## Scale-invariant pattern recognition using a combined Mellin radial harmonic function and the bidimensional empirical mode decomposition

Optics Express, Vol. 17, Issue 19, pp. 16581-16589 (2009)

http://dx.doi.org/10.1364/OE.17.016581

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### Abstract

A novel scale and shift invariant pattern recognition method is proposed to improve the discrimination capability and noise robustness by combining the bidimensional empirical mode decomposition with the Mellin radial harmonic decomposition. The flatness of its peak intensity response versus scale change is improved. This property is important, since we can detect a large range of scaled patterns (from 0.2 to 1) using a global threshold. Within this range, the correlation peak intensity is relatively uniform with a variance below 20%. This proposed filter has been tested experimentally to confirm the result from numerical simulation for cases both with and without input white noise.

© 2009 Optical Society of America

## 1. Introduction

1. D. Mendlovic, E. Maron, and N. Konforti, “Shift and scale invariant pattern recognition using Mellin radial harmonics,” Opt. Commun. **67**, 172–176(1988).
[CrossRef]

2. A. Moya, J. J. Esteve-Taboada, J. Garcia, and C. “Ferreira, Shift- and scale-invariant recognition of contour objects with logarithmic radial harmonic filters,” Appl. Opt. **39**, 5347–5351(2000).
[CrossRef]

3. Yih-Shyang Cheng and Hui-Chi Chen, “Improved performance of scale-invariant pattern recognition using a combined Mellin radial harmonic function and wavelet transform,” Opt. Eng. **46**, 107204 (Oct. 29, 2007)
[CrossRef]

## 2. Theory

1. D. Mendlovic, E. Maron, and N. Konforti, “Shift and scale invariant pattern recognition using Mellin radial harmonics,” Opt. Commun. **67**, 172–176(1988).
[CrossRef]

*f*(

*r*,

*θ*) can be decomposed into a set of

*m*’th order is

*x*

_{0},

*y*

_{0}) is the expansion center. The finite radius

*R*

_{0}covering the pattern and the smallest radius

*r*

_{0}define the range of the expansion, where proper choice of the integer

*L*=ln

*R*

_{0}-ln

*r*

_{0}is required. The

*M*’ th-order RHF is chosen as the RHF filter function:

*g*(

*r*,

*θ*) can be written as

4. J. C. Nunes, Y. Bouaoune, E. Deléchelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vision Comput. **21**, 1019–1026(2003).
[CrossRef]

*I*, a bidimensional IMF as

*BIMF*, and the residue as

*R*. In the decomposition process,

*i*th

*BIMF F*is obtained from its source image

_{i}*S*, where

_{i}*S*is a residue image obtained as

_{i}*S*=

_{i}*I*- ∑

*i*-1

*j*=1(

*F*) and

_{j}*S*

_{1}=

*I*.

*S*when

_{i}*i*=1,

*S*=

_{i}*I*;

*m*of upper and lower envelopes;

*S*-

_{i}*m*=

*h*, and let

*S*=

_{i}*h*

*h*is a

*BIMF, F*;

_{i}*i*=

*i*+1,

*S*=

_{i}*I*- ∑

*i*-1

*j*=1(

*F*) if the stopping criterions are not satisfied, go to 1), otherwise finish the process.

_{j}*BIMFs*, and a residue

*R*. Components superposition reconstructs the data:

*BIMF*s and a residue based basically on the local frequency or oscillation information. The first IMF/BIMF contains the highest local frequencies of oscillation or the highest local spatial scales, the final IMF/BIMF contains the lowest local frequencies of the oscillation and the residue

*R*only contains the trend of the signal. More details about the EMD and BEMD can refer to Ref. [4

4. J. C. Nunes, Y. Bouaoune, E. Deléchelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vision Comput. **21**, 1019–1026(2003).
[CrossRef]

5. C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process Lett. **12**, 701–704 (2005).
[CrossRef]

*β*

^{2}of Eq. (5), a contour object could be used to remove the useless low-frequency components by digital preprocessing to extract the edges of the original block images [2

2. A. Moya, J. J. Esteve-Taboada, J. Garcia, and C. “Ferreira, Shift- and scale-invariant recognition of contour objects with logarithmic radial harmonic filters,” Appl. Opt. **39**, 5347–5351(2000).
[CrossRef]

*BIMF*,

*F*

_{1}(

*f*) of the reference image

*f*(

*x*,

*y*) is used to act as the reference pattern, and is expanded as the combination Eq. (1) of RHF. And one particular order, radial{

*F*

_{1}(

*f*)}, in Eq. (3) is chosen as the filter function. Then this filter and the input pattern

*F*

_{1}(

*g*) of the input image

*g*(

*x*,

*y*) are used to perform the correlation operation.

*BIMF*

*F*

_{1}(

*g*) constitutes most of the noise in the signal [4

4. J. C. Nunes, Y. Bouaoune, E. Deléchelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vision Comput. **21**, 1019–1026(2003).
[CrossRef]

*BIMF F*

_{1}(

*g*) reduces high spatial frequencies. Since BEMD is local in nature, image blurring is reduced. Filtering occurs in time space rather than in frequency space; therefore, any nonlinearity and nonstationarity present in the data are preserved. Although

*F*

_{1}(

*g*) has been observed to contain most of the noise, the first few BIMFs from BEMD still usually contain a lot of the noise in the input image; therefore, removing them and reconstructing the image with the remaining BIMFs tend to denoise the image. The number of

*BIMF*s needed to be removed depends on the level of noise in the image. In this paper, the first three BIMFs of the input image

*g*(

*x*,

*y*) were discarded as

*g*(

*x*,

*y*),

*Edge*(

*Resid*(

*g*⃗)), is used to perform the correlation operation. The Laplacian of Gaussian method is used to find the edge of

*Resid*(

*g*

_{4}⃗).

## 3. Simulation and results

### 3.1. Experimental setup

5. C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process Lett. **12**, 701–704 (2005).
[CrossRef]

6. C. Damerval, “BEMD Toolbox : Bidimensional Empirical Mode Decomposition”. http://ljk.imag.fr/membres/Christophe.Damerval/software.html

### 3.2. Shift and scale invariant

*β*=1, 0.8, 0.5, 0.2, as shown in Fig. 2(a). Using the traditional RHF filter, PORHF and BEMD_RHF of expansion order 1 obtained from the object of Fig. 1(a), the correlation intensity distribution for the input objects of Fig. 2(a) is shown in Figs. 2(b), 2(d) and 2(f). To simplify the comparison of the peak intensity, the corresponding cross-sectional intensity distribution through all the correlation peaks is shown in Figs. 2(c), 2(e) and 2(g).

*β*=1, 0.8, 0.5, 0.2 and one Mig25 with scale factor

*β*=1, as shown in Fig. 3(a). The four scaled versions of the F16 fighter are respectively in the top left corner, bottom left corner, top right corner and bottom right corner of the input image. And the Mig25 is in the center of input image.

*β*of the F16 for the MRHW (

*M*=1,

*a*=0.5) and the BEMD_RHF, we simulate some typical cases and present the results in Fig. 4. In Fig. 4, both of the CPI curves for the MRHW and BEMD_RHF do not depend on the scale factor

*β*in any explicit way. It can be found that the recognition range of the BEMD_RHF is wider than the MRHW almost at the whole range for scale change whether the threshold is taken at any value. The variance for the CPI curve of BEMD_RHF is a better uniformity than the MRHW (

*M*=1,

*a*=0.5). For instance, if the threshold is set at 0.6, the recognition range of the BEMD_RHF is from 0.15 to 1, which is wider than of the MRHW (from 0.25 to 1).

### 3.3. Noise robustness

## 4. Conclusions

*β*<1) using a global threshold. Within this range, the CPI is relatively uniform with a variance below 20%. This proposed BEMD_RHF filter has been tested experimentally to confirm the results from numerical simulation for cases both with and without input white noise.

## Acknowledgements

## References and Links

1. | D. Mendlovic, E. Maron, and N. Konforti, “Shift and scale invariant pattern recognition using Mellin radial harmonics,” Opt. Commun. |

2. | A. Moya, J. J. Esteve-Taboada, J. Garcia, and C. “Ferreira, Shift- and scale-invariant recognition of contour objects with logarithmic radial harmonic filters,” Appl. Opt. |

3. | Yih-Shyang Cheng and Hui-Chi Chen, “Improved performance of scale-invariant pattern recognition using a combined Mellin radial harmonic function and wavelet transform,” Opt. Eng. |

4. | J. C. Nunes, Y. Bouaoune, E. Deléchelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vision Comput. |

5. | C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process Lett. |

6. | C. Damerval, “BEMD Toolbox : Bidimensional Empirical Mode Decomposition”. http://ljk.imag.fr/membres/Christophe.Damerval/software.html |

**OCIS Codes**

(100.0100) Image processing : Image processing

(100.4550) Image processing : Correlators

(100.5010) Image processing : Pattern recognition

**ToC Category:**

Image Processing

**History**

Original Manuscript: February 2, 2009

Revised Manuscript: April 8, 2009

Manuscript Accepted: April 18, 2009

Published: September 2, 2009

**Citation**

Qingbo yin, Liran Shen, Jong-Nam Kim, and Yong-Jae Jeong, "Scale-invariant pattern recognition using a combined Mellin radial harmonic function and the bidimensional empirical mode decomposition," Opt. Express **17**, 16581-16589 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-19-16581

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### References

- D. Mendlovic, E. Maron, and N. Konforti, "Shift and scale invariant pattern recognition using Mellin radial harmonics," Opt. Commun. 67, 172-176(1988). [CrossRef]
- A. Moya, J. J. Esteve-Taboada, J. Garcia, and C. Ferreira, "Shift- and scale-invariant recognition of contour objects with logarithmic radial harmonic filters," Appl. Opt. 39, 5347-5351(2000). [CrossRef]
- Y.-S. Cheng and H.-C. Chen, "Improved performance of scale-invariant pattern recognition using a combined Mellin radial harmonic function and wavelet transform," Opt. Eng. 46, 107204 (Oct. 29, 2007) [CrossRef]
- J. C. Nunes, Y. Bouaoune, E. Deléchelle, O. Niang, and Ph. Bunel, "Image analysis by bidimensional empirical mode decomposition," Image Vision Comput. 21, 1019-1026(2003). [CrossRef]
- C. Damerval, S. Meignen, and V. Perrier, "A fast algorithm for bidimensional EMD," IEEE Signal Process Lett. 12, 701-704 (2005). [CrossRef]
- C. Damerval, "BEMD Toolbox : Bidimensional Empirical Mode Decomposition," http://ljk.imag.fr/membres/Christophe.Damerval/software.html.

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