# 2-D two-fold symmetric circular shaped filter design with homomorphic processing application

## Summary (2 min read)

### 1. INTRODUCTION

- There are many different approaches in designing 2-D filters, each with its pros and cons.
- Their complexity is very low but it is not possible to control the filter characteristics such as passband shape and cut-off frequency.
- There have been continues interest in representing the passband and stopband by trigonometric polynomials that are positive in the region of interest.
- Initially first order trigonometric polynomials have been used to design two-fold symmetric filters [6].
- The resulting SDP formulation is of very high dimension.

### 2. TWO-FOLD SYMMETRIC 2-D FILTER DESIGN

- The frequency response of a zero-phased digital filter is a real valued function and its impulse response is symmetric about the origin h(n, n) = h(−n,−n) [10].
- The design objective is to find the coefficients of the matrix X and Y such that the desired frequency response is obtained.
- There are 2(n + 1)2 design variables, which is twice as that of the fourfold symmetric filter.
- Depending on the application and the data to be processed the filter specifications vary considerably.
- Generally filter specifications can be formulated as a minimum stopband attenuation problem as follows: min X,Y ,δs δs (4a) s.t. |H(Ω)−.

### 3. CIRCULAR SHAPED FILTER DESIGN

- Circular shape has not been attempted in [6] and the polynomials used in [8] do not produce the desired shape.
- The following least squares optimization problem can be used to find the coefficients of (7).
- Thus two trigonometric polynomials can be derived to represent the passband and the stopband.
- The filter performance is improved in two-fold symmetric filters.

### 4. SIMULATION

- The first step is to derive the second order trigonometric polynomials that represent the passband and the stopband.
- Semi-definite program was derived as described in Section 3 and the simulation was performed using optimization software YALMIP [11] and SDPT3 [12] in MATLAB.
- Fig. 1 shows the frequency response of the designed filter in log scale and the performance comparison with different design methods is given in Table 1.
- Highpass circular shaped filters can be designed analogously.

### 5. HOMOMORPHIC PROCESSING SYSTEM

- When images with large dynamic range such as natural scenes are recorded, image contrast can be significantly reduced.
- The reflectance component i(n1, n2) on the other hand is related to the contrast within the image and generally vary rapidly.
- To see these images more clearly visit http://ee.unsw.edu.au/∼z3265024/enhancement.html.
- To evaluate the effectiveness of the homomorphic process the original image given in Fig. 4 was blurred using a gaussian lowpass filter before applying the homomorphic process.

### 6. CONCLUSION

- A very general approach of designing two-fold symmetric circular shaped filters with complexity equal to that of a four-fold symmetric filter was presented in this paper.
- The main advantage is that it successfully produces the desired circular shape with minimum passband and stopband ripple compared to the currently available design methods.
- The coefficients of the polynomial T1(Ω) in Chebyshev recursion have an influence on the filter performance and a method of selecting optimal values for these coefficients is still open for research.

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##### References

7,676 citations

### "2-D two-fold symmetric circular sha..." refers methods in this paper

...Semi-definite program was derived as described in Section 3 and the simulation was performed using optimization software YALMIP [11] and SDPT3 [12] in MATLAB....

[...]

2,022 citations

1,618 citations

856 citations

### "2-D two-fold symmetric circular sha..." refers methods in this paper

...[12] K.C. Toh, M.J. Todd, and R.H. Tutuncu, “SDPT3 - a MATLAB software package for semidefinite programming,” Optimization Methods and Software, 1999....

[...]

...Semi-definite program was derived as described in Section 3 and the simulation was performed using optimization software YALMIP [11] and SDPT3 [12] in MATLAB....

[...]

222 citations

### "2-D two-fold symmetric circular sha..." refers methods in this paper

...McClellan transform [2], [3] can be used to develop non-separable filters using 1-D filters....

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...Specifications Our method [9] [8] [3] Filter order 18 39 25 25 Filter complexity 42 84 96 625...

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...Specifications Our method [9] [8] [3] Filter order 12 12 25 25 Filter complexity 30 30 96 625...

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