Image Processing Reference
In-Depth Information
8
Reconstruction and Approximation
Often we only know gray values of an image on a discrete set of points. This is
increasingly the case due to the ever more affordable digital imaging equipment, e.g.,
consumer still and motion picture cameras. Even when the original image is on film,
it must often be digitized for further processing on digital equipment, e.g., when
analog processing is not available. In this chapter we study the techniques used to
obtain some discrete image processing schemes for the most common mathematical
operations.
We discuss first the interpolation function and the characteristic function as this
concept has a significant impact on how one implements approximation of functions
and operators. Then we study the important class of linear operators and illustrate
them by computing the partial derivatives of images, and affine coordinate transfor-
mations, e.g., rotation and zooming, of images. Subsequently, we contrast the linear
operators with nonlinear operators and discuss “square of an image”, and “multipli-
cation of two images”, which are nonlinear.
The basic approach to approximation of mathematical operators is to reconstruct
the continuous image from its samples and then to apply the operator followed by
an appropriate sampling of the result. While discussing the typical pitfalls, our goal
is to generate discrete signal processing schemes using discrete data to approximate
continuous operators.
8.1 Characteristic and Interpolation Functions in
N
Dimensions
Definition 8.1.
The
characteristic function
on a limited volume
D
in
N
dimensions
(
N
D) is defined as
χ
D
(
r
)=
1
,
if
r
∈D
;
(8.1)
0
,
otherwise
.
This function is also known as the
indicator function
because it is used to indicate
two volumes, one in which the function is zero, and the complementary volume in
which it is not zero.
