-
8/6/2019 Discrete Cosin Transform
1/9
Image Compression Using the
Discrete Cosine Transform
Tran The Linh
Nguyen Duy Tien
Ngo Duc ThanhImage Compression Using the DCT
-
8/6/2019 Discrete Cosin Transform
2/9
Image Compression Using the Discrete
Cosine Transform Abstract
The discrete cosine transform (DTC) is a
technique for converting a signal into elementary
frequency components. It is widely used image
compression. Here we develop some simple
functions to compute the DCT and compress
images.These functions illustrate the power of Matlab in
the prototyping of image processing algorithms
Image Compression Using the DCT
-
8/6/2019 Discrete Cosin Transform
3/9
The one-Dimensional Discrete
Cosine Transform
The One-Dimensional Discrete Cosine Transform
The Discrete Cosine Transform of a list of n real
number x(n), n=1N, is the list of length N givenby:
where:
Image Compression Using the DCT
1
(2 1)( 1)( ) ( ) ( )cos( ) ( 1... )
2
N
n
n k y k w k x n k N
N
T
!
! !
10
( )2
if kN
w k
otherwiseN
!
!
-
8/6/2019 Discrete Cosin Transform
4/9
The one-Dimensional Discrete
Cosine Transform
The example
Image Compression Using the DCT
x=50sin(2pi*t/40);
X=dct(x);
plot(t,x,t,X);legend('x(t)',DCt(x));
-
8/6/2019 Discrete Cosin Transform
5/9
The one-Dimensional Discrete
Cosine Transform
The example
Find how many DCT coefficients represent 99% of the
energy in a sequence:
Image Compression Using the DCT
x = (1:100) + 50*cos((1:100)*2*pi/40);
X = dct(x); [XX,ind] = sort(abs(X));
ind = fliplr(ind); i = 1;
while (norm([X(ind(1:i))
zeros(1,100-i)])/norm(X)
-
8/6/2019 Discrete Cosin Transform
6/9
The Two-Dimensional Discrete
Cosine Transform
The One-Dimensional DCT is useful in
processing on dimensional signals such as
speech waveforms. For analysis of two-
dimensional (2D) signal such as images, we
need a 2D version of the DCT. for an matrix.
The DCT-2D is computed in a simply way: The
DCT-1D is applied to each row of x and the toeach column of the result.
Image Compression Using the DCT
-
8/6/2019 Discrete Cosin Transform
7/9
The Two-Dimensional Discrete
Cosine Transform
Thus, the transform of x given by:
where:
Image Compression Using the DCT
1 1
0 0
0 1(2 1) (2 1)cos( )cos( )
0 12 2
M N
pq p q mn
m n
p Mm p n qB A
q NM
N
T TE E
! !
e e !
e e
11, 0, 0
2 2,1 1 ,1 1
p q
qpNM
and
p M q MM N
E E
!!
! !
e e e e
-
8/6/2019 Discrete Cosin Transform
8/9
The Two-Dimensional Discrete
Cosine Transform
The example
RGB = imread(hnen.bmp');
I = rgb2gray(RGB);
J = dct2(I);
imshow(log(abs(J)),[]),
colormap(jet(64)), colorbar;
Image Compression Using the DCT
-
8/6/2019 Discrete Cosin Transform
9/9
The Two-Dimensional Discrete
Cosine Transform
Now set values less than magnitude 10 in the DCT matrix to
zero, and then reconstruct the image using the inverse DCT
function idct2.
Image Compression Using the DCT
J(abs(J) < 10) = 0;K = idct2(J);
imshow(I) figure,
imshow(K,[0 255]);
