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

TL;DR: The proposed filter outperforms currently available filter design methods and is presented as a performance comparison, as well as a homomorphic processing image enhancement example to illustrate the effectiveness of this method.

Abstract: A design method of a linear-phased, two-dimensional (2-D), two-fold symmetric circular shaped filter is presented in this paper. Although the proposed method designs a non-separable filter, its implementation has linear complexity. The shape of the passband and the stopband is expressed in terms of level sets of second order trigonometric polynomials. This enables the transformation of the filter specifications to a Semi-Definite Program (SDP) of moderate dimension. The proposed filter outperforms currently available filter design methods. We present a performance comparison, as well as a homomorphic processing image enhancement example to illustrate the effectiveness of this method.

## 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

7,174 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

1,493 citations

730 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....

[...]

217 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....

[...]

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

[...]

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

[...]