Image Compression - Bezier and Splines in Image Processing and Machine Vision - page 84

k k

α ( C gi + C ie )+ β c + β + γ

i =1 i =1

R c = , (4.1)

M 2 log 2 L

where C g,i ,i =1 , 2 ,

,k is the number of coecients required for the
i th subimage for variable order global approximation and C ie is the number
of coecients for regions that require residual approximation. We assign α
number of bits for each coecient. β c is the overhead for all patches due to
such correction. β is the overhead due to different orders of approximation of
subimages and γ is the number of bits for contour representation of the image.
If ν is the number of bits required for encoding texture blocks, then the
equation (4.1) becomes

···

k k

α ( C gi + C ie )+ β c + β + ν + γ

i =1 i =1

R c = . (4.2)

M 2 log 2 L

Note that the number of bits for graylevel approximation, in general, is

k k

β gr = α ( C gi + C ie )+ β c + β + ν. (4.3)

i =1 i =1

If the global approximation itself is su cient to meet the desired error crite-
rion so that the approximation of residual error is not needed, then the term
containing C ie and hence β c in equation (4.2) do not contribute anything and
under such conditions, equation (4.2) reduces to

k

α C gi + β + ν + γ

i =1

R c = . (4.4)

M 2 log 2 L

Further, when all global approximations are seen to be of fixed order and local
residual approximations are also of fixed order, we get the total number N c
of coecients as

N c = C g k + C l ( N 1 + N 2 + ··· + N k ) , (4.5)

where C g is the number of coecients required for global approximation of
a subimage and C l is the number of coe cients for local residual surface
approximation of each of the N 1 ,N 2 , ··· ,N k patches. Compression ratio R c

in this case reduces to

R c = αN c + ν + γ
M 2 log 2 L

. (4.6)

Note that when all the regions N 1 ,N 2 ,

N k in all subimages are locally
approximated for their residual surface, we do not need to store information

···