1. University of Tehran School of Electrical and Computer Engineering Custom Implementation of DSP Systems - 2010 By Morteza Gholipour Class presentation for the course: Custom Implementation of DSP Systems Instructor Prof. S. Mehdi Fakhraie * Some materials are copyrights of their respective authors as listed in references

2. Outline Introduction to wavelet Lifting based wavelet transform 2-D wavelet in image compression 3-D wavelet transform An efficient architecture for 3 -D DWT [ 1 ] Conclusion

3. Fourier Analysis Breaks down a signal into constituent sinusoids of different frequencies [ 2 ]. It is extremely useful when the signal's frequency content is of great importance. Drawback: In transforming to the frequency domain, time information is lost. It is impossible to tell when a particular event took place. Can not indicate: drift, abrupt changes, and beginnings and ends of events.

4. Shortcomings of Fourier Transform Basis functions don’t have limited duration, no localization. Information about sharp changes spread across many frequencies and many basic functions. Time information is lost: x f(x)

5. Short-Time Fourier Analysis Dennis Gabor (1946): windowing the signal: analyze only a small section of the signal at a time Short-Time Fourier Transform (STFT), maps a signal into a two- dimensional function of time and frequency. Limited precision indicated by window size. The window size is same for all frequencies. Narrow window -> poor frequency resolution Wide window -> poor time resolution

6. Wavelet Transform An alternative approach to the STFT to overcome the resolution problem. Transformation of a signal into time-frequency representation Different basis and transformations result in different constituents and T-f information Analyze the signal at different frequencies with different resolutions.

7. What is wavelet transform? Wavelets are functions defined over a finite interval and having an average value of zero. The basic idea of the wavelet transform is to represent any arbitrary function ƒ(t) as a superposition of a set of wavelets or basis functions. These basis functions or baby wavelets are obtained from a single prototype wavelet called the mother wavelet, by dilations (scaling) and translations (shifts). Translation Dilation

8. Wavelet Analysis Variable-sized windows. Long time intervals: low-frequency information. Shorter regions: high frequency information. Wavelets have scale aspects and time aspects: [2]

9. Wavelet Transform Definition The wavelet transform of signal S is the family C(a,b), which depends on two indices a (scale) and b (position). If the C is large, the resemblance of the signal to wavelet is strong, otherwise it is slight. The indexes C(a,b) are called coefficients.

10. Continuous vs. Discrete WT CWT: Information is redundant -> reconstruction is concerned. Requires large amount of computation time and resource. DWT: Coefficients are calculated in discrete times and scales. Reduce the computation time. Easier to implement. Provides sufficient information both for analysis and synthesis of the original signal.

11. Continuous vs. Discrete WT

12. Example of an Advantage major advantage afforded by wavelets is the ability to perform local analysis -- that is, to analyze a localized area of a larger signal A sinusoidal signal with a small discontinuity :

13. Filter Bank implementation of DWT The high-pass filter produces detail information and the low-pass filter produces approximations in each level. Multiresolution Analysis (MRA): Analyzes the signal at different frequencies with different resolutions.

14. Lifting Based Wavelet Transform Introduced by [ 3 ]: Sweldens, “The lifting scheme: A new philosophy in biorthogonal wavelet constructions,” 1995. The lifting scheme is essentially an easy way to find new filters, which returns a set of linear algebraic equations.

15. Advantages of Lifting Scheme Perfect reconstruction. The convolution operations can be replaced by any other operation. Easily produces integer-to-integer wavelet transforms for lossless compression. It does not require temporary arrays in the calculation steps. In place: The transformation can be performed immediately in the memory of the input data. The resulting transform is invertible. Requires less computation and less memory. Speedup by a factor of two. Real-time implementation possible.

16. 2-D Discrete Wavelet Transform in JPEG2000 Step 1: Replace each row with its 1 -D DWT Step 2: Replace each column with its 1 -D DWT Step 3: repeat steps (1) and (2) on the lowest subband for the next scale Step 4: repeat steps (3) until as many scales as desired have been completed [ 4 ]

17. 3-D Discrete Wavelet Transform Application of 3 -D DWT: Space: Hyperspectral imaging [5] Data across the electromagnetic spectrum, e.g. IR & UV. Medical: object based coding for 3 -D MRI data [6]. Video coding 3 -D DWT consists of spatial and temporal transforms which could be interchanged. First temporal and the spatial (t+ 2 -D) suffers from spatial scalability for future extensions First spatial and then temporal ( 2 -D+t) in which the reverse method can be equally mapped into hardware. Mantis shrimp [7]

18. Proposed Architecture [ 2 ] Lifting based complete 3 -D wavelet transform without GOP restriction. Spatial transform first, then temporal. Improvement made in SFG.

19. Implementation Strategy Irrational numbers considered up to the finite precision during designing -> lowered PSNR Trade-off between affordable hardware budget and video quality. Coefficient and fractional data precision of 11 and 2 bits. The multipliers are designed through pipelined “shift- n-add” mechanism.

20. Implementation results Xilinx Virtex -4 family XC4VFX140 is used. Simulation is performed by ModelSim and compared to MATLAB model.

21. Results The architecture has no restriction on GOP. Complete 3 -D-DWT architecture. Operating speed of 321 MHz. Standard rate of 30 FPS with frame size 256x256  minimum clock=1.09MHz (for real time) Minimized storage Low latency and power consumption Increased throughput Only a single adder in critical path

22. References [1] A. Das, A. Hazra, and S. Banerjee, “ An Efficient Architecture for 3-D Discrete Wavelet Transform,” IEEE Transaction on Circuits and Systems fro Video Technology, vol. 20, no. 2, Feb. 2010. [2] MATLAB Help. [3] W. Sweldens, “The lifting scheme: A new philosophy in biorthogonal wavelet constructions”, Proceedings of SPIE, 2569, pp.68-79, 1995. [4] A. Aminlou, H. Badakhshannoory, M.R. Hashemi, O. Fatemi, “A New Discrete Wavelet Transform Architecture with Minimum Resource Requirements,” pp. 470- 473. [5] J. E. Fowler and J. T. Rucker, “3-D wavelet-based compression of hyperspectral imagery,” in Hyperspectral Data Exploitation: Theory and Applications, C.-I. Chang, Ed. Hoboken, NJ: Wiley, 2007, ch. 14, pp. 379–407. [6] G. Menegaz and J.-P. Thiran, “Lossy to lossless object-based coding of 3-D MRI data,” IEEE Trans. Image Process., vol. 11, no. 9, pp. 1053–1061, Sep. 2002. [7] http://www.wikipedia.org

23. Thanks