Journal of Optics

, Volume 42, Issue 1, pp 1–7 | Cite as

Selective edge enhancement using shifted anisotropic vortex filter

  • Manoj Kumar Sharma
  • Joby Joseph
  • Paramasivam Senthilkumaran
Research Article

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 mth 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 4f 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 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

Isolated optical vortices are phase singularities characterized by helical wavefronts winding about amplitude zeros where the phase is indeterminate. Around the singularity, the phase variation shows helical structure keeping the singularity at the centre. The phase singularity in an optical beam is also known as optical vortex. Even though, the phase singularity exists at the centre; it affects the phase distribution over the entire beam. Analytically an optical vortex can be defined by a complex field
$$ {V_i}\left( {x,y} \right)=x+iy=r\exp \left( {i\theta } \right) $$
(1)
where 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.
In an anisotropic optical vortex [21] this is not a constant. Consider an anisotropic vortex given by
$$ {{\tilde{V}}_a}\left( {x,y} \right)=x+i\sigma y=r\exp \left( {i\psi \left( {x,y} \right)} \right) $$
(2)
where, the phase is given as
$$ \psi \left( {x,y} \right)={\tan^{-1 }}\left( {\sigma \frac{y}{x}} \right)={\tan^{-1 }}\left( {\sigma \frac{{\sin \theta }}{{\cos \theta }}} \right) $$
(3)

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

The selective edge enhancement can also be achieved by translating an isotropic vortex phase mask in the filter plane. We know that in an isotropic vortex phase distribution of charge 1, any two points opposite to the vortex core have 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.
Fig. 1

The schematic showing the position of the shifted vortex core with respect to the zero frequency component in the filter plane

Fig. 2

The plot of the phase difference introduced by the filter to the diametrically opposite frequency components of the object as a function of azimuthal angle. a For lower frequency components of the object by the shifted vortex filter, (b) For the higher frequency components of the object by the shifted vortex filter and (c) For all the frequency components of the object by an unshifted vortex filter

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.

We consider the anisotropic vortex phase mask [20] given by
$$ S\left( {r,\theta } \right)=\exp \left[ {i\theta \left\{ {\left| {\sin^n \left( {\theta /2} \right)} \right|} \right\}} \right] $$
(4)
for our experiment here. In an anisotropic vortex filter the phase varies in such a manner that it enhances the edges selectively but in radially symmetric manner.
The shifted anisotropic vortex phase mask can be represented as
$$ {S_{sh }}\left( {r,\theta } \right)=S\left( {r,\theta } \right)*\delta \left( {r_0 -R} \right) $$
(5)
where, the convolution with delta function determines the shift R of the filter with respect to the central position r0.
The Fourier transform of the shifted anisotropic vortex phase mask, using the convolution theorem, can be given as
$$ {s_{sh }}\left( {\rho, \varphi } \right)=FT\left\{ {{S_{sh }}} \right\}=s\left( {\rho, \varphi } \right)\times 2\pi R{J_0}\left( {2\pi R\rho } \right) $$
(6)
where \( s\left( {\rho, \varphi } \right) \) is the Fourier transform of the anisotropic vortex filter [20] and the Bessel-Fourier transform of the shifted delta function in polar coordinates is given as [10]
$$ FT\left\{ {\delta \left( {r_0 -R} \right)} \right\}=2\pi R{J_0}\left( {2\pi R\rho } \right) $$
(7)

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

Edge enhancement in a smaller region can be achieved by increasing the power n of sine function. The orientation selection is done by adding θ0 to θ and the spatial shift in phase of the filter provides the enhancement in one side of the centre. More over after adding θ0 to θ the function θ + θ0 is made to lie between − π and π by modulo 2π 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.
Fig. 3

Simulation results for a circular aperture using anisotropic vortex filter with n = 30 and no spatial shift is provided to the phase of the filter (a) phase mask (b) 3D plot of output intensity

Fig. 4

Simulation results for a circular aperture using shifted isotropic vortex mask (a) and (c) are right and left shifted masks each by 20 pixels, (b) and (d) are corresponding edge enhanced 3D plots of output intensity

Fig. 5

Simulation results for edge enhancement of a circular aperture when a shifted anisotropic filter with n = 30 is used. a, c are shifted anisotropic masks with different shifts, (b), and (d) are 3D plots of edge enhanced output intensity

Experimental results

We have implemented the phase masks corresponding to the function 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 2π.
Fig. 6

Experimental setup L1 collimating lens, S sample/object, M Microscope objective, L2 Fourier transforming lens, SLM spatial light modulator in phase mode, L3 imaging lens and CMOS infinity1 camera used to record the output images

Fig. 7

Fork gratings to be displayed on SLM corresponding to (a) Vortex filter (b) shifted vortex filter (c) anisotropic vortex filter (d) shifted anisotropic vortex filter

The computer generated holograms (CGH) corresponding to function S is formed and displayed on the SLM. This CGH is a fork grating formed by interference of anisotropic vortex beam and a tilted plane wave. The CGH corresponding to the proposed function S and Ssh 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.
Fig. 8

Experimental results for selective edge enhancement with (a) Vortex filter (b) Shifted vortex filter, (c) anisotropic (n = 5) and shifted (20 pixels) vortex filter, (d) Anisotropic (n = 30) and shifted (20 pixels) vortex filter

Fig. 9

Experimental results for selective edge enhancement with anisotropy (n = 30) and the shift (20 pixels) is provided towards (a) right (b) left (c) down

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).

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Copyright information

© Optical Society of India 2012

Authors and Affiliations

  • Manoj Kumar Sharma
    • 1
  • Joby Joseph
    • 1
  • Paramasivam Senthilkumaran
    • 1
  1. 1.Department of PhysicsIndian Institute of technology DelhiHauzkhas New DelhiIndia

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