Discrete Splines and Vision - Bezier and Splines in Image Processing and Machine Vision - page 153

Algorithm 1 :
x (0) =0;
x ( k ) =( I M ) x ( k− 1) λh 2 b ( x ( k − 1 ) )+ r ,for k =1 , 2 , ···

The following three points here are worth paying attention to:
(1) Existence and uniqueness of the solution of equation (7.38) were not ad-
dressed.
(2) Convergence of Algorithm 1 was not shown.
(3) Necessary condition did not have dependence on the interior points. The
constraint f ij + g ij < 4 for interior points is not taken into account.
Lee showed that for a range of λ , equation (7.38) has a unique solution that
provides a unique DSS . His proposed algorithm converges to this unique
solution.

− −

7.5.5 Some Important Points About DSS

(1) A DSS minimizes the error expression e ( x ) of equation (7.37) between all
x in the compact set S N .
(2) If R ( f, g ) is continuous, then e ( x ) is a continuous functional of x and its
infimum is in S N . This means DSS exists.
(3) A DSS x is regular, if for all ( i, j )in D i , f ij + g ij < 4.
(4) A regular DSS minimizes expression (7.37) and is an interior point in S N ,
so it satisfies equation (7.38).
(5) One can show that add DSS s are regular.
Theorem 2 :
If the function R in the image irradiance equation (7.34) is continuous, then
discrete smoothing splines exist and are regular.
From Theorem 2, one can tell that a regular DSS x ∗ exists that minimizes
error e ( x ) between all x and also satisfies equation (7.38). Hence we can write,

Mx ∗ = λh 2 b ( x ∗ )+ r . −

Matrix M is symmetric and positive definite, and so it has an inverse M − 1 .
This leads to:

x ∗ = λh 2 M − 1 ( x ∗ )+ M − 1 r . − (7.40)

7.6 A Provably Convergent Iterative Algorithm

To provide the algorithm, Lee rewrote equation (7.38) as

λh 2 M − 1 b ( x )+ M − 1 r , x = − (7.41)

and based on this the algorithm is as follows.
Algorithm 2
x (0) =0;
x ( k ) =

λh 2 M − 1 b ( x ( k− 1) )+ M − 1 r ,

− k =1 , 2 , ···