Computer Generated Quadratic and Higher Order Apertures and Its Application on Numerical Speckle Images

A computer generated quadratic and higher order apertures are constructed and the corresponding numerical speckle images are obtained. Secondly, the numerical images of the autocorrelation intensity of the randomly distributed object modulated by the apertures and the corresponding profiles are obtained. Finally, the point spread function (PSF) is computed for the described modulated apertures in order to improve the resolution.

The intensity pattern of speckle images may be considered as a superposition of the aperture spread function of an optical system and the classical speckle pattern [11,12].The contrast may be affected by the PSF and it may be understood by considering the far-field speckle produced by weak diffuser [13].
Electronic/Digital speckle pattern interferometer (ESPI/ DSPI) is a promising field that having a variety of applications [14][15][16][17], for example in the measurement of displacement/deformation, vibration analysis, contouring, non-destructive testing etc.The capability of ESPI/DSPI in displaying correlation fringes on a TV monitor is one of its distinct features.The digital speckle interferometer [18] (DSI) has many advantages since it does not need the photographic film and the optical dark room which are necessary for the holographic interferometers and the speckle photographs.The DSI has been used to the study of the density field in an acoustical wave for quantitative diagnosis of the speckle intensity.Digital data treatment is based on the direct computer aided correlation analysis of the temporal evolution of dynamic speckle pattern [19].The evaluation procedure uses the autocorrelation analysis of the speckle pattern obtained with FFT and low pass noise filtering to check the statistical function of speckle intensity distribution.
In this paper, the numerical quadratic and higher order apertures are considered as a replacer of the thin film techniques.These apertures are placed nearly in the same plane of the randomly distributed object and the numerical speckle images are obtained.The difference between any two speckle images, for these different apertures, can be visualized by the human eye.Also, the autocorrelation intensity of the diffuser and the profile shapes are plotted.The autocorrelation intensity leads to the recognition of the aperture distribution in particular in case of the deformed aperture [12].

Theoretical Analysis
An aperture of n  distribution is represented as follows: 0 ( ) with 1 x y    is the radial coordinate in the aperture plane.This radial aperture is constructed, using MATLAB program, and is represented as shown in Figure 1, where 2, 4, 6, , etc n   .The point spread function (PSF) or the amplitude impulse response is calculated by operating the two dimensional Fourier transform to get   n h r as follows [3,20]: With the help of recurrence relations and using ntegration by parts [21], we get: Where 1 J is the Bessel function of 1st order, and The complex amplitude located in the focal plane of the lens L is obtained by operating the Fou the complex amplitude rier transform upon ( , ) A x y , Equati ourier Making use of the properties of convolution pr Equation (5) becomes: on (4), to get: Where F.T. refers to F transform operation.oduct, Where   , s u v   is complex amplitude of tern formed in the focal plane of givenby: speckle patthe lens L and is x y , wh tu aperture and is given b ile h(u,v) is amplide impu onse of the imaging system and is calculated by operating the Fourier transform upon the modulated y: The recorded intensity of the sp   , u v is given by : The symbol   * bolic Equation stands for convolution operation.This sym (7) is explicitly tegral form as follows: The difference between any two speckle ima two different modulated apertures of the same nu aperture is obtained by subtraction,using formula (8),as fo ges for merical llows : 1 2 Where 1 I stands for the 1 speckle image and st 2 I stands fo 2 nd image.We can reconstruct either of the diffuser image m pl he n upon Equation ( 6), to get: r the ultiied by t modulated aperture or the autocorrelatio function of the diffuser.
Firstly, in order to reconstruct the diffuser which is modulated by the aperture it is sufficient to operate the inverse Fourier transform   , x y   is the imaging or reconstruction plane.
Substitute from Equation ( 6) in Equa the diffuser function multiplied by the modulated aperture sin ion produc tion (10), we get ce the Fourier transform of the convolut t is transformed into multiplication [21,22].Hence, we get in the Fourier plane   Secondly, in order to reconstruct the autocorrelation function of the diffuser which is affected lated radial aperture we are obliged to operate the Fourier tra by the modunsform upon the intensity distribution of the speckle pattern Equation (7), to get : Special case: If the impulse response of the ima system is approximated by Dirac-Delta distribution i.e.
which is valid only for actly the Fourier transform of iform illumination like that obtained from laser beam by spatial filtering, then the speckle image becomes exthe diffuser function as an object.Hence, Equation (7) becomes: orrelation function of the diffuser is exactly the Fourier transform Equation ( 13): In this case, the reconstructed autoc inverse of And the autocorrelation intensity is computed as the modulus square of Equation ( 14): It is clear that the profile of the speckle image has a resolution which is dependent upon the aperture distribution.It is shown that resolution improvement odulated speckle images, in particular in case of radial distributed apertures.This is attributed due to the improvement occurred in the point spread function of the imaging system.A comparison of the different PSF in case of circular, annular apertures, and radial distributed apertures is given later in Figure 10(d).The reconstructed images of the apertures are obtained from Equation (11) and plotted in Figures 6(a) and (b)and the corresponding profiles are plotted as in Figures 7(a), (b) and (c).

Results and Discussion
The difference between the actual analog image and the quantized digital image is called the quantization error.This quantization error may be minim te is at least as great as the total spectral width w.Thus the critical sampling rate is just w called the Nyquist rate and the critical sampling interval is w -1 which is called the Nyquist interval.
The autocorrelation intensity of the diffuser, in case of modulated apertures Equation (12), is plotted as shown in Figures 8(a), (b) and es, that the diameter of the autocorrelation intensity is twice the diameter of the whole circular aperture.Also, the contrast of the autocorrelation intensity is improved for the radial apertures as compared with the correlation images obtained in case of uniform apertures.The profiles of the autocorrelation intensity are plotted as in Figures 9(a), (b) and (c) which are taken at the slice x = [1,256,128,128], and the slice y = [1,256,128,128].It is shown that two different profiles are obtained for the two different apertures of 2   and 10  distributions.
The point spread function is computed for apertures of ρ n distribution using Equation (2) for different powers of even values of n. we ta n = 2 6, 8, and 10.Th ke , 4, e PSF is represented graphically as shown in Figure 10(a) and  (b).The comparative curves corresponding to circular and annular apertures are plotted as in Figure 10(c).It is shown, referring to the plotted results, that the best resolution is attained as n increases (n = 10) followed by n = 8 etc.Hence, the lowest resolution corresponds to the circular aperture and the best resolution corresponds to higher order aperture of n = 10 while the contrast of the image obtained in case of circular aperture is better than that obtained in case of higher order aperture.The Fig- ure 10(d) shows three curves where the best resolution is attained for annular aperture at the expense of the contrast while the higher order aperture of ρ 10 distribution gives better resolution and contrast as compared with circular aperture.eckle images is recommended.Figure 4(a), (b) and (c) is showing the difference beference between the two speckle images correspondintween images.The difg to circular and quadratic apertures is plotted as in Figure 4(a) and the difference corresponding to the linear and quadratic apertures is plotted as in Figure 4(b) while the difference corresponding to the quadratic and higher order apertures of 10  is shown in Figure 4(c).These images which represent the difference between speckle images shown in Figure 4

Conclusions
We have computed numerically the autocorrelation intensity of the randomly distributed object, using three different apertures, from the speckle images.It is concluded that, from the shape of the autocorrelation intensity for both of the modulated apertures, the diameter of the autocorrelation peak is two times the diameter of the whole aperture as expected.Also, the contrast of the modulated speckle images is affected by the modulated apertures.It is shown that the her order PSF curve is better in resolu-tion than for the circular aperture since the central peak of the PSF for higher order aperture is sharper than the corresponding peak obtained for the circular aperture.The contrast is better for the circular aperture since it depends mainly on the numerical aperture without modulation.
The PSF plot for higher order apertures of ρ n distributions showed a great improvement in resolution as compared with that obtained in case of uniform circular aperture.
of the computerized modufacility of fabrication as compared with the tedious work The potential application ture is better than the contrast obtained in case of higher order aperture.
The radial hig lated apertures on metrological systems, such as digital speckle interferometers and holographic filters, lies in its

MATLAB program is constructed to design two 3 .
Figur distriplotted as in e 1 [1 butions.These digital apertures are (a) and (b) and compared with the uniform circular aperture and the linearly varied aperture 1].Another MATLAB program is constructed to fabricate a diffuser as a randomly distributed object of dimensions 1024 × 1024 pixels shown in Figure 2. The parts of MATLAB program are used to obtain the different Figures (3)-(7).Digital speckle images for the randomly distributed objects which is modulated by the different apertures, using Equation (6), are represented in the Figure 3.The Figure 3(a) is plotted for the speckle image which is modulated by the linear aperture, the Figure 3(b) shows the speckle image modulated by the quadratic aperture, and the Figure 3(c) is given for higher order aperture 10  .It is shown, for the naked eye, that the three speckle images are different since they are modulated by different distributions of impulse responses or point spread func on (PSF) shown in Figures 9. Also, the comparative speckle image, shown in Figure 3(d), obtained for circular aperture is completely different from those speckle images shown in Figure 3(a), (b) and (c).If the difference between the modulated speckle images is not obvious for the naked eye, hence the computation of the difference between any two different modulated sp ti

Figure 5 .
Figure 5. (a) The profile shape of the numerical speckle mage obtained using the linearly distributed aperture; (b) The profile shape of the numerical speckle image obtained using the quadratic aperture; (c) The profile shape of the numerical speckle image obtained in case of ρ 10 distributed aperture; (d) The profile shape of the numerical speckle image obtained in case of uniform circular aperture.

Figure 6 .Figure 7 .FFigure 9 .Figure 10 .
Figure 6.(a) Reconstruction of the quadratic aperture obtained from the modulated speckle image shown in Figure 3(b); (b) Reconstruction of the ρ 10 aperture obtained from the modulated speckle image shown in Figure 3(c).
oint spread function (PSF ) corresponding to five different apertu plotted for the quadratic aperture of n = 2 .The green curve corresponds to n = 4, the red for n = 6, the lowers are plotted for n = 8 , and n = 10.The range of w equals [-6, 6]; (b) The same plot shown in Figure 9 but in the range of w extends from [-2,2] or the sake of clarity and comparison.The resolution is improved by increasing the order n quadraticall f a contrast for quadratic aper-