Abstract

A method is presented for the representation of (pictures of) faces. Within a specified framework the representation is ideal. This results in the characterization of a face, to within an error bound, by a relatively low-dimensional vector. The method is illustrated in detail by the use of an ensemble of pictures taken for this purpose.

© 1987 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972).
  2. R. B. Ash, M. F. Gardner, Topics in Stochastic Processes (Academic, New York, 1975).
  3. N. Ahmed, M. H. Goldstein, Orthogonal Transforms for Digital Signal Processing (Springer-Verlag, New York, 1975).
    [CrossRef]
  4. B. B. Mandelbrot, The Fractral Geometry of Nature (Freeman, San Francisco, Calif., 1982).
  5. F. Riesz, B. Sz.- Nagy, Functional Analysis (Ungar, New York, 1955).
  6. R. Bellman, Introduction to Matrix Analysis (McGraw-Hill, New York, 1960).

Ahmed, N.

N. Ahmed, M. H. Goldstein, Orthogonal Transforms for Digital Signal Processing (Springer-Verlag, New York, 1975).
[CrossRef]

Ash, R. B.

R. B. Ash, M. F. Gardner, Topics in Stochastic Processes (Academic, New York, 1975).

Bellman, R.

R. Bellman, Introduction to Matrix Analysis (McGraw-Hill, New York, 1960).

Fukunaga, K.

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972).

Gardner, M. F.

R. B. Ash, M. F. Gardner, Topics in Stochastic Processes (Academic, New York, 1975).

Goldstein, M. H.

N. Ahmed, M. H. Goldstein, Orthogonal Transforms for Digital Signal Processing (Springer-Verlag, New York, 1975).
[CrossRef]

Mandelbrot, B. B.

B. B. Mandelbrot, The Fractral Geometry of Nature (Freeman, San Francisco, Calif., 1982).

Nagy, B. Sz.-

F. Riesz, B. Sz.- Nagy, Functional Analysis (Ungar, New York, 1955).

Riesz, F.

F. Riesz, B. Sz.- Nagy, Functional Analysis (Ungar, New York, 1955).

Other (6)

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972).

R. B. Ash, M. F. Gardner, Topics in Stochastic Processes (Academic, New York, 1975).

N. Ahmed, M. H. Goldstein, Orthogonal Transforms for Digital Signal Processing (Springer-Verlag, New York, 1975).
[CrossRef]

B. B. Mandelbrot, The Fractral Geometry of Nature (Freeman, San Francisco, Calif., 1982).

F. Riesz, B. Sz.- Nagy, Functional Analysis (Ungar, New York, 1955).

R. Bellman, Introduction to Matrix Analysis (McGraw-Hill, New York, 1960).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Average face based on an ensemble of 115 faces. In this, as in the other plates, we have refrained from filtering out the high frequencies produced by the digitization. A pleasanter picture can be had by the usual trick of squinting or otherwise blurring the picture.

Fig. 2
Fig. 2

Sample face on top and its caricature below it.

Fig. 3
Fig. 3

Cropped faces: upper, the average; middle, a sample face; and bottom, its caricature.

Fig. 4
Fig. 4

First eight eigenpictures starting at upper left, moving to the right, and ending at lower right.

Fig. 5
Fig. 5

Approximation to the exact picture (middle panel of Fig. 3) using 10, 20, 30, and 40 eigenpictures.

Fig. 6
Fig. 6

Percent error versus number of eigenpictures used in the approximation. Solid curve is for picture shown in Fig. 2 (see also Fig. 5). Dashed curve is average over 10 different sample faces.

Fig. 7
Fig. 7

Probability projection along the eigenpicture directions; see Eq. (27).

Fig. 8
Fig. 8

Percent error versus number of eigenpictures for the full face shown in Fig. 2. The dashed curve is the corresponding cropped face and is the same as in Fig. 6.

Equations (27)

Equations on this page are rendered with MathJax. Learn more.

φ ( x ) φ i j = ( φ ) i j ,
φ ( x ) m , n N a m n exp [ 2 π i ( n x 1 + m x 2 ) ] ,
φ ¯ = φ = 1 M n = 1 M φ ( n ) .
ϕ ( n ) = φ ( n ) - φ ¯
( u ( n ) , u ( m ) ) = δ m n ,
λ 1 = 1 M n = 1 M ( u ( 1 ) , ϕ ( n ) ) 2
λ k = 1 M n = 1 M ( u ( k ) , ϕ ( n ) ) 2 = ( u ( k ) , ϕ ) 2
( u ( k ) , u ( l ) ) = δ k l ,             l k .
C = 1 M n = 1 M ϕ ( n ) ϕ ( m ) ,
C u ( n ) = λ ( n ) u ( n ) .
C = ϕ ϕ ,
C ( x , y ) = ϕ ( x ) ϕ ( y ) = 1 M n = 1 M ϕ ( n ) ( x ) ϕ ( n ) ( y ) ,
C ( x , y ) = n = 1 λ n u ( n ) ( x ) u ( n ) ( y ) ,
ϕ ( x ) = n = 1 a n u ( n ) ( x )
u = k = 1 M a k ϕ ( k ) ,
n = 1 M L m n a n = λ a m ,
L m n = ( ϕ ( m ) , ϕ ( n ) ) .
C ( x , y ) u ( y ) d y = λ u ( x ) .
u ( x ) = n = 1 M a n ϕ ( n ) ( x ) .
L m n = ϕ ( m ) ( x ) ϕ ( n ) ( x ) d x .
φ ˜ ( x ) = I r ( x ) .
φ ( x ) = I 0 φ ˜ ( x 0 ) φ ˜ ( x ) .
φ = φ ¯ + n = 1 M a n u ( n ) ,
a n = ( u ( n ) , φ - φ ¯ ) .
φ φ ¯ + n = 1 N a n u ( n ) = φ N
E N = φ - φ N / φ .
p n = λ n k λ k .

Metrics