Sub-windowed laser speckle image velocimetry by fast fourier transform technique Abstract In this work, laser speckle velocimetry, a unique optical method for velocity measurement of fluid flow has been described. A laser sheet is developed and is illuminated on microscopic seeded particles to produce the speckle pattern at the recording plane. Double frame- single-exposure speckle images are captured in such a way that the second speckle image is shifted exactly in a known direction. The auto-correlation method has the ambiguity of direction of flow. To rectify this, spatial shift of the second image has been premeditated. Cross-correlation of sub interrogation areas is obtained by Fast Fourier Transform technique. Four sub-windows processed to obtain the velocity information with vector map analysis precisely.

1. 1
Sub-windowed laser speckle image velocimetry by fast
fourier transform technique
R. Balamurugan1*, G. Rajarajan2* and S. Jayanthi3
1 Department of Physics, Kumaraguru College of Technology, Coimbatore 641 049
Tamilnadu, India
2 Department of Physics, Hindustan Institute of Technology & Science,
Chennai 603 103 Tamilnadu, India
3 Department of Electronics and Communication Sciences, Hindustan College of Arts and Science,
Chennai 603 103 Tamilnadu, India
*Email: balamurugan.r.sci@kct.ac.in & grajarajan@yahoo.com
Abstract
In this work, laser speckle velocimetry, a unique optical method for velocity measurement
of fluid flow has been described. A laser sheet is developed and is illuminated on microscopic
seeded particles to produce the speckle pattern at the recording plane. Double frame-
single-exposure speckle images are captured in such a way that the second speckle image is shifted
exactly in a known direction. The auto-correlation method has the ambiguity of direction of flow. To
rectify this, spatial shift of the second image has been premeditated. Cross-correlation of sub
interrogation areas is obtained by Fast Fourier Transform technique. Four sub-windows processed
to obtain the velocity information with vector map analysis precisely.
Keywords: Laser speckle; seeded particles; sub-windows; fast fourier transform; velocity measurement; vector
maps

2. 1. Introduction
It is indispensable in fluid mechanics that the measurements of flow properties like
velocity, vortices, and pressure field in many applications. Seeded particle is also known as the
tracer particle photography method has become a suitable method for the determination of the
velocity of fluid flow. The light scattering tracer particles put off in the flow to provide the velocity
information to track the instantaneous fluid flow is the principle of Particle Image Velocimetry
method. Particle Tracking Velocimetry perfect for low image density [1], the intermediate-range
image density is called Particle Image Velocimetry and Laser Speckle Velocimetry suitable for high
image density of seeded particles. Melling [2] reported that variety of seeded particles is used in
liquid and gas image velocimetry experiments and for spraying in the fluid flow. Particle Image
Velocimetry method is preferred for both liquid and airflow referred as correlation image
velocimetry [3]. It consists of test area encompassing seeded particles, laser sheet to illuminate the
area of interest, charge-coupled device to capture images with software for velocity information by
the position of seeded particles [4-5]. Laser Speckle Velocimetry (LSV), first demonstrated by Barker
and Fourney [Barker D B and Fourney M E. ‘Measuring flow velocities with speckle patterns’. Opt
Lett, 1:135–137 (1977)], with novelty of measuring the flow velocity in a less intrusive way than that
offered by Laser Doppler Velocimetry (LDV) which is based on the measurement of the velocity of
the visualised fluid-markers, with the additional advantage of being whole-field.
To extract the displacement of a group of particles over a short time period, two successive
images are compared. Two- in- one approach of fluid temperature and flow measurements by
thermo-sensitive polymer particles is reported [6]. A high resolution spatial and temporal turbulent
flow measurement by instantaneous time snapshots is reported in [7]. A cost-effective,
single-camera 3D velocimetry method called as endoscopic tomographic velocimetry has been
developed [8]. The central-wall-end design on the fluid characteristics is experimentally verified in
a laboratory-scale pond [9].
In this work, Fast Fourier Transform technique [Cooley J W and Tukey O W. ‘An algorithm for
the machine calculation of complex fourier series’. Math Comput, 297-301:19 (1965)] for correlation
has been used to evaluate the double frame-single exposure recordings of Laser Speckle
Velocimetry. A small domain of first speckle image, named as sub-window is compared with a
sub-window of the same coordinates in the second image by cross-correlation. Interrogation areas

3. are the same size and hence the velocity map acquired from PIV presents vectors arranged on an
identical grid. Depends on the window size, the strength of the correlation peak increases and
constructing a valid measurement.
2. Materials and Methods
The experimental setup is shown in figure.1.
Fig. 1 Schematic experimental setup of Laser Speckle Velocimetry
The Nd-YAG laser of energy 100 mJ, the wavelength of 532nm and pulse duration of 10ns is
illuminated in the plane of investigation. It develops a bright sheet with a constant thickness
without aberration. The pulsed laser is transformed as a sheet with the help of a cylindrical lens. The
fluid flow direction and the light sheet has been aligned satisfactorily. The speckle patterns are
recorded by a 10 Megapixel resolution and 30 fps CCD camera which is positioned perpendicular to
the laser plane sheet.High concentration seeded particles are used to scatter the laser light and is
captured by two separate frames with a short inter-frame period of 0.01s. The two major images of
speckle patterns LSI-A and LSI-B are split into small interrogation areas a1, a2, a3, a4 and b1, b2, b3,
b4 respectively as shown in figures 2&3.

4. Fig. 2 LSI-A and sub-windows a1, a2, a3 and a4
Figure.3. LSI-B and its sub-windows b1, b2, b3 and b4.
2.1. Image evaluation by double frame single exposure
There are many ways to extract the mean displacement of the particles in sub-interrogation
spots from the speckle images. The autocorrelation technique is used for a multi-exposed single
image. The cross-correlation method is appropriate for single exposed but multiple images. Laser
speckle cross-correlation of two sequential images using FFT technique because it is more clear-cut
as well as accurate method [10]. Seeded polyurethane granule images are compared successively at
short interval time to find the displacements. A sub-interrogation area of the first image is
compared with the second image at the same location of sub-interrogation area through double
frame by single exposure method. The principle of double frame single exposure is illustrated in
Figure.4.

5. Fig.4 Double frame/single exposure cross-correlation
This interrogation was performed manually by divided into many parts of images. This
furnishes the possible displacement vector map for a particular -sub-window pattern. The little
image density is preferred for 3D great speed flows, but it is not suitable generally for the
high-density tracer particles or in two-phase flows is investigated.
2.2. Fast Fourier Transform
Cross-correlation is not computed for the two main images, but for each sub interrogation
windows. The value of the displacement is embodied by the peak value of the output image. Thus,
we can calculate the velocity by applying the above equation (1). The position vector (Xi) and the
image position vector (xi) of a particle are related to the first exposure is given by:
Xi = xi * M. … (1)
Here, M is magnification factor. First exposure image intensity field expressed by:
I(x) = ∑ Vo(Xi)τ(x − xi)
𝑁
𝑖=1
… (2)
Where, Vo(Xi) is the transfer function of a particle i in the interrogation windows and τ(x) is the
point spread function. ΔX is the displacement vector between two exposures. Second exposure
image intensity field is:
I′(x) = ∑ V′o(Xj + ΔX )τ(x − xj − δx)
𝑁
𝑗=1
… (3)
Where, δx is the displacement of the particle is obtained by:
X = δx/M. …. (4)
R = IFT • I’FT* … (5)

6. Where IFT and I’FT are the Fourier transforms of the functions I and I’ respectively. The
de-convolution of the image pair helps to find the local displacement function. In the speckle
pattern image processing, each sub-image is transformed from the real to the complex domain by
Fast Fourier Transforms (FFT). Then, complex conjugate multiplication between the transformed
results of both speckle images is computed. Finally, the product is reversed to the real domain by
applying inverse FFT as shown in Figure 6.
Fig. 6 Principle of Fast Fourier Transform
3. Result and discussions
Two images are recorded with a time interval, in which the second image was captured a
known time Δt, after the first image. In Fourier Transform analysis of the image pairs, each
sub-image is transformed from the real to the complex domain using fast Fourier transforms. In the
complex domain a conjugate multiplication between the transform results from both images takes
place, and the product is transformed back to the real domain using an inverse FFT. This yields after
normalization about the same image representation of the 2D probability–density function of the
level of matching between the two sub-images. Increasing the computational speed by optimizing
FFT practices with the help of lookup tables, data re-ordering, weighting coefficients and
fine-tuning the machine level code. LSI-A and LSI-B images are shown in Figure 7&8. The FFT
correlation of these speckle image is shown in Figure.9. with peak value.

7. Fig. 7 First speckle image (LSI-A)
Fig. 8 Second speckle image (LSI-B)
Fig. 9 Correlation of speckle images by FFT (LSI-A&LSI-B)

8. The suitable method for cross-correlation analysis is the division of square
sub-interrogation window with side length N = 2n here, n is an integer. N is occupied as a power of
2 to take the improvement in cross-correlation analysis of the frequency domain by Fast Fourier
transform and is explained [11]. N = 32 and an image size of mR x nC pixels, where mR and nC are
the number of rows and columns respectively. Each image has divided into four images without
non-overlapping m x n sub-windows. After that, we perform the 2D cross-correlation of each
sub-window pair in the image pair. The cross-correlation function is defined as:
R(s, t) =
1
N2
∑ ∑ FI,J
′
(i, j)FI,J
′′
(i + s, j + t)
N−1
j=0
N−1
i=0 … (6)
Where R is the recurrent cross-correlation among sub-windows I, J in the first image of the image
pair (F') and the next image of the image pair (F"), i,j is the pixel location within sub-window I, J, and
s,t are the 2-D cyclic lag for that cross-correlation computing is mentioned in the equation(1). R is
habitually calculated in the spectral domain can be written as:
R(s, t) = ℱ−1
[ℱ∗
{FI,J
′
)} ℱ{FI,J
′′
(i + s, j + t)} … (7)
Where ℱ and ℱ−1
are the Fourier and Inverse Fourier Transform operators. The star denotes
complex conjugate. The cross-correlation of two tasks is the same to a complex conjugate
multiplication of their Fourier transforms. Fourier transform is efficiently implemented for discrete
data by using the Fast Fourier transform, which reduces the computation from O[N2] operations to
O[N log2 N] operations. Match up to O [N4] for the direct computation of the 2D correlation the
process is abridged to O [N2 log2 N] operations. The computational efficiency of this
implementation can be increased by observing the symmetry properties between real value
function and their Fourier transform, named as real part of the transform (which is symmetric) and
imaginary part (anti-symmetric). The vector maps for different windows are shown in Figure.10.
and the colour scale graph is shown in Figure.11.

9. Fig. 10 Vector map-Interrogation (64 pixels) of the LSI-A & LSI-B (including all 4 windows)

10. Fig. 11 Colour scale graph
Table 1 Correlation of speckle images and their corresponding parameters
Measurements
In speckle
images
Correlated Speckle Images as sub interrogation windows
LSI-A&LSI-B a1&b1 a2&b2 a3&b3 a4&b4
R_mean 0.96007 0.96025 0.95303 0.96013 0.92923
R_max 0.99109 0.99048 0.98885 0.99053 0.99083
R_min 0.55576 0.81751 0.39073 0.73668 0.14011
|v|_mean 6.8838 10.744 4.7175 7.2709 5.7359
|v|_max 26.8701 16.8878 9.132 11.4195 22.4722
|v|_min 0.51423 5.5167 0 3.7004 0
vx_mean 1.7332 7.9208 3.7516 0.59335 -4.064
vx_max 13.935 12.8532 8.7995 4.6786 19
vx_min -19 3.3061 0.97388 -5.1313 -9.0944
vy_mean 2.5333 7.0852 -1.4053 6.7982 -1.4603
vy_max 19 10.9541 2.9285 10.9144 19
vy_min -6.7153 3.197 5.8845 2.582 -17.6651

11. Table 2 Comparison of parameters of speckle images
Correlation of
Speckle images
Peak differences
The difference of correlation of
speckle images with LSI-A&LSI-B
LSI-A&LSI-B 0.96-0.60 0.36
a1&b1 0.98-0.82 0.16
a2&b2 0.90-0.40 0.50
a3&b3 0.94-0.74 0.20
a4&b4 0.90-0.20 0.70
Various values of speckle images are obtained as a database from the major images LSI-A,
LSI-B and sub interrogation images. X,Y axes values, minimum, maximum and mean values are
given in the table-1. It is explicit that a3&b3 is the best matching because short gap between the
LSI-A&LSI-B value of 0.36 with 0.20 difference as shown in table-2. It requires only 0.16 value to
matching the major image. The pair a1&b1 is the better matching because medium gap between the
LSI-A&LSI-B value of 0.36 with 0.16 difference. It requires 0.20 to match the major image. The pair
a2&b2 is good matching because the gap between LSI-A&LSI-B value of 0.36 with 0.50 difference. It
means -0.14 is needed to coincide with the major image. The negative sign indicates the direction of
vector map deviated with error. The pair a4&b4 is not appropriately matching because large gap
between the LSI-A&LSI-B value of 0.36 with 0.70 difference. It means -0.34 is needed to coincide
with the major image. The negative sign indicates that the large mismatching of the direction of
vector map with more error as shown in arrow marks in Figure.10.Displacement range limitation,
periodicity of data and bias error, all these points are handled properly, FFT algorithm provides the
necessary correlation data from which the displacement data can be retrieved. Inside the
cross-correlation domain, the peak’s location resultant to the normal shift of particles inside the
sub-window area is identified. The pixel shift is converted into a velocity through calibration
parameters.
A high density seeded particles are required for the PIV vector maps, predominantly for
experimental data analysis and in numerical calculations. In airflow method, it is uneasy to achieve
a high image density because beyond some level, the seeded particle density cannot be increased
further in the flow. This yields the cross-correlation output of the 2D probability–density function of

12. the level of matching between the 2 sub-images. This method is suitable when the noise in the
signals is negligible or insignificant.
But, the noise due to different capturing conditions makes the degradation of the output
yield data. Sharp signal peak gives the displacement in sub-pixels level. The displacement function
d is measured by using the best match between the images in a statistical way of speckle pattern
[12]. All seeded particles confined by the interrogation small areas having no uniformities in
velocity due to various factors like turbulence, velocity gradients, particle size etc., The common
FFT implementation needs the input data to have a base-2 Dimensions (i.e. 32 × 32 pixel or 64 × 64
pixel or 128x128 pixels samples). This process carried out for the pairs of a1 & b1, a2 & b2, a3 & b3
and a4 & b4. Advantage of this method is to identify the offset error in both samples of images
according to the mean displacement of the seeded particles between the two illuminations. This
reduces loss of correlation in-plane and therefore increases the correlation peak strength. The light
scattering effect of seeded tracer particles and their photographic images are studied [13].
The individual particles in the 1-10 micron range would produce more detectable images at
lower mass loadings than the large numbers. But the fine particles would be needed to produce
speckle. Multiple scattering causes for broadened light-sheet makes the poor resolution; this effect is
negligible because majority of the Mie scattering is in forward direction together with the direction
of transmission of the laser beam reported by Christopher Abram and et al [14]. The irregular
pattern of this portion may due to the variation of seeding particle concentration. If the density of
the seeded particles is low, then the pattern is resolved, but here the concentration is increased, so
the images overlap and interfere to develop a random cigar like bright and dark image pattern.
Certain errors cause for the poor result in some part of the analysis are mentioned below and their
impact involved in this experimental study:
1. Random error due to noise in the recorded images.
2. Bias error arising from the process of computing the signal peak location to sub-pixel
accuracy [15].
3. Gradient error resulting from rotation and deformation of the flow within an
interrogation spot leading to loss of correlation. These are minimized by careful
selection of experimental conditions.
Laser Speckle Velocimetry gives a suitable tool for studying the fluid flow in two dimensional,
but it is still limited as they are confined to measurements within a plane. These errors can be

13. rectified by a cautious selection of experimental components and make the suitable ambient
conditions. Though the experimental conditions are perfect, Laser speckle velocimetry producing
vector map consists of ‘bad’ vectors called as false or spurious vectors. These are readily identifiable
when the vector field is re-plotted after subtracting the mean. Magnitudes and directions of bad
vectors are significantly different from their neighbours. Bad vectors are due to the -sub-window
areas, in which Signal to Noise Ratio (SNR) is less than unity because noise peak is higher than the
signal peak. Usually, less than 2% of vectors considered as bad. This is due to lack of particle pairs in
the interrogation area by inadequate seeding density or excess of tracer particles are out-of-plane
motion such that the particles exoduses the light sheet between laser pulses. At irregular intervals, a
piece of fragments or flare from physical boundaries or objects, drawn-out into the measured area.
Hence it causes for overwhelming response of the interrogation area leading to a bad vector.
In this case, a blurry imaged particle just below the focused plane makes the adverse effect of
result. The sequence of the colour red and green wavelength enhancing the particle streak
velocimetry (CSPSV) is reported [16]. The speed of the process may be increased by downsampling
the images during the interrogation passes. The use of smaller interrogation samples is evaluated
much faster. A constant interrogation window size can be used regardless of the image resolution. It
means a 4X down-sampled image interrogated by a 32x32 pixel sampling window corresponds to a
128 x128 pixel sample at the initial image resolution.
4. Conclusions
Double frame by single exposure laser speckle velocimetry method for the measurement of
fluid flow has been described with the help of sub-divided interrogation areas. Determination of the
displacement of seeded particle within the interrogation areas gives the velocity of fluid flow and
various pixel size vector maps have been reported. This non-intrusive method uses four equal
sub-windows from the consecutive speckle images and computed by FFT algorithm.
Acknowledgements
The author Dr. RB would like to thank the management of Kumaraguru College of Technology,
who continuously encouraging for this area of research work. The authors are highly grateful to the
support of Indian Institute of Technology, Madras for the development of LSV technology.
Conflicts of Interest: The author declares no conflict of interest.

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