Discrete Splines and Vision - Bezier and Splines in Image Processing and Machine Vision - page 150

the interior of D , respectively. The silhouette for an object in the image plane
provides the outline of projection. If the object has a smooth surface, then
the silhouette provides occluding boundaries, where the surface orientation is
known. Though it has a problem, even then surface orientations can be made
known depending on the nature of the reflectance map, e.g., if the reflectance
map is a strictly monotonic function of gradients along x and y axes. David
Lee, however, considered the surface orientations known on the boundary ∂D
of the image domain that contains the occluding boundary.

7.5.1 Image Irradiance Equation

For a Lambertian surface illuminated by a single distant point source, the
reflectance map is

1+ pp s + qq s
1+ p 2 + q 2 1+ p s + q s

R ( p, q )= . (7.32)

R ( p, q ) is the function of surface gradient ( p, q ) and the gradient ( p s ,q s ) speci-
fies the direction of the source. The reflectance map tells the relation of image
brightness on surface orientation. In the image plane at a particular point
( x, y ), we record the image irradiance I ( x, y ). It is proportional to the image
radiance at the corresponding point on the surface. R ( p, q ) is known as the
image radiance. Hence, by normalizing, we get the image irradiance equation
as

R ( p, q )= I ( x, y ) . (7.33)

If we take f ( x, y )and g ( x, y ) as two different functions for p and q , then the
reflectance map can also be written as

R ( f ( x, y ) ,g ( x, y )) = I ( x, y ) . (7.34)

In the present case, ( x, y ) D . The function R ( f, g ) can be determined the-

oretically or experimentally if distribution of light sources, viewing geometry,
and intrinsic reflecting properties of the materials composing the surface are
known [80]. One easily note that in stereographic projection, the Northern
Hemisphere is projected into a plane, namely the fg plane, tangent to the
Gaussian sphere at the North Pole with the South Pole as the center of pro-
jection. As it is a bijection of the Northern Hemisphere onto a disc S of radius
2inthe fg -plane, points in S provide surface orientations of visible parts of
the object's surface. Points on the circumference of S are, therefore, the ori-
entations of the points on the occluding boundaries. Therefore, for any point
( x, y ) on the occluding boundary, we must have,

∈

f 2 ( x, y )+ g 2 ( x, y )=4 .

(7.35)

One can assume for interior points ( x, y )

∈ D i , I ( x, y ) > 0 and, for occluding

boundary points, ( x, y )

∈ ∂D , I ( x, y )=0.