Motion detection using extended fractional Fourier transform and digital speckle photography

Basanta Bhaduri; C. J. Tay; C. Quan; Colin J. R. Sheppard

doi:10.1364/OE.18.011396

1. Introduction

Digital speckle photography is a useful means for determining motion, such as in-plane translation and out-of-plane rotation or tilt of an optically rough surface, from the speckle shift that takes place at the recording plane [1–9]. The technique involves recording of the speckle intensity patterns from the surface both before and after the motion. In translation measurement, a Fourier transform (FT) of the sum or difference of the two sequential images is performed numerically. The resulting FT yields a cosinusoidal fringe pattern with a spacing which is inversely proportional to the surface displacement with the direction of motion being perpendicular to the fringe pattern [4]. For the measurement of tilt, the optical Fourier transform (OFT) of the waveform from the reflected surface fields is captured sequentially, and further the Fourier transform of the sum or difference of the captured images results in a pattern whose fringe spacing is inversely proportional to the magnitude of the rotation [1]. The former image recording technique is insensitive to tilting motion, whereas the latter OFT is insensitive to translation, and thus the two operations are necessary to measure both types of surface motion. Moreover neither of the above two operations allows the user to determine the direction of the motion. An optical system which produces neither pure optical imaging nor OFT is sensitive to both surface titling and translation.

A fractional Fourier transform (FRT) is a generalization of Fourier transform which was mathematically defined by Condon in 1937 [10]. Lohmann suggested two basic types of setup using lenses for implementing optical FRT [11]. It is possible to completely determine both the in-plane translation and the tilting motion of a rigid body using speckle correlation, if the motion is captured simultaneously in two different fractional Fourier domains. Two scale-invariant FRTs, either parallel with two CCDs [7] or simultaneous with a single CCD [8], are needed in order to detect direction and magnitude of both translation and tilt. The former case involves many optical components and hence errors could occur due to misalignment. In the latter case, the positions of the cross-correlation peaks are indistinguishable and hence one needs to know a-priori of the correlation peak for a particular FRT order to differentiate them. The measurement range and sensitivity also cannot be changed easily using a particular optical setup. Digital speckle photography in combination with optical linear canonical transform (LCT) has been used to measure small surface tilting or translation either individually or simultaneously [9]. The system uses a limiting aperture, and the varying curvature of the illumination field effectively changes the output domain. This system needs to make four recordings, two for each illumination, and is thus time consuming and hence not suitable for dynamic motion detection.

In this paper we propose a simple correlation based speckle photographic system that uses a single lens and detector to simultaneously detect both the magnitude and direction of translation and tilting motion by capturing only two frames: one before and the other after the object motion. The proposed technique is superior to digital speckle photography which is suitable mainly for the measurement of in-plane translation; any information about the tilt can’t be obtained.

A Michelson type optical setup has been used where two arms produce two simultaneous optical extended fractional Fourier transforms (EFRTs) of different orders. As the EFRT, a general case of FRT [12], is sensitive to both translation and tilting motion, the present system simultaneously determines both translational motion and surface tilting. The dynamic range and sensitivity of the measurement can be varied readily by shifting either of the mirrors along the optic axis. Theoretical analysis and experiment results are presented.

2. Theoretical analysis

2.1 Extended fractional Fourier transform

The extended fractional Fourier transform (EFRT) is a general case of FRT, and the most general case of FT where the input and output distances need not necessarily be the same. The EFRT of an input field is given by [12]

U_{a, θ, b} (q) = E F R T {f (p)} = C (q) \int_{- \infty}^{\infty} f (p) \exp (\frac{i π a^{2} p^{2}}{\tan θ} - \frac{i 2 π a b p q}{\sin θ}) d p,

where

a^{2} = \frac{1}{λ} \frac{\sqrt{f - l^{'}}}{\sqrt{f - l}} \frac{1}{{[f^{2} - (f - l) (f - l^{'})]}^{1 / 2}},

θ = arc \cos (\frac{\sqrt{f - l} \sqrt{f - l^{'}}}{f}),

b^{2} = \frac{1}{λ} \frac{\sqrt{f - l}}{\sqrt{f - l^{'}}} \frac{1}{{[f^{2} - (f - l) (f - l^{'})]}^{1 / 2}},

C (q) = \frac{\exp [i (π / 4 - θ / 2)]}{{(2 π \sin θ)}^{1 / 2}} \exp (\frac{i π b^{2} q^{2}}{\tan θ}) = \exp [i ϕ (q)],

a, b, θ are the extended fractional orders,

C (q)

is a complex phase factor with

ϕ (q)

being the phase, l is the input distance i.e. the distance of the object from the lens,

l^{'}

is the output distance i.e. the distance of the detector plane from the lens, and f is the focal length of the lens. If the input field is translated in the X-direction by ξ and tilted by a small angle

α,

which is equivalent to a shift in the spatial frequency of amount

κ (\approx 2 α)

, then the input field can be written as [8]

f^{'} (p) = f (p - ξ) \exp (i κ p) .

The EFRT of the field after undergoing small tilt and translation can be written as

\begin{array}{l} {U^{'}}_{a, θ, b} (q) = E F R T {f^{'} (p)} = C (q) \int_{- \infty}^{\infty} f (p - ξ) \exp (i κ p) \exp (\frac{i π a^{2} p^{2}}{\tan θ} - \frac{i 2 π a b p q}{\sin θ}) d p \\ = C (q) \int_{- \infty}^{\infty} f (p - ξ) \exp (i κ p) \exp {\begin{cases} \frac{i π a^{2} {(p - ξ)}^{2}}{\tan θ} - \frac{i π a^{2} ξ^{2}}{\tan θ} \\ + \frac{i 2 π a^{2} p ξ}{\tan θ} - \frac{i 2 π a b p q}{\sin θ} \end{cases}} d p \\ = C (q) \exp (- \frac{i π a^{2} ξ^{2}}{\tan θ}) \int_{- \infty}^{\infty} f (p - ξ) \exp {\begin{cases} \frac{i π a^{2} {(p - ξ)}^{2}}{\tan θ} \\ - i (p - ξ) [\frac{2 π a b q}{\sin θ} - \frac{2 π a^{2} ξ}{\tan θ} - κ] \\ - i ξ [\frac{2 π a b q}{\sin θ} - \frac{2 π a^{2} ξ}{\tan θ} - κ] \end{cases}} d p . \end{array}

If we use

t = p - ξ

in the above Eq. (7), we have

\begin{array}{l} {U^{'}}_{a, θ, b} (q) = C^{'} (q) \int_{- \infty}^{\infty} f (t) \exp {\frac{i π a^{2} t^{2}}{\tan θ} - \frac{i 2 π a b t}{\sin θ} (q - \frac{a ξ \cos θ}{b} - \frac{κ \sin θ}{2 π a b})} d t \\ = C^{'} (q) \int_{- \infty}^{\infty} f (t) \exp (\frac{i π a^{2} t^{2}}{\tan θ} - \frac{i 2 π a b t (q - Q)}{\sin θ}) d t, \end{array}

where

\begin{array}{l} C^{'} (q) = C (q) \exp (- \frac{i π a^{2} ξ^{2}}{\tan θ}) \exp {- i ξ [\frac{i 2 π a b q}{\sin θ} - \frac{i 2 π a^{2} ξ}{\tan θ} - κ]} \\ = \frac{\exp [i (π / 4 - θ / 2)]}{{(2 π \sin θ)}^{1 / 2}} \exp {\frac{i π}{\tan θ} (b^{2} q^{2} + a^{2} ξ^{2} - 2 a ξ b q \cos θ + κ ξ \tan θ / π)} \\ = \exp [i ϕ^{'} (q)], \end{array}

and

Q = \frac{a ξ \cos θ}{b} + \frac{κ \sin θ}{2 π a b} = s ξ + t κ = s ξ + t^{'} α,

where

s = \frac{a \cos θ}{b}

,

t = \frac{\sin θ}{2 π a b}

and

t^{'} = 2 t

; s and

t^{'}

being the sensitivities for the translation and tilt measurements, respectively i.e. the shift of the cross-correlation (CC) peak with respect to the auto-correlation (AC) peak for unit translation or tilt. Thus the extended fractional Fourier domain parameter q has been shifted by an amount Q.

2.2 Simultaneous motion detection

Figure 1 shows the schematic diagram of the proposed system, which is similar to a Michelson interferometric setup. However the contributions from both the arms will not interfere as they are orthogonally polarized with each other. Also the distances from lens to the CCD plane in arms 1 and 2 ( $l_{1}^{'}$ and $l_{2}^{'}$ ), as shown in Fig. 1, are different from each other. Collimated light from a laser illuminates a diffuse plane surface placed at distance l from the lens. As $l_{1}^{'}$ and $l_{2}^{'}$ are different from l, there are two EFRTs with extended fractional orders ( $a_{1}, b_{1}, θ_{1}$ ) and ( $a_{2}, b_{2}, θ_{2}$ ) which superpose at the detector plane. In order to avoid interference between the light fields from arms 1 and 2, linear polarizers (P₁ and P₂) are placed at orthogonal orientation.

Fig. 1 Schematic of the Michelson type optical arrangement for obtaining two simultaneous extended factional Fourier transforms: BS₁, BS₂, Beam splitters; L, lens; S, Aperture stop; M₁, M₂, Mirrors; P₁, P₂, Polarizers; CCD, charge coupled device.

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The recorded intensity at the detector plane before the object motion is the sum of each individual intensities and is given by

I = {| U_{a_{1}, θ_{1}, b_{1}} (q_{1}) |}^{2} + {| U_{a_{2}, θ_{2}, b_{2}} (q_{2}) |}^{2} = I_{1} (q_{1}) + I_{2} (q_{2}),

where we have assumed

I_{1} (q_{1}) = {| U_{a_{1}, θ_{1}, b_{1}} (q_{1}) |}^{2}

and

I_{2} (q_{2}) = {| U_{a_{2}, θ_{2}, b_{2}} (q_{2}) |}^{2}

.

Likewise, the recorded intensity at the detector after the object motion is given by

\begin{array}{l} I^{'} = {| U_{a_{1}, θ_{1}, b_{1}}^{'} (q_{1}) |}^{2} + {| U_{a_{2}, θ_{2}, b_{2}}^{'} (q_{2}) |}^{2} = {| U_{a_{1}, θ_{1}, b_{1}} (q_{1} - Q_{1}) |}^{2} + {| U_{a_{2}, θ_{2}, b_{2}} (q_{2} - Q_{2}) |}^{2} \\ = I_{1} (q_{1} - Q_{1}) + I_{2} (q_{2} - Q_{2}), \end{array}

where

I_{1} (q_{1} - Q_{1}) = {| U_{a_{1}, θ_{1}, b_{1}} (q_{1} - Q_{1}) |}^{2}

and

I_{2} (q_{2} - Q_{2}) = {| U_{a_{2}, θ_{2}, b_{2}} (q_{2} - Q_{2}) |}^{2}

.

If we assume the intensity of the field is a perfectly random function (though not exactly in practice), the cross-correlation between Eqs. (11) and (12) will cause two independent peaks arising at $q_{1} = Q_{1}$ and at $q_{2} = Q_{2}$ . Due to the assumption of the random nature of the fields, all other correlations will be zero. These Q’s, which are locations of CC peaks for different EFRTs, can be written as

Q_{1} = s_{1} ξ + {t^{'}}_{1} α,

and

Q_{2} = s_{2} ξ + t_{2}^{'} α,

where

s_{1} = \frac{a_{1} \cos θ_{1}}{b_{1}}, t_{1}^{'} = \frac{\sin θ_{1}}{π a_{1} b_{1}}

and

s_{2} = \frac{a_{2} \cos θ_{2}}{b_{2}}, t_{2}^{'} = \frac{\sin θ_{2}}{π a_{2} b_{2}}

. Thus, we can obtain the translation and tilt by solving Eq. (13) and Eq. (14) (

ξ = (Q_{2} t_{1}^{'} - Q_{1} t_{2}^{'}) / (s_{2} t_{1}^{'} - s_{1} t_{2}^{'})

and

α = (Q_{1} s_{2} - Q_{2} s_{1}) / (s_{2} t_{1}^{'} - s_{1} t_{2}^{'})

) if s_j and t'_j (j = 1,2) are known. The values of s_j and t'_j can be obtained experimentally through calibration. Thus with only two recordings; one before and another after the object motion we can measure the translation and tilt simultaneously.

3. Experimental work

3.1. Experimental setup

Figure 1 shows the schematic diagram of the experimental arrangement. We have used a He-Ne laser (Coherent Optics, 70 mW output power, 632.8 nm wavelength) as the source and a CCD camera (Pulnix 62EX, $768 \times 576$ pixels, $11.6 μ m \times 11.2 μ m$ pixel size) as the detector. The test object is an aluminum plate of size 60 mm x 60 mm and thickness 5 mm. The plate was coated with a white matt paint, and mounted on a translational/rotational stage which is able to provide precise motion. A convex lens with a 250 mm focal length and 50 mm diameter is used. A circular aperture stop (S) of size 17 mm is placed in front of the lens in order to ensure that the speckles are comparable to the CCD pixel size. The effect of the path delay by the beam splitters is taken into consideration while the system is being aligned. Matrox Meteor-II frame grabber card is used to record the images and sub-images sizes of 256x256 are used for calculating the cross-correlation coefficients using the Maltab platform.

3.2. Controlling the measurement sensitivity and range

Using the values of a, θ and b from Eqs. (2)–(4) in Eq. (10) we can get

Q = s ξ + t^{'} α = (1 - l^{'} / f) ξ + \frac{λ}{π} (l + l^{'} - l l^{'} / f) α .

Thus the sensitivities for translation and tilt measurement can be given, respectively, by

s = (1 - l^{'} / f),

and

t^{'} = \frac{λ}{π} (l + l^{'} - l l^{'} / f) .

It is evident from Eq. (16) that the sensitivity for translation measurement (s) is independent of the input distance l, and changes linearly as a function of the ratio of the output distance to the focal length ( $l^{'} / f$ ). Figure 2(a) shows a plot of the sensitivity s, as a function of the ratio $l^{'} / f$ , when the focal length (f) remains unchanged. It can be seen from Fig. 2(a) that s decreases from a positive to a negative value as $l^{'} / f$ increases and attains a zero value at $l^{'} = f$ . Further we have plotted the sensitivity for tilt measurement (t') as a function of the ratio $l^{'} / f$ , when the focal length (f) remains unchanged with the input distance (l) fixed at $2 f$ . Figure 2(b) shows such a plot where the sensitivity t', also decreases lineally from a positive to a negative value as $l^{'} / f$ increases and attains a zero value at $l^{'} = 2 f$ . We have also determined experimentally the sensitivities s and t'. The shift of the CC peak (in pixels) from the center (location of the auto-correlation peak) for a unit magnitude of translation (mm) or tilt (arcmin) is a measure of the sensitivity. A certain translation or tilt was prescribed while the output distance, $l^{'}$ , was varied for a fixed focal length and input distance. The focal length (f) of the lens used in the setup was 250 mm and the input distance, l, was fixed at 500 mm. Figure 3(a) shows the variation of sensitivity for translation measurement (s) while Fig. 3(b) shows the variation of sensitivity for tilt measurement (t') with the ratio of the output distance to the focal length ( $l^{'} / f$ ). As can be seen, the plots for both the sensitivities are similar to those shown in Fig. 2. At higher sensitivities, the system becomes sensitive to small motion; however, the measurement range decreases as a small motion of the object would be magnified in the output plane resulting in de-correlation. Conversely at lower sensitivities a larger measurement range is achieved. Negative values of the sensitivity indicate shifting of the CC peaks in the opposite direction.

Fig. 2 Plot of the theoretical sensitivity for (a) translation and (b) tilt as a function of the ratio of the output distance to the focal length ( $l^{'} / f$ ).

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Fig. 3 Plot of the experimental sensitivity for (a) translation and (b) tilt as a function of the ratio of the output distance to the focal length ( $l^{'} / f$ ).

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When either of the mirrors M₁ or M₂ in Fig. 1 is shifted along the optic axis by an amount D, due to reflection geometry the output path in the EFRT configuration changes by 2D. One can see from Eqs. (2)–(4) that the extended fractional orders will vary even though the focal length and the input distance remain unchanged. Thus one can readily change the measurement range and sensitivity simply by shifting either or both of the mirrors M₁ and M₂.

3.3. Experimental results

We first demonstrate the applicability of the system for the measurement of translation and tilt individually followed by their simultaneous measurement. The input distance was fixed at 500 mm and the output distances were 409 mm and 573 mm in arms 1 and 2, respectively. Thus we have first an EFRT with extended fractional orders $a_{1} = 3
.0216$ , $b_{1} = 3
.6751$ , $θ_{1} = 2
.536$ and another EFRT with extended fractional orders $a_{2} = 2
.578 - 2
.578i$ , $b_{2} = 2
.268 - 2
.268i$ , $θ_{2} = 3
.141 - 0
.517i$ . The sensitivities for translation and tilt measurements in arm 1 are 30 pixels/mm (s₁) and 4.17 pixels/arcmin (t'₁), respectively whereas for arm 2 the corresponding values are 56 pixels/mm (s₂) and −4 pixels/arcmin (t'₂), respectively.

3.3.1 Pure translation measurement

For the measurement of in-plane translation, both the waves from arm 1 and arm 2 are allowed to superpose at the CCD plane without any interference as shown in Fig. 1. Two intensity fields are recorded at the CCD: one before (I₁) and another after translating the object along X-direction (I₂). The cross correlation of the two recorded intensity fields produces two peaks due to both of the EFRTs. Figure 4(a) shows a 2D plot of the CC coefficient when the object was translated by 600 µm along the X-direction. Since the two CC peaks look identical, from the figure it is not possible to distinguish between them. Hence to identify each of them, we have initially recorded the individual intensity of each arm by blocking the other arm, which we call the reference intensities. Each of the reference intensities are then individually cross-correlated with the intensity recorded after the translation (I₂). The CC produces a peak for the particular arm or the particular EFRT. Figure 4(b) and 4(c) are the individual 2D plots of the CC coefficient for arm 1 and arm 2, respectively obtained with the reference intensities. White arrows in the figures indicate the CC peaks due to either of arm 1 or arm 2 (CCP1 and CCP2, respectively). Using the location of the CC peaks and solving two simultaneous equations, the measured translation is 592.92 µm in the X-direction with some negligible tilt (3 arcsec). This value shows an excellent agreement with the prescribed translation and the discrepancy is within 1.2%.

Fig. 4 Translation measurement: (a) 2D plot of cross correlation coefficient, (b) 1st CC peak, and (c) 2nd CC peak.

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3.3.2 Pure tilt measurement

As in translation measurement, two intensity fields are recorded at the CCD: one before and another after a small tilt of the object. The cross correlation of these two recorded intensity fields produces two peaks due to both of the EFRTs. Figure 5 shows a 2D plot of the CC coefficient when the surface was tilted clockwise by 3.6 arcmin. Here, as in the translation measurement, we have used reference intensities to identify the peaks for a particular EFRT. The arrows in Fig. 5 show the CC peaks due to either of the EFRTs (CCP1 and CCP2). Using the location of the CC peaks and solving two simultaneous equations, the measured tilt is 3.56 arcmin in the clockwise direction with very negligible translation (4.7 pm). Here also we get an excellent agreement with the prescribed tilt and the discrepancy is around 1.1%.

Fig. 5 Tilt measurement: 2D plot of cross correlation coefficient. The white arrows show the CC peaks due to EFRTs.

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3.3.3 Simultaneous translation and tilt measurement

As in the previous measurements, here we record two intensities at the CCD: one before and another after the object motion without any interference. The surface was translated by 900 µm along the negative X-direction and tilted anticlockwise by 4.8 arcmin. Figure 6 shows a 2D plot of the CC coefficient. As explained previously, we have used reference intensities to identify the peaks for a particular EFRT. The arrows in the Fig. 6 show the CC peaks for both the EFRTs (CCP1 and CCP2). Using the location of the CC peaks and solving two simultaneous equations, the measured translation is 893.4 µm along the negative X-direction and the tilt is 4.99 arcmin in the anticlockwise direction. Again the agreements with the prescribed values in both cases are excellent. The discrepancy for translation is less than 1% while that for the tilt is 4%.

Fig. 6 Simultaneous translation and tilt measurement: 2D plot of cross correlation coefficient. The white arrows show the CC peaks due to EFRTs.

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4. General discussions

The main aim of the present study is to demonstrate the effectiveness of the technique to measure simultaneously the tilt and translation. As the spatial resolution is not the main objective at this stage, we have not used the interpolation scheme on the recorded images to obtain sub-pixel accuracy [13].

The proposed technique is suitable for two dimensional translations as well. In that case instead of solving one set of 2 simultaneous equations, we need to solve two sets of 2 simultaneous equations, each in either of x or y directions. The shift of the cross-correlation peaks needs to be determined separately in both x and y directions for the same.

We have investigated the effect of sub-image size for calculating the CC coefficients, for which we have started with a sub-image size of $64 \times 64$ till $512 \times 512$ pixels for different translations and tilts. It is found that the shift of the CC peak should be around the 50% of the sub-image size in order to distinguish the peak from rest of the CC values and hence for reliable measurement. Though the peak can still be observed with lower CC value for a shift larger than 50% of the sub-image size, one needs to increase the sub-image size for accurate detection of the CC peaks. However, increase in the size of the sub-image also increases the CC computation time. We have also tested the repeatability of the results with different sub-image sizes. A small translation or tilt of the object is prescribed and the shift in the CC peak with different sub-image sizes found not varies with the sub-image sizes. We have used $256 \times 256$ pixels as the sub-image size in this paper.

The system can be used for dynamic measurement of motion by recording any number of frames during the motion and cross-correlating each of them with the single frame recorded before the motion. However, if the translation or tilt is higher than the system range, the reference images would be not usable for peak identification.

There can be an alternative method for unambiguous measurement when only the static situation is considered. In this method the intensity of arm 1 is recorded (I₁) while arm 2 is blocked. This is followed by recording of arm 2 intensity (I₂) while arm 1 is blocked. Then after the motion of the object, the intensity of arm 1 is recorded (I₃) while arm 2 is blocked. Subsequently the intensity of arm 2 is recorded (I₄) while arm 1 is blocked. Now by cross-correlating the intensities (I₁ and I₃) and (I₂ and I₄) we can measure the CC peak shifts for a particular EFRT, and then solving simultaneous equations we can measure translation and tilt without any ambiguity. The CC peak values obtained using this method show a much higher values compared to the case when two fields are superposed at the CCD plane.

It is worth mentioning that one can change the extended fractional orders and hence the sensitivities s and t' by changing the focal length (f) alone as can be evident from Eqs. (16) and (17). However, it is difficult to remount the lenses every time with different focal lengths. Use of spatial light modulator (SLM) as Fresnel lens can be useful in those circumstances.

5. Conclusions

We have presented a speckle correlation based technique to determine the direction and magnitude of a surface motion simultaneously. A Michelson type system has been proposed where two arms produce two different extended fractional Fourier transforms. The measurement range and sensitivity of the system can be easily varied by shifting the mirrors used in the setup. The technique is suitable for measuring dynamic motion of an object as it requires only two recordings: one before and the other after or during the object motion. A reference intensity method has been introduced to identify the CC peaks. The system is also able to determine pure translation or tilt by individually blocking one of the arms during the recording. One limitation of the work is that the system cannot be used to measure motion of object which is not flat.

Acknowledgement

The financial support provided by the Singapore Millennium Foundation (SMF) is gratefully acknowledged.

References and links

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3. M. Sjödahl, “Electronic speckle photography: measurement of in-plane strain fields through the use of defocused laser speckle,” Appl. Opt. 34(25), 5799–5808 (1995). [CrossRef] [PubMed]

4. K. J. Gåsvik, Optical Metrology, 3rd ed., (John Wiley & Sons Ltd, Chichester, 2002).

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6. J. M. Diazdelacruz, “Multiwindowed defocused electronic speckle photographic system for tilt measurement,” Appl. Opt. 44(12), 2250–2257 (2005). [CrossRef] [PubMed]

7. D. P. Kelly, B. M. Hennelly, and J. T. Sheridan, “Magnitude and direction of motion with speckle correlation and the optical fractional Fourier transform,” Appl. Opt. 44(14), 2720–2727 (2005). [CrossRef] [PubMed]

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9. D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O’Neill, and J. T. Sheridan, “Paraxial speckle-based metrology systems with an aperture,” J. Opt. Soc. Am. A 23(11), 2861–2870 (2006). [CrossRef]

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Motion detection using extended fractional Fourier transform and digital speckle photography

Abstract

1. Introduction

2. Theoretical analysis

2.1 Extended fractional Fourier transform

2.2 Simultaneous motion detection

3. Experimental work

3.1. Experimental setup

3.2. Controlling the measurement sensitivity and range

3.3. Experimental results

3.3.1 Pure translation measurement

3.3.2 Pure tilt measurement

3.3.3 Simultaneous translation and tilt measurement

4. General discussions

5. Conclusions

Acknowledgement

References and links

Cited By

Figures (6)

Equations (17)

Optics Express