# Selective edge enhancement using shifted anisotropic vortex filter

## Abstract

We propose a new method for selective edge enhancement using shifted anisotropic vortex phase mask. The shifted anisotropic phase mask is generated by introducing controllable anisotropy in conventional vortex mask [\( \exp \left( {i\theta } \right) \)] with the help of sine function and shifting the singularity away from the zero frequency component in the filter plane. The shifted anisotropic vortex mask is capable of enhancing the edges of the given object selectively in any desired direction.

### Keywords

Anisotropic vortex Selective edge enhancement CGH Shifted anisotropic vortex## Introduction

An optical vortex is an isolated point singularity, in a wavefront phase distribution. These vortices can be generated using holograms [1, 2]. Multiple vortices arranged in a lattice can also be realized using interference of vortex free beams [3, 4, 5, 6]. Vortices are useful in metrology [7, 8] and are generally avoided in diffractive optical designs [9]. Optical vortices can be used for isotropic edge enhancement of an object. In image processing, to understand the images, it is always preferred to enhance the edges. For edge enhancement, the spatial filtering operation is employed by Fourier transforming the object and this Fourier transform is manipulated with the help of a filter function before taking the inverse Fourier transform [10, 11].

An optical phase singularity with topological charge *m* can be used to perform *m*^{th} order Hankel transform [12]. Davis et al have used a vortex phase mask, \( \exp \left( {i\theta } \right);0 < \theta < 2\pi \) as a spatial filter in the 4*f* geometry. To achieve radially symmetric edge enhancement the charge of the vortex *m* is unity and for selective edge enhancement *m* is non integer. If the phase profile of the vortex mask is analyzed, it is obvious that there is a phase difference of *mπ* at a symmetric position in any radial line with respect to the vortex core. Similar characteristics can be seen in the 1D-Hilbert transform [13, 14, 15]. Therefore, vortex filters can be used to perform radially symmetric edge enhancement. The spiral phase filtering using an spiral phase plate (SPP), which is characterized by function *exp* (*imθ*) performs symmetric Hilbert transform and is regarded as radial Hilbert phase mask with *m* as order of the radial Hilbert transform. Generally the edge enhancing effect is isotropic, when spiral phase plate (SPP) of topological charge 1 is used as a radial Hilbert phase mask. The radial Hilbert transform which is effectively the vortex spatial filtering, does the edge enhancement by redistributing the intensity in a symmetric manner because the radial Hilbert mask is symmetric. Therefore, to enhance the selective edges in a particular desired direction, one has to break this symmetry of Hilbert mask.

The methods reported for selective edge enhancement and phase contrast enhancement, use fractional vortex mask and the shifted vortex mask [16, 17]. The fractional Hilbert transform [12, 18] method uses modified Hilbert transform in which the phase difference between two radial points on either side of the vortex core is a fractional multiple of *π*. In another study, Guohai et al [19] have proposed selective edge enhancement using fractional spiral phase filter with additional offset angle and by positioning the vortex core away from the zero frequency component in the filter plane. Very recently we have proposed anisotropic vortex phase mask [20] for selective edge enhancement which can control the selectivity. In this paper we demonstrate a new method for the selective edge enhancement in any desired direction using shifted anisotropic vortex phase mask. Apart from vortex anisotropy induced selective edge enhancement we have exploited the concept of positioning the vortex core away from the zero frequency component to achieve better selectivity.

## Anisotropic vortex filter

*r*is the distance from the vortex center and \( \theta =\arctan \left( {y/x} \right) \) is the azimuthal angle. The phase distribution \( \psi \left( {r,\theta } \right)=\theta \) and the rate of change of the phase around the vortex is constant.

Here *σ* is anisotropy parameter which determines the internal structure of the optical vortex. When we use this type of filter for edge enhancement the property of realizing the effect of radial Hilbert mask is affected because the filter can no longer represent signum function in any radial direction except for a small range of azimuthal directions. As a consequence the edge enhancement becomes selective.

## Shifted isotropic vortex filter

*π*phase difference. Hence isotropic edge enhancement is possible only if the dark core of the filter as well as the zero frequency component of the object Fourier transform coincide. This is because of the presence of the signum filters all around the zero frequency component of the spectrum of the object. If the dark core of the vortex filter doesn’t coincide with the zero frequency component of the spectrum of the object the signum functions are not available along all azimuthal directions with respect to the zero frequency component of object spectrum. The schematic showing the position of the vortex core with respect to the zero frequency component in the filter plane is shown in Fig. 1, and the plot of the phase difference introduced by the shifted vortex filter on the frequency components \( \left( {f_x, {f_y}} \right) \) and \( \left( {-{f_x},-{f_y}} \right) \) which are radially opposite to the zero frequency, is shown in Fig. 2.

Let \( F=\sqrt{{\left( {{f_{xs}}^2+{f_{ys}}^2} \right)}} \) is the shift that represents the separation between the zero frequency of the object and the vortex core. The phase difference introduced by the shifted filter between the diametrically opposite frequency components, say between \( \left( {f_x, {f_y}} \right) \) and \( \left( {-{f_x},-{f_y}} \right) \) about the zero frequency are computed and plotted as a function of azimuthal angle in the interval between 0 and *π*. The frequency components of the object that lie within the region \( \sqrt{{\left( {f_x^2+{f_y}^2} \right)}} < F \) do not acquire phase shifts by the shifted vortex filter that can be termed as Hilbert like as it is evident from the curve ‘a’ in Fig. 2. On the other hand for the frequencies \( \left( {f_x, {f_y}} \right) \) satisfying the condition \( \sqrt{{\left( {f_x^2+{f_y}^2} \right)}} > F \), the phase shifts introduced by the filter are such that the phase difference between diametrically opposite frequencies follows curve ‘b’. Note that only at one azimuthal direction the phase difference is *π* that is responsible for selective edge enhancement, whereas for other directions the Hilbert transform is fractional. For a vortex filter in which the zero frequency component of the object spectrum coincides with the vortex core, the phase shifts acquired by the diametrically opposite frequency components of the object follow curve ‘c’. From the curves of the Fig. 2 one can see that a shifted vortex leaves the lower frequency components less disturbed while the higher frequency components experience phase shifts leading to anisotropic edge enhancement of the object.

## Shifted anisotropic vortex filter for selective edge enhancement

We have seen that the selective edge enhancement is possible either by anisotropy or by shifting the filter in the FT plane. Hence we believe that a shifted anisotropic vortex filter will perform selective edge enhancement more effectively.

*R*of the filter with respect to the central position

*r*

_{0}.

The filter function specified by Eq. (5) yields radially non symmetric selective edge enhancement after spatial filtering operation. The selectivity is controlled by parameter *n*, which is a measure of anisotropy in the filter function and the shift in the phase of the anisotropic vortex filter function provides the enhancement in one side of the centre of the object [19]. The output of the filtering operation can be given by the convolution of the object with the spectrum of the filter function given by Eq. (6).

Anisotropy induced selectivity enhances edges in ± *ϕ* directions; whereas shift induced selective edge enhancement happens either in + *ϕ* or in − *ϕ* direction, depending on the direction of the shift. On the other hand in the anisotropy induced selectivity one can control the range of ∆*ϕ* whereas in shifted vortex filtering such a selection on ∆*ϕ* is not seen.

## Simulation results

*n*of sine function. The orientation selection is done by adding

*θ*

_{0}to

*θ*and the spatial shift in phase of the filter provides the enhancement in one side of the centre. More over after adding

*θ*

_{0}to

*θ*the function

*θ*+

*θ*

_{0}is made to lie between −

*π*and

*π*by modulo 2

*π*operation. This is done to preserve the helical shape of the wavefront. Figure 3 shows the simulation results of edge enhancement for a circular aperture, using anisotropic vortex function

*S*and it can be seen that the edge enhancement is selective but the same region of the edges is enhanced on either side of the centre. In our simulation the grid size is taken equal to 600 × 600 pixels and the size of circular aperture is kept equal to 150 pixels. Figure 4 shows the simulation results for edge enhancement of a circular aperture by the shifted vortex mask. The 3D plots of the intensity versus azimuthal angle, in Fig. 5, show the angular selectivity of edge enhancement using the shifted anisotropic vortex mask. Increased enhancement selectivity at different orientations is clearly visible.

## Experimental results

*S*with the help of reflective Spatial Light Modulator (SLM), Holoeye LC-R 2,500 with resolution 1,024 × 768, pixel pitch 19

*μm*. The object used is a circular aperture of size 200

*μm*. The experimental setup is shown in Fig. 6 and the fork grating corresponding to the different phase masks have been shown in Fig. 7. The object is illuminated by collimated beam from He-Ne laser (632.8 nm) and Fourier transformed with the help of Newport 10X microscopic objective and the Fourier transform is imaged on the SLM with 4X magnification by a lens of focal length 135 mm. The SLM is operated in phase mode keeping the polarizer at angle 170° to get the phase shift up to 2

*π*.

*S*and

*S*

_{sh}are formed in MATLAB keeping the resolution same as that of the SLM and the grating period has been kept equal to the six pixels of the SLM. The incident light wave is then diffracted by the fork grating displayed on the SLM, and only the light diffracted at the first diffraction order is used. The undesired diffraction orders are blocked. Imaging is done with help of a lens of focal length 200 mm, kept in between SLM and infinity-1 CMOS camera. The experimental results recorded for different values of anisotropy and for a given shift, are shown in Fig. 8 for circular aperture and the experimental results a given shift and for different values of angle of rotation are shown in Fig. 9. The experimental results are in well support of simulated results.

## Conclusion

We have proposed a new method for selective edge enhancement, capable of selecting desired region at any required redial direction. The proposed function provides the controllable anisotropy and shift in the phase of the vortex function and hence it is possible to enhance only the region of interest. Using a high resolution spatial light modulator (SLM) for displaying the phase masks corresponding to the proposed shifted anisotropic vortex mask, it is possible to get selective edge enhancement for an object. We have successfully implemented the method for the selective edge enhancement of a circular aperture. The method is efficient and useful in image processing when selective region of edges of the objects are important. More over the selective edge detection of phase objects is also possible using the same phase mask. The application of such filters is possible in microscopy to detect the edges of small biological objects selectively and in the anisotropic signal processing.

## Notes

### Acknowledgement

Manoj Kumar Sharma would like to thankfully acknowledge council of scientific and industrial research of India (CSIR) for senior research fellowship (SRF).

### References

- 1.N.R. Heckenberg, R. McDuff, C.P. Smith, A.G. White, Generation of phase singularities by computer generated holograms. Opt. Lett.
**17**, 221 (1990)ADSCrossRefGoogle Scholar - 2.M.R. Reicherter, T. Haist, E.U. Wagemann, H.J. Tiziani, Optical particle trapping with computer-generated holograms written on a liquid-crystal display. Opt. Lett.
**24**, 608 (1999)ADSCrossRefGoogle Scholar - 3.S. Vyas, P. Senthilkumaran, Interferometric optical vortex array generation. Appl. Opt.
**46**, 2893 (2009)ADSCrossRefGoogle Scholar - 4.S. Vyas, P. Senthilkumaran, Vortex array generation by interference of spherical waves. Appl. Opt.
**46**, 7862 (2009)ADSCrossRefGoogle Scholar - 5.D.P. Ghai, S. Vyas, P. Senthilkumaran, R.S. Sirohi, Vortex lattice generation using interferometric techniques based on lateral shearing. Opt. Commun
**282**, 2692 (2009)ADSCrossRefGoogle Scholar - 6.B.K. Singh, G. Singh, P. Senthilkumaran, D.S. Mehta, Generation of optical vortex array using single element reversed-wave front folding interferometer. Int. J. Opt.
**2012**(689612), 7 (2012). doi:10.1155/2012/689612 - 7.P. Senthilkumaran, Optical phase singularities in detection of laser beam collimation. Appl. Opt.
**42**, 6314 (2003)Google Scholar - 8.P. Senthilkumaran, J. Masajada, S. Sato, Interferometry with vortices. Int. J. Opt.
**2012**(517591), 18 (2012). doi:10.1155/2012/517591 - 9.P. Senthilkumaran, F. Wyrowski, H. Schimmel, Vortex stagnation problem in iterative Fourier transform algorithm. Opt. and lasers in Eng.,
**43**, 43–56 (2005)Google Scholar - 10.J.W. Goodman,
*Introduction to Fourier optics*(Roberts and Co, Colardo, 2007)Google Scholar - 11.S.N. Khonina, V.V. Kotlyar, M.V. Shinkaryev, V.A. Soifer, G.V. Uspleniev, The phase rotor filter. J. Mod.Optics
**39**, 1147 (1992)ADSCrossRefGoogle Scholar - 12.J.A. Davis, D.E. Mcnamara, D.M. Cottrel, J. Campos, Image processing with the radial Hilbert transform: theory and experiments. Opt. Lett.
**25**, 99 (2000)ADSCrossRefGoogle Scholar - 13.R.B. Bracewell,
*The Fourier transform and its application*(McGraw-Hill, New York, 1965)Google Scholar - 14.J.A. Davis, D.E. McNamara, D.M. Cottrell, Analysis of the fractional Hilbert transform. Appl. Optics
**37**, 6911 (1998)ADSCrossRefGoogle Scholar - 15.J.A. Davis, D.A. Smith, D.E. McNamara, D.M. Cottrell, J. Campos, Fractional derivatives—analysis and experimental implementation. Appl. Optics
**40**, 5943 (2001)ADSCrossRefGoogle Scholar - 16.A.W. Lohmann, D. Mendlovic, Z. Zalevsky, Fractional Hilbert transform. Opt. Lett.
**21**, 281 (1996)ADSCrossRefGoogle Scholar - 17.G. Situ, M. Warber, G. Pedrini, W. Osten, Phasecontrast enhancement in microscopy in microscopy using spiral phase filtering. Opt. Commun.
**283**, 1273–1277 (2010)Google Scholar - 18.A.W. Lohmann, E. Tepichı’n, J.G. Ramı’rez, Optical implementation of the fractional Hilbert transform for two-dimensional objects. Appl. Optics
**36**, 6620 (1997)ADSCrossRefGoogle Scholar - 19.S. Guohai, P. Giancarlo, O. Wolfgang, Spiral phase filtering and orientation-selective edge detection/enhancement. J. Opt. Soc Am. A
**26**, 1788 (2009)CrossRefGoogle Scholar - 20.M. K. Sharma, J. Joseph, P. Senthilkumaran, Selective edge enhancement using anisotropic vortex filter. Appl. Optics
**50**, 5279 (2011)ADSCrossRefGoogle Scholar - 21.K. Guang-Hoon, L. Hae June, K. Jong-Uk, S. Hyyong, Propagation dynamics of optical vortices with anisotropic phase profiles. J. Opt. Soc Am. B
**20**, 351 (2003)CrossRefGoogle Scholar