1. Digital Image Processing Final Project Compression Using DFT, DCT, Hadamard and SVD Transforms Zvi Devir and Assaf Eden

2. The project’s criterions A.The Applied transform. B.The size of the blocks the image is divided to. C.The quality of the compression. D.Adaptive compression (later)

3. The DFT Transform n The 2D Discrete Fourier Transform (DFT) is defined by: n The 2D Discrete Fourier Transform (DFT) is defined by: n The transform is linear and separable. It maps a real function (matrix) f(x,y) into a complex phase plane u,v - denoted F(u,v). n The inverse transform is: n Can be calculated with O (n² log n).

4. The DCT Transform n The Discrete Cosine Transform (DCT) is defined as: N is the vector size n Unlike the DFT matrix, the DCT matrix is real and orthogonal. That is n The DCT transform can be calculated in O (n² log n). n The property of the DCT matrix together with the fact that it is a fast transform has made it useful substitute for the KL transform of highly correlated first order Markov sequences.

5. The Hadamard Transform n The Hadamard transform is defined by the sorted Hadamard matrix. n The Hadamard matrix has only binary values of ±1. n The Hadamard matrices are defined recursively: n The sorted Hadamard matrix can be calculated recursively too. n The Hadamard transform is real symmetric and orthogonal. That is n The transform can be calculated in O (n² log n).

6. Compression Using DCT, DFT and Hadamard n After transforming the image to the Phase plane, the significant data is given in a bounded area. n Our compression methods takes only the part that is significant and ignore the insignificant pixels (of the phase plane).

7. The SVD Transform (For Comparison) n The Single Value Decomposition (SVD) of a matrix is the factorization of Where U and V are orthogonal matrices and is a diagonal sorted real matrix. n No fast algorithm...

8. Compression using SVD n When we use SVD for compression we take the singular values and work with them in our matrix. n The importance of the SVD is that the solution it provides for an image approximation by separable basis images.

9. Implementation of DFT, DCT and Hadamard n Transforming the given image to the phase plane. n Sample a part of the transformed image: –DFT - Taking rectangles near the four corners. –DCT and Hadamard - Taking triangle from the top left corner. n Transforming the image back.

10. Discrete Fourier Transform Discrete Cosine Transform Hadamard Transform

11. Our Implementation for the Transformations n First we wrote a 1D version for each of the three transformation. n Then we implemented a 2D versions of the transforms using the 1D versions. n We also used a matrix multiplication for the transforms, since the original versions were too slow... n In the actual implementation we used Matlab’s fft2() and ifft2() functions for DFT, and pre-calculated matrices for DCT and Hadamard. n Using the pre-calcualated matrices is much faster since a 2D tranform is only a multiplication of three matrices.

12. Using Adaptive Compression n In this algorithm we will find out how much we should cut from each block so most of the image is maintained. n From Parseval theorem, We can measure the error in the phase plane.

13. Using Adaptive Compression n Using the MSE of each block for each ratio, we can decide how much to compress the block. n We increment the MSE wage for the first steps.

14. Conclusions n After making some tests we found out that the DCT transform work better, on equal ratio, then the other transforms. Works best for correlated data. n Hadamard is useful for small block sizes. n When using DFT, the blocks are noticeable. Therefore we might consider using larger blocks the overlaps. n DCT, Hadamard and SVD handle sharp borders better than DFT.

15. Conclusions (continued) n Adaptive compression with DFT or DCT does not handle well sharp edges. n The smaller the block size, the adaptive compression gives better results.

16. Additional Directions n Other methods for reducing image size –Quantization –Other masks n Testing results on blocks with adaptive size (Like quadtree subdivision). n Checking different methods that were not discussed here. –Haar, Slant, KL... –Actually any orthonormal base