28-07-2012, 04:38 PM
Discrete Cosine Transform and Image Compression
[attachment=29427]
Overview
One-dimensional DCT
Least Squares Approximation
Two-dimensional DCT
Image Compression (grayscale, color)
One-dimensional DCT
Definition: Let n be a positive integer. The one-dimensional DCT of order n is defined by an n x n matrix C whose entries are
Interpolation with DCT
Why interpolate with DCT?
What about Lagrange or Splines?
DCT interpolation gives terms already arranged in terms of importance to the human visual system !!
First terms are most important, last terms are least important
2-D Least Squares
Done in the same way as with 1-D
Implement a low pass filter (drop terms)
Delete the “high-frequency” components
Two-Dimensional DCT
Idea 2D-DCT: Interpolate the data with a set of basis functions
Organize information by order of importance to the human visual system
Used to compress small blocks of an image
(8 x 8 pixels in our case)
Image Compression
Image compression is a method that reduces the amount of memory it takes to store in image.
We will exploit the fact that the DCT matrix is based on our visual system for the purpose of image compression.
This means we can delete the least significant values without our eyes noticing the difference.
Now we have found the matrix Y = C(CXT)T
Using the DCT, the entries in Y will be organized based on the human visual system.
The most important values to
our eyes will be placed in the
upper left corner of the matrix.
The least important values
will be mostly in the lower
right corner of the matrix.
[attachment=29427]
Overview
One-dimensional DCT
Least Squares Approximation
Two-dimensional DCT
Image Compression (grayscale, color)
One-dimensional DCT
Definition: Let n be a positive integer. The one-dimensional DCT of order n is defined by an n x n matrix C whose entries are
Interpolation with DCT
Why interpolate with DCT?
What about Lagrange or Splines?
DCT interpolation gives terms already arranged in terms of importance to the human visual system !!
First terms are most important, last terms are least important
2-D Least Squares
Done in the same way as with 1-D
Implement a low pass filter (drop terms)
Delete the “high-frequency” components
Two-Dimensional DCT
Idea 2D-DCT: Interpolate the data with a set of basis functions
Organize information by order of importance to the human visual system
Used to compress small blocks of an image
(8 x 8 pixels in our case)
Image Compression
Image compression is a method that reduces the amount of memory it takes to store in image.
We will exploit the fact that the DCT matrix is based on our visual system for the purpose of image compression.
This means we can delete the least significant values without our eyes noticing the difference.
Now we have found the matrix Y = C(CXT)T
Using the DCT, the entries in Y will be organized based on the human visual system.
The most important values to
our eyes will be placed in the
upper left corner of the matrix.
The least important values
will be mostly in the lower
right corner of the matrix.