Hardware Implementation of 3-D Discrete Wavelet Transform

University of Tehran

School of Electrical and Computer Engineering

Custom Implementation of DSP Systems - 2010

ByMorteza Gholipour

Class presentation for the course: Custom Implementation of DSP Systems

InstructorProf. S. Mehdi Fakhraie

* Some materials are copyrights of their respective authors as listed in references

OutlineIntroduction to waveletLifting based wavelet transform2-D wavelet in image compression3-D wavelet transformAn efficient architecture for 3-D DWT [1]Conclusion

Fourier AnalysisBreaks 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.

Shortcomings of Fourier TransformBasis 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)

Short-Time Fourier AnalysisDennis 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

Wavelet TransformAn 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.

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

Wavelet AnalysisVariable-sized windows.

Long time intervals: low-frequency information.

Shorter regions: high frequency information.Wavelets have scale aspects and time aspects:

[2]

Wavelet Transform DefinitionThe 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.

Continuous vs. Discrete WTCWT:

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.

Continuous vs. Discrete WT

Example of an Advantagemajor 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 :

Filter Bank implementation of DWTThe 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.

g~ 2

h~

2

S

di

si

g~ 2

h~

2

di+1

si+1

g~

h~

Lifting Based Wavelet TransformIntroduced 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.

Advantages of Lifting SchemePerfect 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.

2-D Discrete Wavelet Transformin JPEG2000Step 1: Replace each row with its 1-D

DWTStep 2: Replace each column with its 1-D

DWTStep 3: repeat steps (1) and (2) on the

lowest subband for the next scaleStep 4: repeat steps (3) until as many

scales as desired have been completed [4]

3-D Discrete Wavelet TransformApplication 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 extensionsFirst spatial and then temporal (2-D+t) in

which the reverse method can be equally mapped into hardware.

Mantis shrimp [7]

Proposed Architecture [2]Lifting based complete 3-D wavelet transform

without GOP restriction.Spatial transform first, then temporal.Improvement made in SFG.

Implementation StrategyIrrational numbers considered up to the finite

precision during designing -> lowered PSNRTrade-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.

Implementation resultsXilinx Virtex-4 family XC4VFX140 is used.

Simulation is performed by ModelSim and compared to MATLAB model.

ResultsThe 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 storageLow latency and power consumptionIncreased throughputOnly a single adder in critical path

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

Thanks