Discrete Splines and Vision - Bezier and Splines in Image Processing and Machine Vision - page 152

1

One can assume that m

m . Now if we designate the surface
orientation at the grid point ( i, j )by( f ij ,g ij ) and the image brightness by
I ij , then ( f ij ,g ij )areknownif( i, j

≤ n and h =

∂D ). Consider a vector x of surface
orientations at an interior grid point D i as

∈

) T .

x =( ··· ,f ij , ··· , , ··· ,g ij , ···

Then x is defined on a compact set S N , where S is disc of radius 2 in the
fg -plane and N is the number of interior grid points in D i . A corresponding
smoothing spline or DSS minimizes the following error between all x :
e ( x )=
i,j

( 1

f i,j ) 2 +( f i,j +1 − f i,j ) 2

h 2 (( f i +1 ,j −

(7.37)

g i,j ) 2 +( g i,j +1 − g i,j ) 2 + λ ( R ( f i,j ,g i,j ) I i,j ) 2 ) .

+( g i +1 ,j − −

The term ( i, j ) is included in the sum if and only if { ( i, j ) , ( i +1 ,j ) , ( i, j +1) }∈

D . The minimization is subject to the condition { f ij ,g ij }∈ ∂D , so that f ij

and g ij are known.

7.5.4 Necessary Condition and the System of Equations

One can find the necessary condition for a DSS and hence the system of
equations by computing the partial derivatives of e ( x ) in equation (7.37) with
respect to f ij and g ij for all ( i, j )in D i . Equating these derivative to zero,
one gets in a generalized form the necessary condition,

λh 2 b ( x )+ r .

Mx = − (7.38)

N Laplacian Matrix of D i . Here, M = diag ( A, A ) where A is the N ×

I ij ) ∂R ( f ij ,g ij )
∂f ij

b ( x )=( ··· , ( R ( f ij ,g ij ) − , ··· ,

I ij ) ∂R ( f ij ,g ij )
∂g ij

··· , ( R ( f ij ,g ij ) − , ··· ) T

) T .Now, r ij = 0 when all the four neighbors of ( i, j ) th
pixel are within the region D i , otherwise r ij

and r =( ··· ,r ij , ···

= 0 and its value depends on the
number of pixels lying outside the region. Obviously, there can be a number
of situations; for example, suppose the grid points at ( i 1)

are the boundary points and f i− 1 ,j , f i,j− 1 , g i− 1 ,j ,and g i,j− 1 are known. This
provides r ij = f i− 1 ,j + f i,j− 1 . For details see the article by David Lee [101].
The remaining cases can be treated similarly. Equation (7.38) is equivalent to

− 1 ,j ) and ( i, j −

λ h 2 b ( x )+ r , x =( I − M ) x − (7.39)

where I is the identity matrix of size 2 N . An algorithmic approach to solve
equation (7.39) is described below [81].