Image Compression - Bezier and Splines in Image Processing and Machine Vision - page 85

for β and β c . So these two terms do not contribute anything. Kunt et al. [97]
observed small errors in data approximation when each surface is represented
by its r pixels. These r pixels on the surface are used to recompute the co-
ecients. The only possible error appears in the quantization of each pixel.
We followed the same strategy and examined both the cases in our work.
Since each pixel can be represented by log 2 L bits, the equation (4.2) can be
rewritten as

R c = log 2 LN pix + β c + β + ν + γ
M 2 log 2 L

, (4.7)

where N pix is the total number of surface pixels. The number of bits required
for graylevel approximation in this case is

β gr = log 2 LN pix + β c + β + ν. (4.8)

In the following section, we discuss the choice of weights in the least square
approximation for the proposed coding scheme.

4.2.1 Approximation and Choice of Weights

Subimages obtained through the segmentation scheme as described in Chap-
ter 1 were used for compression. Readers interested in details can consult [24].
The approximation algorithms are exactly the same as used for segmenta-
tion, but the weights are different from unity. For compression, weights are
chosen in a way described below. For completeness and clear understanding,
we first briefly state the approximation problem. Bezier-Bernstein polynomial
has been used because our segmentation algorithm was basically designed for
image compression, and Bezier-Bernstein polynomial provides a number of
merits in compression and reconstruction. The Bezier-Bernstein surface is a
tensor product surface and is given by

p q

φ rp ( u ) φ zq ( v ) V rz s pq ( u, v )=

r =0 z =0

(4.9)

p q

B rp D zq u r (1 u ) p−r v z (1 v ) q−z V rz ,

− −

=

r =0 z = o

p ! q !

where u, v ∈ [0 , 1] and B rp =

( p−r )! r ! , D zq = ( q−z )! z ! . p and q define the

order of the Bezier-Bernstein surface.
To approximate an arbitrary image surface f ( x, y )ofsizeM

M, f ( x, y )
should be defined in terms of a parametric surface (here s pq with the pa-
rameters u and v both in [0 , 1]. Therefore, the function f ( x, y )canbe
thought in terms g ( u, v ) where u =

×

( i− 1)
( M− 1) ; i =1 , 2 ,..

( j− 1)
( M− 1) ;

··· M and v =

j =1 , 2 ,.. M .

We choose the weighted least square technique for estimation of parame-
ters V rz to be used for reconstruction of the decoded surface. Although the

···