IMAGE PROCESSING BY 2D DELTA WINDOW APPROXIMATION In ...

BIOPHYSICS

IMAGE PROCESSING BY 2D DELTA WINDOW APPROXIMATION V. I. R. NICULESCU1, RODICA MARIANA ION2, C. MARDARE3 1

National Institute for Laser, Plasma and Radiation Physics, Bucharest, Romania 2 Institute for Chemical Research, ICECHIM, Bucharest, Romania 3 Physics Faculty, Bucharest Univ., Bucharest, Romania Received December 14, 2005

Physics applications in medicine diagnosis (PDT – photodynamic therapy of cancer) and biology need different numerical processing of images. Usually the tumor images was enhanced to increase the image significance. A more deep caracterisations of different tissue zones can be done using wavelets. We introduced a new type of bidimensional windows and wavelets. Key words: window delta approximation, wavelet delta approximation, PDT.

1. INTRODUCTION

In medicine diagnosis (PDT – photodynamic therapy of cancer) and biology different numerical processing of images were neded [1, 2]. In processing signals and images, apart the well known Fourier methods, a more powerful instrument is offered by the wavelet theory. In this preliminary report we obtain a 2D window and 2D wavelet, which take in account one of delta approximation. The advantage of wavelets is given by the ortogonal base on a compact support. The disavantage is implied by the necessity to pass to a space augmented by one in each direction. 2. METHOD

The wavelet can be constructed in many different ways. The approximated delta window is given by:

f ( ε; t ) =

ε2

ε 1. + t2 π

(1a)

where for ε tending to zero the limit is the Dirac function δ ( t ) . 

Paper presented at the National Conference of Physics, 13–17 September, 2005, Bucharest, Romania. Rom. Journ. Phys., Vol. 51, Nos. 5– 6 , P. 663–666, Bucharest, 2006

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Also for Dirac function we have (Mexican hat):

( ) ⎥⎦

(1b).

ε 1 2 ε2 + ( t − d ) π

(2a)

⎡ f ( ε; t ) = 1 exp ⎢ − t ⎣ ε ε π

2⎤

The approximated [3] delta window

g ( ε, d ; t ) =

was given, where d represents the window length. In Fig. 1 the normalized delta approximated windows (2a) are represented for different values of ε, d: ( ε = 0.1; ε = 0.3; d = 0 ) .

Fig. 1

The mother wavelet which is the first derivative of the window expression (1a) can be evaluated. The set of daughter wavelets from the mother wavelet by shift operations are generated. This type of wavelets satisfy admissibility condition (the Fourier transform and the integral from the wavelet equal zero). The wavelet transform is a correlation operation between the signal f (t) and the shifted and scaled mother wavelet. In Fig. 2 the mother wavelets (ε = 0.1, ε = 0.3, d = 0) were also shown. An 2D window as sum of two term products of 1D windows like (1a or 2a) can be constructed. In Fig. 3 an approximated delta 2D window constructed in arbitrary units was generated which represents an adult brain tumor. We noticed that for a tumor image description a good one is given by a sum with for terms. Each term was given by the product of two 1D windows for each direction.

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Image processing by 2D delta window approximation

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Fig. 2

Fig. 3

3. CONCLUSIONS

In the window computation only usual arithmetic operations are used. For the same number of parameters less operations in delta wavelet evaluation than for mexican hat one was obtained. So the necessary computing time for this wavelet by the absence of exponential evaluations was reduced. The images can be approximated by 1D windows or wavelets sums products. So an analytical description by Cauchy bidimensional windows is more tractable than the Fourier ones. This types of windows and wavelets are more versatile ones than the Morlet wavelets.

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REFERENCES 1. Rodica Mariana Ion, Porphyrins and photodynamic therapy of cancer, Ed. FMR, Bucharest, 2003. 2. V. I. R. Niculescu, V. Babin, Mihaela Dan, A Sigmoid Wavelet, Rom. J. Phys. 43, (Supplement I), (2003) 271–275. 3. V. I. R. Niculescu, V. Babin, C. Stancu, Claudia Stancu, Annual Scientific Conference, University of Bucharest, Faculty of Physics, May 30, 2003.