Nonlinear Image Processing

406 CHRISTOPH ZETZSCHE AND GERHARD KRIEGER

Figure 14.1: Typical examples for the three basic classes of intrinsic dimensionality in

natural images. Each local region of the image can be classified according to its intrinsic

dimensionality. Two aspects become intuitively apparent from this example: (1) There

will be a gradual decrease of the probability of occurrence from i0D to i2D signals (i2D

signals are "rare events"). (2) There is a hierarchy of intrinsic dimensionality in terms of

redundancy, since knowledge of the i2D signals can be used to predict the more redundant

ilD and i0D signals.

14.2

Transdisciplinary Relevance of Intrinsic

Dimensionality

14.2.1 Definition

We are mainly interested in local signal properties and, accordingly, in operators

with limited spatial support, but for convenience, we use signals with infinite spa-

tial extent in all our definitions and derivations. For any operator with spatially

limited support it is simply irrelevant whether the input signal extends outside

the support region.

Intrinsic dimensionality is directly related to the degrees of freedom being used

by an actual image signal [Zet90b]. A signal

u(x, y)

can be classified as a member

of one of the classes i0D, ilD, or i2D according to the following rule:

{i0D}

u(x, y) ~

{ilD}

{i2D}

for u (x, y) = const,

for

u(x,

y) = func(ax +

by), (a . O) v (b .

0),

otherwise.

(14.1)

The first class is given by all signals that are constant, that is, do not depend on

(x, y). The second class comprises all signals that can be written as a function of

one variable in an appropriately rotated coordinate system, where the constants

a and b are the coefficients of the corresponding affine transform. Signals that

cannot be classified as either i0D or ilD belong to the third class.

Typical local image configurations with different types of intrinsic dimension-

ality are illustrated in Fig. 14.1, which shows a simplified view of a natural image.

Some basic features, such as the substantially differing probabilities or the dif-

ferent degrees of predictability associated with the three types of local signals,

CHAPTER 14: INTRINSIC DIMENSIONALITY

407

are already evident from this simple example. A detailed statistical evaluation is

given in the next section. Most important for the current context is the fact that

i2D signals are the most significant ones.

Since we make extensive use of the spectral representation of signals and op-

erators in subsequent derivations, we shall also give the corresponding Fourier-

domain classification rules for the intrinsic dimensionality, i0D signals may be

represented as

U(fx, fy) = K. 6(fx, fy)

where K is a constant and 6 denotes the

Dirac delta function. To derive a spectral representation of ilD signals, we start

from the subset of ilD signals that can be written as

u(x,

y) = ~p(x), Vx, y ~ R, (14.2)

where ~ (x) is an arbitrary function of one variable x. qJ (x) is independent of y

and therefore constant for all sections parallel to the ),-axis. Equation (14.2) can

be expressed in the spatial frequency domain as

U(fx,fy) : I~_oo f~-oo *(x)e-j2rr(f~x+fYY) dx dy : ~f(fx) " 6(fy),

(14.3)

where

xg(fx)

is the 1-D Fourier transform of ~p(x). Hence, the ilD signal of Eq.

(14.2) corresponds to a modulated Dirac line in the frequency domain.

Applying an affine coordinate transform to

u(x, y)

in Eq. (14.2) (which includes

rotation as a special case) and using the Fourier correspondence [Barn89]

1

tf ,

u(Ax) e=, det(A) "

U ((A -1 )

) det(A) ~: 0, (14.4)

where

A = ( all

\

a~21

we obtain

)

a12 x = f =

0,,22 ' y ' fy '

~(allX +a12y) e=,

I det(A) I

(14.5)

(a22 a21 ) (axx y a12 ) det n det n

Hence, every signal which can be written as u(x, 3') = tP(allx

+ a.12Y),

i.e., which

is constant along the direction perpendicular to the

vector (all)ax2 ,

has as its Fourier

transform a modulated Dirac delta line through the origin. This permits us to give

a definition of intrinsic dimensionality in the frequency domain:

{i0D} for

U(fx,fy) = K. 6(fx,fy),

U(fx,fy) ~

{ilD} for

U(fx,fy) = U(fx,fy) 9 6(afx + bfy), (a e: 0 v b ~: 0),

{i2D} otherwise.

(14.7)

In the following sections we provide some general arguments supporting the rel-

evance of the concept of intrinsic dimensionality for communication engineering

and biological vision systems. We further provide a short discussion of earlier

approaches related to the processing of i2D signals.