158 DIGITAL IMAGE PROCESSING
4.4 DISCRETE SINE TRANSFORM
popular transforms used for image
Discrete sine transform(DST) is one of the
The ID DST for the sequence f(x) is given
as follows: compression
2
N-1
sin n(n +1)(k + 1) 0sxsN-1
F(k) = VN+1-xXf) N+1
x=0
follows:
The ID inverse discrete sine transform (IDST) is given as
N-1
2 sin z(n+1)(k +1) 0sksN-1
f()=x
VN+1
F(K) N+1
k=0
9
4.
The term 2 sin(n+1)(k +1)T 0<k.nsN-1 is the matrix Acalledkernel matriy
VN+1 N+1
For example, 2 x 2 kernel matrix A(k, x) for k = 0,1 and x =0,1 is given as follows:
2T
Sin sin
3 3
A=
2T 4T
sin sin
3
Like the previous transforms, the forward discrete sine transformn is given as
F= Af
where fis the input sequence, A is the kernel and F is the transformed sequence. 1h
matrix A is real, symmetrical, and unitary. Therefore, A* =A. The
inverse is given as
f= A* F=AF
Therefore, the forward and its inverse are same. This is not the real part of DST. Is
energy compaction is good and hence can be used
Like in other transforms, the for image compression.
2D DST is given as
follows: aforementioned logic can be extendedto 2D transfom. The
2 N-IN-]
F(k.)=,VN+1N+1
2Xfu)sin T(k+l)(x+1)
x0 y)
where k and / range between 0 to
N+1
T(+1)(y+1)
N+1
N- 1.
The inverse DST is
given as follows:
f(x,y) = 2 N-N
N+1VNN-iF(k,)
+1 k=0 l=0 sin (k+1)(x+1)
+lXx +1) sin+IXyy+)
N+1 N+1
TRANSFORM
WALSH4.5
The The compaction is
DST As where Like where
erm Walsh
transform A
A other
in x
h(x, transform =A, and
is
y) good, the
the transforms, y
is kernel range
n is
forward
is DST fast
given matrix from
as isand
very and A'AFAFA = the
as its transformation to0
inverse and AfAF*
useful
computational N
N 1
X=0 N-1
the -1.
for original
sine
compressing of
i=0 n-1 transforms
can image
complexity
be
recovered
images. are x, DIGITAL
TRANSFORMS
y) IMAGE
same.
is can
O(n be
as
performed
log
n).
As using
its
energy DST
159
as
