Perish, Berkeley, Calif., 1979), Vol. 1.

Appl. Opt. (1)

J. Math. Psychol. (1)

W. C. Hoffman, “The Lie algebra of visual perception,” J. Math. Psychol. 3, 65–98 (1966);errata, 4, 348–349 (1967).
[CrossRef]

Math. Biosci. (1)

W. C. Hoffman, “Higher visual perception as prolongation of the basic Lie transformation group,” Math. Biosci. 6, 437–471 (1970).
[CrossRef]

Opt. Eng. (1)

D. Casasent, D. Psaltis, “Scale invariant optical transform,” Opt. Eng. 15, 258–261 (1976).
[CrossRef]

Other (9)

L. V. Ovisiannikov, Group Analysis of Differential Equations (Academic, New York, 1982).

A. Rosenfeld, A. C. Kak, Digital Picture Processing, 2nd ed. (Academic, Orlando, Fla., 1982).

N. J. Ferrier, “Invariance coding in pattern recognition,” M.Sc. thesis (University of Alberta, Edmonton, Alberta, Canada, 1987).

H. W. Guggenheimer, Differential Geometry (McGraw-Hill, New York, 1963).

P. J. Olver, Application of Lie Groups to Differential Equations (Springer-Verlag, Berlin, 1986).
[CrossRef]

J. S. Mondragón, K. B. Wolf, eds., Lie Methods in Optics (Springer-Verlag, Berlin, 1985).

G. W. Bluman, J. D. Cole, Similarity Methods for Differential Equations (Springer-Verlag, Berlin, 1974).
[CrossRef]

B. Schutz, Geometrical Methods of Mathematical Physics (Cambridge U. Press, Cambridge, 1980).

M. Spivak, A Comprehensive Introduction to Differential Geometry (Publish or Perish, Berkeley, Calif., 1979), Vol. 1.

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Figures (1)

Fig. 1
Fig. 1

Invariant functions of R, ℒD, and RD (see the text), which result in amplitude spectra that are invariant to rotations and size changes of the image: (x2 + y2)Re[w(u, υ; x, y)] = cos{log[(x2 + y2)1/2]u + tan−1(y/x)υ}, for orders in u and υ. Across: υ = 0; u = 0, l, 2, 4, 8, 16, 31 (picture cycles) invariants of R. Down: u = 0, υ = 0, 1, 2, 4, 8, 16, 31 (cycles/revolution) invariants of D.

Equations (51)

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O [ f ( x , y ) ] = g ( u , υ ) = + f ( x , y ) w ( u , υ ; x , y ) d x d y
g ( u , υ ) = A ( u , υ ) exp [ i ϕ ( u , υ ) ] ,
O [ T a f ( x , y ) ] = A ( u , υ ) exp { i [ ϕ ( u , υ ) + u a ] } .
O { T a [ S b ( f ) ] } = A ( u , υ ) exp { i [ ϕ ( u , υ ) + ( u a + υ b ) ] } .
O [ T a S b f ( x , y ) ] = g a , b ( u , υ ) = exp ( iau ) g 0 , b ( u , υ ) ,
O [ S b T a f ( x , y ) = g a , b ( u , υ ) = exp ( i b υ ) g a , 0 ( u , υ )
O [ f ( ξ , η ) ] = g ( u , υ ) = k + f ( η , ξ ) exp [ i ( η u + ξ υ ) ] d η d ξ ,
a η = 1 , a ξ = 0 ,
b η = 0 , b ξ = 1 .
a = a 1 ( x , y ) x + a 2 ( x , y ) y
b = b 1 ( x , y ) x + b 2 ( x , y ) y ,
a ( | w ( u , υ ; x , y ) | d x d y ) = 0 ,
b ( | w ( u , υ ; x , y ) | d x d y ) = 0 ,
w ( u , υ ; x , y ) = h ( x , y ) exp [ i γ ( u , υ ; x , y ) ] .
a ( h ( x , y ) d x d y ) = 0 ,
b ( h ( x , y ) d x d y ) = 0 .
a γ ( u , υ ; x , y ) = u ,
b γ ( u , υ ; x , y ) = υ .
a [ h ( ξ , η ) | J ( x , y η , ξ ) | d η d ξ ] = η [ h ( ξ , η ) | J ( x , y η , ξ ) | d η d ξ ] = 0 , b [ h ( ξ , η ) | J ( x , y η , ξ ) | d η d ξ ] = ξ [ h ( ξ , η ) | J ( x , y η , ξ ) | d η d ξ ] = 0 ,
η [ h ( ξ , η ) | J ( x , y η , ξ ) | ] = 0 ξ [ h ( ξ , η ) | J ( x , y η , ξ ) | ] = 0 ;
h ( η , ξ ) = 1 | J ( x , y η , ξ ) |
h ( x , y ) = | J ( η , ξ x , y ) | .
g ( u , υ ) + f ( x , y ) | J ( η , ξ x , y ) | × exp { i [ ( η ( x , y ) u + ξ ( x , y ) υ ] } d x d y
w ( u , υ ; x , y ) = | J ( η , ξ x , y ) | exp { i [ η ( x , y ) u + ξ ( x , y ) υ ] } .
Re [ w ( u , υ ; x , y ) ] + i Im [ w ( u , υ ; x , y ) ] ,
Re [ w ( u , υ ; x , y ) ] = | J ( η , ξ x , y ) | cos ( η u , ξ υ )
Im [ w ( u , υ ; x , y ) ] = | J ( η , ξ x , y ) | sin ( η u , ξ υ ) .
1 | J ( η , ξ x , y ) | Re [ w ( u , υ ; x , y ) ] ,
1 | J ( η , ξ x , y ) | Re [ w ( u , υ ; x , y ) ]
= υ a + u b .
{ 1 | J ( η , ξ x , y ) | Re [ w , υ ; x , y ) ] } = cos ( η u + ξ υ ) = υ sin ( η u + ξ υ ) a ( η u + ξ υ ) u sin ( η u + ξ υ ) b ( η u + ξ υ ) = sin ( η u + ξ υ ) ( υ u u υ ) = 0 .
= α x + β y , = α x + β y ,
η = 1 ( β α β / α ) ( y β / α x )
ξ = 1 ( β α β / α ) ( y β / α x ) .
T R = y x + x y , T D = x x + y y .
y x η ( x , y ) + x y η ( x , y ) = 1 , y x ξ ( x , y ) + x y ξ ( x , y ) = 0 , x x η ( x , y ) + y y η ( x , y ) = 0 , x x ξ ( x , y ) + y y ξ ( x , y ) = 1 .
x x ξ ( x 2 + y 2 ) + y y ξ ( x 2 + y 2 ) = 1 .
x z ξ ( z ) 2 x + y z ξ ( z ) 2 y = 2 x 2 z ξ ( z ) + 2 y 2 z ξ ( z ) = 1
ξ z = 1 2 z and ξ ( x , y ) = ξ ( x 2 + y 2 ) = log [ ( x 2 + y 2 ) 1 / 2 ] .
y 2 x 2 z η ( z ) + z η ( z ) = 1 , η ( x , y ) = tan 1 ( y / x ) .
| J ( η , ξ x , y ) | = 1 / ( x 2 + y 2 ) ;
w ( u , υ ; x , y ) = 1 x 2 + y 2 exp ( i { log [ ( x 2 + y 2 ) 1 / 2 ] u + tan 1 ( y / x ) υ } ) ,
Re [ w ( u , υ ; x , y ) ] = 1 x 2 + y 2 cos { log [ ( x 2 + y 2 ) 1 / 2 ] u + tan 1 ( y / x ) υ } .
( x 2 + y 2 ) Re [ w ( u , υ ; x , y ) ] = cos { log [ ( x 2 + y 2 ) 1 / 2 ] u + tan 1 ( y / x ) υ }
R D cos { log [ ( x 2 + y 2 ) 1 / 2 ] u + tan 1 ( y / x ) υ } ] = 0 ,
R D = υ R + u D = υ ( y x + x y ) + u ( x x + y y ) .
exp ( a a ) exp [ i γ ( u , υ ; x , y ) ] = exp [ i γ ( u , υ ; x , y ) ] + a a × exp [ i γ ( u , υ ; x , y ) ] + a 2 2 a 2 exp [ i γ ( u , υ ; x , y ) ] + ,
a exp [ i γ ( u , υ ; x , y ) ] = i exp [ i γ ( u , υ ; x , y ) ] a γ ( u , υ ; x , y ) ,
a 2 = exp [ i γ ( u , υ ; x , y ) ] [ a γ ( u , υ ; x , y ) ] 2 + i exp [ i γ ( u , υ; ; x , y ) ] a 2 [ γ ( u , υ ; x , y ) ] .
exp ( a a ) exp [ i γ ( u , υ ; x , y ) ] = exp [ i γ ( u , υ ; x , y ) ] ( 1 i a a γ ( u , υ ; x , y ) a 2 2 { [ a γ ( u , υ ; x , y ) ] 2 i a 2 γ ( u , υ ; x , y ) } ) = exp [ i γ ( u , υ ; x , y ) ] ( 1 iau a 2 2 u 2 )
a γ ( u , υ ; x , y ) = u .

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