Digital image transforms Digital image processing 4 DIGITAL

Digital image transforms Digital image processing 4. DIGITAL IMAGE TRANSFORMS 4. 1. Introduction 4. 2. Unitary orthogonal two-dimensional transforms Separable unitary transforms 4. 3. Properties of the unitary transforms Energy conservation Energy compaction; the variance of coefficients De-correlation Basis functions and basis images 4. 4. Sinusoidal transforms The 1 -D discrete Fourier transform (1 -D DFT) Properties of the 1 -D DFT The 2 -D discrete Fourier transform (2 -D DFT) Properties of the 2 -D DFT The discrete cosine transform (DCT) The discrete sine transform (DST) The Hartley transform 4. 5. Rectangular transforms The Hadamard transform = the Walsh transform The Slant transform The Haar transform 4. 6. Eigenvectors-based transforms The Karhunen-Loeve transform (KLT) The fast KLT The SVD 4. 7. Image filtering in the transform domain 4. 8. Conclusions

Digital image processing Digital image transforms

x 2 x’ 2 v 2 l 2=u(0, 1) v 1=0 v 1 x’ 2 v 2 U l 1=u(0, 0) x’ 1 x 1

Digital image processing Digital image transforms

Digital image processing Digital image transforms

Digital image processing Digital image transforms

Digital image processing Digital image transforms

Digital image transforms Digital image processing Basis functions and basis images KLT Haar Walsh Slant DCT Basis functions (basis vectors) Basis images (e. g. ): DCT, Haar, ….

= + + + + … Keeping only 50% of coefficients + +

Digital image processing Digital image transforms

Digital image processing Digital image transforms

Digital image processing Digital image transforms

Digital image processing Digital image transforms

Digital image processing Digital image transforms

Digital image processing Digital image transforms

Digital image processing Digital image transforms

Digital image processing Digital image transforms

Digital image processing Digital image transforms

Digital image processing Basis vectors for the Walsh-Hadamard transform Digital image transforms

Digital image transforms Digital image processing Original image Ordered Hadamard Non-ordered Hadamard

Digital image processing Digital image transforms

Digital image processing Digital image transforms

Digital image transforms Digital image processing Applying the Haar transform at block level (e. g. 2× 2 pixels blocks => Hr[2× 2]): Block transform: Rearrange coefficients: Applying the Haar transform at block level for a 4× 4 pixels blocks => Hr[4× 4]: Block transform: Rearrange coefficients:

Digital image processing Digital image transforms

Digital image processing Digital image transforms

KLT (PCA) Eigenimages – examples: Facial image set Corresponding “eigenfaces” 3 eigenimages and the individual variations on those components Face aproximation, from rough to detailed, as more coefficients are added

Digital image processing Digital image transforms

DFT Original image = (white square, grey background) + aditive noise DFT = sinc 2 -D for the square + cst. (for noise) LPF 2 -D IDFT

The 2 -D spectrum of the image and the filters applied: Noisy image; periodic noise as vertical lines In the regions corresponding to the vertical lines frequencies Image restoration through filtering

Digital image processing Digital image transforms