Received 4 February 2016; accepted 27 May 2016; published 30 May 2016
1. Introduction
Corona viruses are an important family of human and veterinary pathogens that can cause enteric and respiratory infections. Corona virus infection can lead to respiratory failure, gastroenteritis, nephritis, and hepatitis.
A novel methodology of single particle image analysis is applied to select virus features in order to obtain detailed model of oligomer state and spatial relationships among viral structural proteins [1] .
The addition of electronics, computers, and software to interferometry has enabled enormous improvements to optical metrology. Phase-shifting interferometry is used for getting data into a computer so the data can be analyzed [2] - [9] . Image processing of uniform objects and modified apertures was outlined [10] - [14] .
In this paper, the refractive index distribution of the corona virus images is computed from the phase shift method. The corona virus fringe shift with respect to the background interference is computed to get useful information about the phase shift of the image leading map the height depth and the refractive index distribution. The results and discussions are given followed by a conclusion. The former work concerning the digital fringe shift is limited by λ/2 the inter-fringe spacing. Hence, the computation of refractive index is dependent on the fringe spacing either using two or multiple beam interference.
2. Analysis
The complex amplitude of the corona virus as an object can be represented as follows:
(1)
where a―is the amplitude of the image, and 
is its phase for object depth z. Equation (1) can be written in a discrete matrix form as follows:
(2)
where a square matrix of dimensions N ´ N = 512 ´ 512 pixels (N = M) is assumed.
This work focuses on the main technique for phase evaluation of the corona virus image using the phase shifting method.
Coherent addition of a reference laser beam 
to the above object beam is considered to fabricate the modeled interference pattern. The laser beam is spatially filtered using a pinhole located in the focal plane of a converging lens in order to get uniform illumination. In this case, a plane wave is obtained. The pinhole pass only the central peak from the whole diffraction pattern and suppress all the diffraction legs and then the converging lens which is placed a distance (f) from the pinhole passes parallel rays of uniform intensity which is considered as a plane wave.
Then the intensity of the two beam interference obtained in the detector plane can be expressed as the modulus square as follows:
(3)
where 
is the intensity of the modulated interference field at the point 
for object depth z, 
, is the function that characterizes the mean intensity of the interference pattern and
, is the function that determines the modulation of the interference signal. In this case, the obtained trigonometric function has straight line fringes modulated by the object phase information. The modulated intensity is rewritten in matrix form as follows:
(4)
Certainly, the d. c term in equation (3),
.
R is the amplitude of the coherent laser beam and that Y appeared in equation (3) its phase. The equation (4) is used in the fabrication of the phase-shifted images outlined in equation (5).
Since the distance between any two fringes = l/2. Consequently, according to the phase shift technique [7] [9] , the phase information of the image is governed by the following equation:
(5)
where the range of the interference phase 
extends from 0 up to 2π for a height z, I1 is the intensity given in equation (4) at a phase
, I2 has
, and I3 has phase![]()
. Then, three equations are solved to get equation (5).
Once the phase is determined across the interference field, the corresponding height distribution h(x, y) on the surface of corona virus can be determined [2] as follows:
![]()
(6)
We have assumed the surface measured at normal incidence. Almost all interferometers used to measure surface height variations use phase-shifting techniques.
The refractive index of the corona virus m is computed as follows:
Since the phase of the wave cumulates traveling a distance L in a medium is
![]()
(7)
Then, the same wave that propagates over two equivalent paths L in corona virus medium and in vacuum gives the phase difference as follows (Figure 1):
![]()
(8)
where k = w/c = 2p/l is the propagation wave number in a medium of refractive index m while k0 is the propagation constant in vacuum.
By differentiation w.r.t. the path l = z, the refractive index distribution of the corona virus image is computed as follows:
![]()
(9)
Since the angular frequency is related to the wavelength as ![]()
and ![]()
then the above equation becomes:
![]()
(10)
The optical path difference represents the height variation of the image, namely ![]()
then equation (10) becomes:
![]()
(11)
The differentiation of the height distribution ![]()
with respect to z give the differential fringe shift and the amplitude of the planar image a(x, y). Consequently, we finally get equation (12).
![]()
(12)
The fringe shift is ![]()
with respect to inter-fringe spacing ![]()
at constant x, the fringes are assumed located in the x-y plane and z is the axis normal to the fringe system which represents the height depth and a(x, y) represents the amplitude of the image. In equation (12),![]()
.
Computation of the Contrast
Consider the visibility expression to represent the fringe contrast as follows:
![]()
. Substitute in equation (3), we get this result for the contrast:
![]()
Figure 1. The propagation of light in a medium of refractive index μ compared with air.
![]()
(13)
While the contrast given in case of multiple beam interference is extracted from the transmitted intensity distribution [15] :
![]()
(14)
The parameter ![]()
and the corresponding contrast is given by the formula (11). The maximum and minimum intensities are obtained from equation (13) as:
![]()
for![]()
, m is integer.
![]()
for ![]()
It is known that the refractive index has a direct relation with the polychromatic spectral distribution of illuminating light according to the Cauchy formula as follows:
![]()
where a, and b are constants. Then the fringe shift and the refractive index are affected by the change of the wavelength.
3. Results and Discussion
The Corona virus image used in the processing is shown as in the Figure 2. It has dimensions of 512 ´ 512 pixels. The fringe shift occurred within the corona virus w.r.t. the background shift is plotted as in the Figure 3 at frequency f = 1/32. This interferometer plotis obtained from equation (3) written in discrete form where M= N = 512 pixels. The yellow discontinuous horizontal lines are taken at 10, 80, 140, 200, 260, 320, 380, 440 pixels. Only 7 fringes are shown in this image. The shift of the corona virus cells are computed as compared with the inter-fringe spacing. The Mat Lab code is used for the computation of the modulation term cos (y-(1/32) A(i, j)). While the background two beam interference is only cos (y). It is represented by straight line fringes modulated by the object matrix A (M, N). From the Figure 3, eight plots are shown in the Figure 4 at 10, 80,140, and 200 pixels in the 1st column, and at 260, 320, 380, and 440 pixels in the 2nd column. The upper left plot at 10 pixels show uniform straight line fringes which is compared with the shifted fringes corresponding to the image geometry shown at the mentioned lines. Hence, we can get phase information about the image as plotted in the Figure 5 using equation (8) and height information from equation (9).
Four different modulated fringes shifts are shown in the Figure 6. Interferometry images of the corona virus
![]()
Figure 2. Corona virus image used in the processing. It has dimensions of 512 ´ 512 pixels.
![]()
Figure 3. Two beam interference of corona virus image of dimensions 512 ´ 512 pixels. The yellow discontinuous lines are taken at 10, 80, 140, 200, 260, 320, 380, 440 pixels. 7 fringes are shown in this image. The shift of the corona virus cells are computed compared with the inter-fringe spacing.
![]()
Figure 4. Eight plots from the Figure 3 are shown at 10, 80,140, and 200 pixels in the 1st column in the left, and at 260, 320, 380, and 440 pixels in the 2nd column.
using the cosine function to represent the phase at four different spatial frequencies at 1/32, 1/64, 1/96, and 1/128. The original image is multiplied by a factor of α = 1/32 in the interference modulated terms as follows: for a) cos (y-(1/32)A (i, j)), b) cos (2y-(1/32) A (i, j)), c) cos (3y-(1/32) A (i, j)), and d) cos (4y-(1/32) A (i, j)).
The multiple beam interferometry images of the corona virus using the Airy function to represent the phase at four different background frequencies is plotted as in the Figure 7. The original image is multiplied by a factor of α = 1/32 as in the Figure 6.
Finally, comparing the fringe shift in the corona virus image at different frequencies f = 1/32, 1/64, 1/96, and 1/128, it is shown that the fringe shift is not resolved at frequency greater than f = 1/128.
The profile of the corona virus image taken at constant x = 150 pixels extracted from the image shown in the Figure 3 is plotted as in the Figure 8(a) and the profile taken at x = 330 pixels is plotted in the Figure 8(b). In the left of both plots, seven fringes are shown. The profiles are affected by the fringe shift resulted from the phase change of the image under consideration.
![]()
Figure 5. The Phase map of the corona virus of background spatial frequency f = 5; where α = 1/32.
![]()
Figure 6. The interferometry images of the corona virus using the cosine function to represent the phase at four different spatial frequencies at 1/32, 1/64, 1/96, and 1/128. The original image is multiplied by a factor of α = 1/32.
Effect of mirror reflection coefficient upon the multiple beam interference images is shown as in the Figure 9. Refer to the derived equation from the Airy distribution (14); the contrast of the image is sharper for higher reflectivity, equation (13). The images shown in the Figure 9 are in good agreement with the theoretical results since higher contrast is attained at R = 80% as compared with that obtained at R = 40%.
The values of refractive index extracted from the Figure 10 are given in the Table 1 and plotted for horizontal line at 330 pixels as in the Figure 11. The modulated interference image of corona virus as in the Figure 12 where 32 fringes are plotted is used in the segmentation.
The upper cell segment selected from the image in the Figure 12 is shown as in the Figure 13(a) and the corresponding modulated fringe system of 11 fringes shown in the Figure 13(b) is investigated. The values of refractive index extracted from the Figure 13(b) for the cell are computed from equation (12) given in the Tables 2-4 and plotted for horizontal lines at 60, 80, and 100 pixels as in the Figure 14.
![]()
Figure 7. The multiple beam interferometry images of the corona virus using the Airy function to represent the phase at four different background frequencies as in the Figure 6. The original image is multiplied by a factor of α = 1/32.
![]()
(a)![]()
(b)
Figure 8. (a): The profile of the corona virus image taken at constant x = 150 pixels extracted from the image shown in the Figure 3. Seven fringes are shown; (b) the profile of the corona virus image taken at constant x = 330 pixels extracted from the image shown in the Figure 3. Seven fringes are given.
![]()
![]()
Figure 9. Effect of mirror reflection coefficient upon the multiple beam interference images is shown.
![]()
Figure 10. The corona virus image modulated by fourteen fringes used in the computation of the refractive index at 150 and 330 pixels as in the Table 1.
![]()
Table 1. The refractive index values as a function of the Z coordinate at certain horizontal line at 330 pixels.
![]()
Figure 11. plot of the refractive index versus the horizontal distance at constant line x = 330 pixels. Three cells from the corona virus are scanned in the image. Inter-fringe spacing ∆Z = 34 pixels.
![]()
Figure 12. The modulated interference image of corona virus where 32 fringes are shown.
![]()
Figure 13. In the left (a), a segment from the image shown in the Figure 12 is taken at i = 220: 390 pixels and j = 10:180 pixels. The modulated two beam interference of the cell (segment) shown in (b) is investigated. Both segments in a, b have dimensions of 170 × 170 pixels.
![]()
Table 2. The refractive index values as a function of the Z coordinate at certain horizontal line at 60 pixels.
![]()
Table 3. The refractive index values as a function of the Z coordinate at certain horizontal line at 80 pixels.
![]()
Table 4. The refractive index values as a function of the Z coordinate at certain horizontal line at 100 pixels.
![]()
Figure 14. Three plots of the refractive index variation with Z at constant horizontal lines at x = 60, 80, 100 pixels computed from the Figure 13(b) which has only one cell.
The map of the refractive index distribution computed from equation (10) is the final object of this work.
4. Conclusions
The phase shift of corona virus images deduced from the interferometer images. The interferometer images using multiple beam interference gave better contrast than the corresponding images with the two-beam interference as expected. The effect of mirror reflection coefficient upon the multiple beam interference images discussed.
Useful information obtained from studying this virus using interferometry is extracted from the fringe shift of the modulated interference pattern namely the refractive index distribution of the whole image. Consequently, detailed and precise information about the virus may be extracted from the refractive index distribution. In addition, since the refractive index has a direct relation with the polychromatic spectral distribution of illuminating light according to the Cauchy formula it allows observe the diameter of the virus cell accurately as it changes with the wavelength of light.