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Poster Presentation - Wavelet Analysis, Image Processing and Electromagnetics

Presentation · July 2014

DOI: 10.13140/RG.2.1.1202.5764

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Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

Wavelet Analysis, Image Processing and Electromagnetics

Emanuel Guariglia eguariglia@unisa.it

University of Salerno

22nd Summer School on Image Processing, July 9-18, Zagreb, Croatia

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

Summary

1

Introduction

2

Wavelets and Electromagnetics

3

FDTDW - Simulation on the transmission coefficient

4

RWG with a wavelet approach

5

Future Developments and Conclusions

6

References

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

Why Wavelets and Image Processing?

1st reason No need to block the image.

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

Why Wavelets and Image Processing?

1st reason No need to block the image. 2nd reason More robust under transmission errors.

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

Why Wavelets and Image Processing?

1st reason No need to block the image. 2nd reason More robust under transmission errors. 3rd reason Facilitates progressive transmission of the image (Scalability).

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

Electromagnetics and Image Processing 1/2

As many imaging systems and object recognition applications, there exists a need for pre-processing raw data provided by the image to improve the quality of the measure scene

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

Electromagnetics and Image Processing 1/2

As many imaging systems and object recognition applications, there exists a need for pre-processing raw data provided by the image to improve the quality of the measure scene Examples SAR, Biomedical Processes.

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

Electromagnetics and Image Processing 1/2

As many imaging systems and object recognition applications, there exists a need for pre-processing raw data provided by the image to improve the quality of the measure scene Examples SAR, Biomedical Processes.

Electromagnetics needs Image Processing

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

Electromagnetics and Image Processing 2/2 QUESTION

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

Electromagnetics and Image Processing 2/2 QUESTION Can Electromagnetics and Image Processing use the same contemporary mathematical theories?

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

Electromagnetics and Image Processing 2/2 QUESTION Can Electromagnetics and Image Processing use the same contemporary mathematical theories?

OF COURSE

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

Method of Moments and Sparsity

In Electromagnetics, the Method of Moments works with dense matrices.

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

Method of Moments and Sparsity

In Electromagnetics, the Method of Moments works with dense matrices.

Wavelet Analysis

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

Method of Moments and Sparsity

In Electromagnetics, the Method of Moments works with dense matrices.

Wavelet Analysis Sparse Matrices

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

Method of Moments and Sparsity

In Electromagnetics, the Method of Moments works with dense matrices.

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

MRTDM and FDTDM 1/2

Using Multiresolutional Analysis, we can improve the classical FDTDM.

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

MRTDM and FDTDM 1/2

Using Multiresolutional Analysis, we can improve the classical FDTDM. Weaknesses of the FDTDM are as follows.

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

MRTDM and FDTDM 1/2

Using Multiresolutional Analysis, we can improve the classical FDTDM. Weaknesses of the FDTDM are as follows.

1 The accuracy of results due to the discretization. Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

MRTDM and FDTDM 1/2

Using Multiresolutional Analysis, we can improve the classical FDTDM. Weaknesses of the FDTDM are as follows.

1 The accuracy of results due to the discretization. 2 Systematic errors due to the discretization. Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

MRTDM and FDTDM 1/2

Using Multiresolutional Analysis, we can improve the classical FDTDM. Weaknesses of the FDTDM are as follows.

1 The accuracy of results due to the discretization. 2 Systematic errors due to the discretization. 3 You can work only with reduced spatial domains. Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

MRTDM and FDTDM 1/2

Using Multiresolutional Analysis, we can improve the classical FDTDM. Weaknesses of the FDTDM are as follows.

1 The accuracy of results due to the discretization. 2 Systematic errors due to the discretization. 3 You can work only with reduced spatial domains. 4 You have to need the boundary conditions. Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

MRTDM and FDTDM 2/2

You can expand the differential operators using wavelets, so you can not use in the derivation the concept of finite difference.

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

MRTDM and FDTDM 2/2

You can expand the differential operators using wavelets, so you can not use in the derivation the concept of finite difference. Examples Telegraph equations.

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

MRTDM and FDTDM 2/2

You can expand the differential operators using wavelets, so you can not use in the derivation the concept of finite difference. Examples Telegraph equations. MRTDM greatly improves the problem of the FTFDM numerical dispersion, but it has an high computational cost.

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

MRTDM and FDTDM 2/2

You can expand the differential operators using wavelets, so you can not use in the derivation the concept of finite difference. Examples Telegraph equations. MRTDM greatly improves the problem of the FTFDM numerical dispersion, but it has an high computational cost.

A good solution could be using FDTD with a wavelet approach.

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

FDTDW Simulation on the transmission coefficient 1/4

Now we consider this stratification:

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

FDTDW Simulation on the transmission coefficient 1/4

Now we consider this stratification:

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

FDTDW Simulation on the transmission coefficient 1/4

Now we consider this stratification: From the Electromagnetic Fields Theory, it is know that

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

FDTDW Simulation on the transmission coefficient 1/4

Now we consider this stratification: From the Electromagnetic Fields Theory, it is know that Γ1,2 = Z2 − Z1 Z2 + Z1 e T1,2 = 2Z2 Z2 + Z1

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

FDTDW Simulation on the transmission coefficient 2/4

If we call x1 the point of the observation, we cannot calculate the transmission coefficient as

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

FDTDW Simulation on the transmission coefficient 2/4

If we call x1 the point of the observation, we cannot calculate the transmission coefficient as E t

z (x1, t)

E i

z(x1, t)

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

FDTDW Simulation on the transmission coefficient 2/4

If we call x1 the point of the observation, we cannot calculate the transmission coefficient as E t

z (x1, t)

E i

z(x1, t)

because this coefficient is a concept of the frequency domain, thus the division of these two signals has no sense in the time domain. Choosing Z0 = Z1 and Z2 = Z0/3, we have

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

FDTDW Simulation on the transmission coefficient 2/4

If we call x1 the point of the observation, we cannot calculate the transmission coefficient as E t

z (x1, t)

E i

z(x1, t)

because this coefficient is a concept of the frequency domain, thus the division of these two signals has no sense in the time domain. Choosing Z0 = Z1 and Z2 = Z0/3, we have T1,2 = 2Z2 Z2 + Z1 = 2✚

✚ ❩ ❩

Z0 3

• ❅

❅

Z0 + ✚

✚ ❩ ❩

Z0 3

= 1 2

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

FDTDW Simulation on the transmission coefficient 2/4

If we call x1 the point of the observation, we cannot calculate the transmission coefficient as E t

z (x1, t)

E i

z(x1, t)

because this coefficient is a concept of the frequency domain, thus the division of these two signals has no sense in the time domain. Choosing Z0 = Z1 and Z2 = Z0/3, we have T1,2 = 2Z2 Z2 + Z1 = 2✚

✚ ❩ ❩

Z0 3

• ❅

❅

Z0 + ✚

✚ ❩ ❩

Z0 3

= 1 2 that is frequency invariant.

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

FDTDW Simulation on the transmission coefficient 3/4

If N1 = 80 (i.e. the distance between x1 and the discontinuity), we take 8192 temporal steps, and Nf is the frequency index, it’s simply to prove that

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

FDTDW Simulation on the transmission coefficient 3/4

If N1 = 80 (i.e. the distance between x1 and the discontinuity), we take 8192 temporal steps, and Nf is the frequency index, it’s simply to prove that TFDTD = e j

4πN1Nf

8192

ˆ

E t

z (x1, t)

ˆ E i

z(x1, t)

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

FDTDW Simulation on the transmission coefficient 3/4

If N1 = 80 (i.e. the distance between x1 and the discontinuity), we take 8192 temporal steps, and Nf is the frequency index, it’s simply to prove that TFDTD = e j

4πN1Nf

8192

ˆ

E t

z (x1, t)

ˆ E i

z(x1, t)

where as source of field we use a discretized mexican hat.

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

FDTDW Simulation on the transmission coefficient 3/4

If N1 = 80 (i.e. the distance between x1 and the discontinuity), we take 8192 temporal steps, and Nf is the frequency index, it’s simply to prove that TFDTD = e j

4πN1Nf

8192

ˆ

E t

z (x1, t)

ˆ E i

z(x1, t)

where as source of field we use a discretized mexican hat. The purpose of simulation It is interested to compare its behavior with that of a sinusoidal

• source. We have plotted the real part of the transmission coef-

ficient (that is a function of the points per wavelength) when N1 = 80 and N1 = 4 .

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

FDTDW Simulation on the transmission coefficient 4/4

Mexican Hat (N1 = 80) Sinusoidal Source (N1 = 80)

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

FDTDW Simulation on the transmission coefficient 4/4

Mexican Hat (N1 = 4) Sinusoidal Source (N1 = 4)

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

FDTDW Simulation on the transmission coefficient 4/4

Conclusion The trend of the (real part) of our coefficient is better when we use a Mexican hat because it has less problems with the frequency than the sinusoidal source.

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

FDTDW Simulation on the transmission coefficient 4/4

Conclusion The trend of the (real part) of our coefficient is better when we use a Mexican hat because it has less problems with the frequency than the sinusoidal source. Problem It is impossible to eliminate the dependence of the transmission coefficient from the frequency: indeed, it depends on the numerical dispersion of an FDTD grid (that is certainly not negligible).

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

RWG with a wavelet approach

One of the variants of the method of the most recently used is based on the functions of Rao-Wilton-Glisson (RWG).

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

RWG with a wavelet approach

One of the variants of the method of the most recently used is based on the functions of Rao-Wilton-Glisson (RWG).

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

RWG with a wavelet approach

One of the variants of the method of the most recently used is based on the functions of Rao-Wilton-Glisson (RWG). Andriulli, Tabacco e Vecchi have improved this method using Multiresolution Analysis.

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

RWG with a wavelet approach

One of the variants of the method of the most recently used is based on the functions of Rao-Wilton-Glisson (RWG).

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

Future Developments 1/2

Rough Surface Using coiflets you can study numerically it. The computational cost can be reduced and treated four times the area.

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

Future Developments 1/2

Rough Surface Using coiflets you can study numerically it. The computational cost can be reduced and treated four times the area. EBG Antennas The EBG materials have band where the waves can not propagate: their utilization have improved the antennas performance: mathematically, this antennas are studied using wavelets and fractals.

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

Future Developments 2/2

Near-to-Far-Field Transformation A wavelet approach can increase the ability to store data, so you have more information on near-field.

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

Future Developments 2/2

Near-to-Far-Field Transformation A wavelet approach can increase the ability to store data, so you have more information on near-field. Semiconductor Devices The Boltzmann transport equation can be solve simply by wavelets, so we have several advantages for MOS devices and High-Frequency Semiconductor Devices.

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

Future Developments 2/2

Near-to-Far-Field Transformation A wavelet approach can increase the ability to store data, so you have more information on near-field. Semiconductor Devices The Boltzmann transport equation can be solve simply by wavelets, so we have several advantages for MOS devices and High-Frequency Semiconductor Devices. Super IFS for the RWG - wavelet approach In self-similarity problems, the RWG functions are built using classical IFS for generating fractal: can we use the SuperIFS?

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

Wavelets: Image Processing vs Electromagnetics

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

Wavelets: Image Processing vs Electromagnetics

Image Processing Wavelets are been used in image processing to solve different problems: today you cannot work in image processing without having a good knowledge of Wavelet Analysis. This was been really a successful marriage.

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

Wavelets: Image Processing vs Electromagnetics

Image Processing Wavelets are been used in image processing to solve different problems: today you cannot work in image processing without having a good knowledge of Wavelet Analysis. This was been really a successful marriage. Electromagnetics We are only at the beginning, many significant results could be found in the coming years.

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

References I

• G. W. Pan

Wavelets in Electromagnetics and Device Modeling 1st edition, John Wiley & Sons, Inc., Hoboken (New Jersey), 2003.

• F. P. Andriulli, A. Tabacco, G. Vecchi

A multiresolution approach to the electric field integral equation in antenna problems Technical Report, Department of Mathematics, Polytechnic University of Turin, n. 30, November 2004.

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

References II

• P. Pirinoli, L. Matekovits, G. Vecchi

Multiresolution analysis of printed antennas and circuits: a dual-isoscalar approach IEEE Trans. on Antennas and Propagation, Vol. 49, , June 2001, pp. 858-874.

• F. Vipiana, P. Pirinoli, G. Vecchi

A multiresolution method of moments for triangular meshes IEEE Trans. on Antennas and Propagation, Vol. 53, July 2005,

• pp. 2247-2258.

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

References III

• U. S. Inan, R. A. Marshall

Numerical Electromagnetics: The FDTD Method 1st edition, Cambridge University Press, New York, 29 April 2011.

• M. Sadiku

Numerical Techniques in Electromagnetics 2nd edition, CRC Press, Boca Raton (Florida), 2001.

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

Introduction Wavelets and Electromagnetics FDTDW - Simulation on the transmission coefficient RWG with a wavelet approach Future Developments and Conclusions References

If you like the beauty, Zagreb will forever stay in your heart!

Emanuel Guariglia Wavelet Analysis, Image Processing and Electromagnetics

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