Even Discrete Cosine Transform The Chinese University of Hong Kong Math 3360: Mathematical Imaging Lecture 11: More about DFT & Even Discrete Cosine Transform Prof. Ronald Lok Ming Lui Department of Mathematics, The Chinese University of Hong Kong

Recap: Properties of DFT

Recap: Algorithm of FFT

Elementary images of DFT decomposition Recap

Elementary images of DFT decomposition Recap

Real example Truncate smallest X% Fourier coefficients Truncate 70% of Fourier coefficients Truncate 50% of Fourier coefficients Truncate 30% of Fourier coefficients Truncate 10% of Fourier coefficients

Real example Truncate smallest X% Fourier coefficients Difference Image: Original minus Compressed image Truncate ~63% of Fourier coefficients Original

Real example Truncate smallest X% Fourier coefficients Difference Image: Original minus Compressed image Truncate 10% of Fourier coefficients

Real example Truncate smallest X% Fourier coefficients Truncate 30% of Fourier coefficients

Real example Truncate smallest X% Fourier coefficients Truncate 50% of Fourier coefficients

Real example Truncate smallest X% Fourier coefficients Truncate 70% of Fourier coefficients

Even Discrete Cosine Transform 1D and 2D Even Discrete Cosine Transform: 1D: 2D: For details, please refer to Lecture Note Chapter 2

Inverse Even Discrete Cosine Transform Image Decomposition

Elementary images of EDCT decomposition

Reconstruction w/ EDCT decomposition = using 1x1 elementary images (first 1 row and first 1 column elementary images; = using 2x2 elementary images (first 2 rows and first 2 column elementary images…and so on…

Other similar transforms Odd Discrete Cosine Transform: Even Discrete Sine Transform: Odd Discrete Sine Transform: All of them have explicit formula for their inverses. (For details, please refer to Lecture Note Chapter 2)

Other similar transforms Inverse Odd Discrete Cosine Transform: Inverse Even Discrete Sine Transform: Inverse Odd Discrete Sine Transform: (For details, please refer to Lecture Note Chapter 2)

Elementary images of ODCT decomposition

Reconstruction w/ ODCT decomposition = using 1x1 elementary images (first 1 row and first 1 column elementary images; = using 2x2 elementary images (first 2 rows and first 2 column elementary images…and so on…

Elementary images of EDST decomposition

Reconstruction w/ EDST decomposition = using 1x1 elementary images (first 1 row and first 1 column elementary images; = using 2x2 elementary images (first 2 rows and first 2 column elementary images…and so on…

Elementary images of ODST decomposition

Reconstruction w/ ODST decomposition = using 1x1 elementary images (first 1 row and first 1 column elementary images; = using 2x2 elementary images (first 2 rows and first 2 column elementary images…and so on…

Comparison of errors The flower example:

More example on DCT decomposition

More example on DCT decomposition Original image DCT

More example on DCT decomposition Original image DCT

More example on DCT decomposition Original image DCT

More example on DCT decomposition

More example on DCT decomposition

More example on DCT decomposition

More example on DCT decomposition

More example on DCT decomposition

More example on DCT decomposition