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Master Thesis
on
High Accuracy Sub-Pixel Based Correspondence Point Matching
Between Two Images
Submitted
For the the award of degree
of
Master of Technology
In
Electrical Engineering
(With the specialization Instrumentations and Signal Processing)
Submitted By
Sunil Kumar Yadav
M-Tech(2nd Year)
p11528020q
Under the Guidance of
Dr.-Ing. Olaf Hellwich
Professor
Computer Vision and Remote Sensing
Technical University Berlin
And
Dr. R.S. Anand
Professor
Department of Electrical Engineering
IIT Roorkee
Statement of Originality
I have written this master thesis independently and none other than the spec-
ified sources and aids were used and that any citations have been marked.
Sunil Kumar Yadav
IIT Roorkee
i
Acknowledgements
This work is thankful to two institutes, Technical University Berlin, Computer
Vision and Remote Sensing and Indian Institute of Technology Roorkee, De-
partment of Electrical Engineering for providing all the academic background,
Lab and experimental set up. I would like to thank DAAD (German Academic
Exchange Service) for funding this work.
I take this opportunity to express my profound gratitude and deep regards
my to supervisor Professor Dr.-Ing Olaf Hellwich from Technical University
Berlin and Professor Dr. R.S. Anand from Indian Institute of Technology
Roorkee, for approving this thesis topic and for their exemplary guidance,
monitoring and constant encouragement throughout the course of thesis.
I would also like to thank my friends for their immense support by keeping
a close observation and review of thesis.
I dedicate this thesis to my family who unremittingly supported me during
my years of study. They made this work possible.
ii
Abstract
Correspondence Matching is an important and necessary step in the Image
Registration. As we know image registration has wide applications in the
field of medical imaging, stereo vision and 3D measurement, remote sensing,
geographic information system, weather forecasting, environment monitoring
etc. The most important factor regarding image registration is, accuracy of
corresponding point measurement. This work is intended to estimate the cor-
responding point with high accuracy at sub pixel accuracy level. In this work,
we are investigating the minimum shifting between two image in terms of
sub pixels, whereas images are shifted in micrometer range. We are using
Area Based Matching, so computational complexity is more. Another focus of
this work is to reduce the computational complexity for calculating the cor-
responding point between two images. To reduce computational complexity
we are using 1D BLPOC (Band Limited Phase Only Correlation), in which
we search for corresponding point along the epipolar line only. So this al-
gorithm has lower computational complexity in comparison to other existing
algorithm. Additionally, we are using the different kind of spectral weighting
functions to reduce the effect of noise and enhance the estimation accuracy.
Finally we compared this algorithm with two existing algorithm, Normalized
Cross Correlation and 2D POC considering two factors, computational time
and estimation accuracy. And then we have discussed about some application
of this algorithm in different area where high accuracy estimation is necessary.
Contents
1 Introduction 1
1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Literature Survey 7
2.1 Feature Based Correspondence Matching . . . . . . . . . . . . 8
2.1.1 Feature Detection . . . . . . . . . . . . . . . . . . . . . 8
2.1.2 Feature matching . . . . . . . . . . . . . . . . . . . . . 12
2.1.3 Feature Mapping . . . . . . . . . . . . . . . . . . . . . 12
2.1.4 Re-sampling and Transformation . . . . . . . . . . . . 13
2.2 Area Based Correspondence Matching . . . . . . . . . . . . . 14
2.2.1 Normalized Cross Correlation . . . . . . . . . . . . . . 14
2.2.2 Sequential Similarity Detection Algorithm . . . . . . . 15
2.2.3 Fourier Method . . . . . . . . . . . . . . . . . . . . . . 16
3 1-D Band Limited Phase Only Correlation 17
3.1 Image Rectification . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Frequency Domain Transformation . . . . . . . . . . . . . . . 21
3.3 The Windowing Technique . . . . . . . . . . . . . . . . . . . . 22
3.3.1 Bartlett Triangular Window . . . . . . . . . . . . . . . 23
3.3.2 Generalized Raised Cosine Windows . . . . . . . . . . 24
3.3.3 Kaiser Window . . . . . . . . . . . . . . . . . . . . . . 26
3.4 Phase Only Correlation . . . . . . . . . . . . . . . . . . . . . . 26
3.5 Spectral Weighting . . . . . . . . . . . . . . . . . . . . . . . . 27
3.5.1 Rectangular Low Pass Weighting Function . . . . . . . 28
iv
3.5.2 Triangular Weighting Function . . . . . . . . . . . . . 29
3.5.3 Gaussian Weighting Function . . . . . . . . . . . . . . 29
3.6 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.6.1 Digital Interpolation . . . . . . . . . . . . . . . . . . . 31
3.6.2 Linear Interpolation . . . . . . . . . . . . . . . . . . . 32
3.6.3 Bilinear Interpolation . . . . . . . . . . . . . . . . . . . 32
3.6.4 Cubic Interpolation . . . . . . . . . . . . . . . . . . . 33
3.7 Time Domain POC . . . . . . . . . . . . . . . . . . . . . . . . 34
3.8 Flow Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Experimental Set-Up 40
4.1 Translational Stage . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3 Wooden Cube . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4 Complete Set Up . . . . . . . . . . . . . . . . . . . . . . . . . 43
5 Results 44
5.1 Comparison with Normalized Cross Correlation . . . . . . . . 48
5.2 Comparison with 2D POC . . . . . . . . . . . . . . . . . . . . 50
6 Summary 55
6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
v
List of Figures
1.1 Application of Computer Vision in Different area . . . . . . . 3
2.1 Different Image Segments . . . . . . . . . . . . . . . . . . . . 9
3.1 Epipolar Geometry and Image Rectification Image Source: Wikipedia 20
3.2 Image Rectification. Image Source:IEEE Xplore . . . . . . . . 21
3.3 Bartlett Window . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Different Window Functions . . . . . . . . . . . . . . . . . . . 25
3.5 Rectangular Weighting Filter . . . . . . . . . . . . . . . . . . 28
3.6 Shape of Gaussian Weighting Filter . . . . . . . . . . . . . . 29
3.7 Interpolation technique implementation . . . . . . . . . . . . 31
3.8 Flow Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.1 Micrometer Translational Stage TCS42 . . . . . . . . . . . . 41
4.2 Micrometer Translational Stage TCS42 with Wooden Cube . 41
4.3 Canon EOS 600D Camera with Remote . . . . . . . . . . . . 42
4.4 Complete Experimental Set Up . . . . . . . . . . . . . . . . . 43
5.1 Input Image Pair . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Time Domain POC . . . . . . . . . . . . . . . . . . . . . . . 47
5.3 Deviations between Disparity and Regression Line . . . . . . 48
5.4 Disparities as gray level image . . . . . . . . . . . . . . . . . 49
5.5 Disparity image with Rectangular Weighting Factor and win-
dows size are 101ˆ 11 and 101ˆ 21 . . . . . . . . . . . . . . . 50
vi
5.6 Disparity image with Gaussian Weighting Factor and windows
size are 101ˆ 11 and 101ˆ 21 . . . . . . . . . . . . . . . . . . 51
5.7 Input Image Pair . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.8 Input Image Pair (Cropped) . . . . . . . . . . . . . . . . . . . 51
5.9 Shift in pixels in the image Vs shift in micrometer stage with
step size 20 micrometer . . . . . . . . . . . . . . . . . . . . . . 52
5.10 Shift in pixels in the image Vs shift in micrometer stage with
step size 20 micrometer . . . . . . . . . . . . . . . . . . . . . 52
5.11 Input Image Pair (Cropped) . . . . . . . . . . . . . . . . . . . 53
5.12 Shift in pixels in the image Vs shift in micrometer stage with
step size 20 micrometer . . . . . . . . . . . . . . . . . . . . . 53
5.13 Normalized Cross Correlation Coefficient Vs Shift in the image 54
5.14 2D POC Vs shift in image . . . . . . . . . . . . . . . . . . . . 54
vii
Chapter 1
Introduction
“There are two mistakes one
can make along the road to
truth, not going all the way,
and not starting”
-Buddha
Researchers in different fields use growing computer techniques to model
different events and visualize phenomena that cannot be observed directly.
Weather forecasting, medical imaging and diagnostic,mechanical assembly de-
sign, Computer graphics and mathematical relationships are only some of the
uses to which virtual reality can be put to[1].As depicted in figure 1.
Computer vision, one of the growing discipline in engineering is concerned
with the theory behind artificial systems that extract information from images.
The image data can take many forms, such as video sequences, views from mul-
tiple cameras, or multi-dimensional data from a medical scanner.The growth
in this field has been both in breadth and depth of concepts and techniques[2].
Computer Vision techniques are being applied in areas ranging from medical
imaging to remote sensing, industrial inspection to document processing, and
nanotechnology to multimedia databases.
1
Computer vision generally includes methods for acquiring the input data,
processing data, analysing data, and understanding images and, in general,
high-dimensional data from the real world in order to produce numerical or
symbolic information, e.g., in the forms of decisions.A theme in the devel-
opment of this field has been to duplicate the abilities of human vision by
electronically perceiving and understanding an image.This image understand-
ing can be seen as the disentangling of symbolic information from image data
using models constructed with the aid of geometry, physics, statistics, and
learning theory.Computer vision has also been described as the enterprise of
automating and integrating a wide range of processes and representations for
vision perception[3].
Applications range from tasks such as industrial machine vision systems
which, say, inspect bottles speeding by on a production line, to research into
artificial intelligence and computers or robots that can comprehend the world
around them. The computer vision and machine vision fields have significant
overlap. Computer vision covers the core technology of automated image anal-
ysis which is used in many fields. Machine vision usually refers to a process
of combining automated image analysis with other methods and technologies
to provide automated inspection and robot guidance in industrial applications.
In a Computer vision system, however, a computer receives a grid of num-
bers from the camera or from disk, and that’s it. For the most of part, there
is no built-in pattern recognition, no automatic control of focus and aperture,
no cross- associations with years of experience. In any picture what computer
sees is just a grid of numbers. Any given number within that grid has a rather
large noise component and so by itself gives us little information, but this grid
of numbers is all computer sees[4].
2
Figure 1.1: Application of Computer Vision in Different area
1.1 Problem Statement
Image matching with high accuracy is an important fundamental task in
many fields, such as computer vision,remote sensing, medical imaging,Geo-
informatics etc. Especially for such applications as stereo-vision 3D mea-
surement and super resolution imaging (that reconstructs a high resolution
image from multiple low-resolution images). Over the years, various tech-
niques for image registration have been developed.Now a days the demand
of 3D measurement with proper accuracy is rapidly growing in a variety of
computer vision applications, for instance, robot vision, human-computer in-
terface, biometric authentication,Surveillance and security etc. Existing 3D
measurement techniques are classified into two major types.
1. Active
3
2. Passive
In general, active measurement employs structure illumination (structure pro-
jection, phase shift, moire topography, etc.) or laser scanning,and require
complex hardware and instrumentation which is not necessarily desirable in
many applications.
On the other hand, passive 3D measurement techniques based on stereo vi-
sion have the advantages of simplicity and applicability, since such techniques
require simple instrumentation.
However, poor reconstruction quality still remains as a major issue for pas-
sive 3D measurement, due to the difficulty in finding accurate correspondence
between stereo images; this problem is generally known as “correspondence
problem ”. As a result, application of passive stereo vision to high-accuracy
3D measurement system for capturing 3D surfaces of free form objects is still
weakly reported in the published literature.
The overall accuracy of passive 3D measurement is mainly determined by:
1. The baseline length between two cameras and
2. The accuracy of estimated disparity between corresponding points
Conventional approaches to passive 3D measurement employ wide-baseline
camera pairs combined with feature-based correspondence matching. How-
ever, in such approaches only a limited number of corresponding points can
be used for 3D reconstruction.
On the other hand, area-based correspondence matching (which must be
combined with narrow-baseline stereo cameras to avoid projective distortion
between stereo images) makes possible to increase the number of correspond-
ing points. However, the accuracy of 3D measurement becomes severely re-
stricted when the baseline is narrow.
4
Area based image matching is an important fundamental task in a variety
of image processing applications, such as stereo vision, motion analysis, im-
age sequence analysis, etc. Although in some applications pixel level image
matching may be adequate,image matching with sub-pixel accuracy is becom-
ing essential in recent applications.
In response to this need, many methods have been developed to estimate
the translational displacement between two images with high accuracy.
The most common stereo correspondence techniques employ Sum of Ab-
solute Differences (SAD) or Sum of Squared Differences (SSD), where corre-
sponding points between stereo images can be obtained by minimizing SAD
or SSD in area-based block matching. Although SAD and SSD exhibit low
computational cost, a major drawback is their low accuracy.
Therefore, we focus on the techniques for high-accuracy stereo correspon-
dence in order to overcome the limitation of measurement accuracy in narrow-
baseline stereo vision.
On the other hand, image matching methods using 2D Phase Only Cor-
relation (POC) exhibit much better matching performance than the methods
using SAD and SSD in general.A drawback of POC-based approach is its high
computational cost in computing 2D POC function for correspondence search,
which limits the potential area of applications.
Addressing this problem, we are using a technique for high accuracy cor-
respondence search between stereo images using 1D version of Band-Limited
POC (BLPOC). The correspondence search between stereo images can be
reduced to 1D search through image rectification. However, conventional ap-
proach is to employ block matching with 2D rectangular image blocks for
5
finding the best matching point.
By using this method we are finding how much accurately we can find out
minimum shift between two images.and also we are consider different window
size of image and we are using different type of image having different texture
so that we can use these parameter to define up to what extent our algorithm
is able to find out the shifting between image and what factors are affecting
the accuracy.
Our work is mainly focussed on:
1. Minimum shift between two images at the sub pixel level.
2. Factors affecting the accuracy
3. Low Computational cost
4. Use this algorithm for different application
6
Chapter 2
Literature Survey
“The task is not so much to
see what no one has yet seen,
but to think what nobody has
yet thought, about that which
every- body sees”
-Erwin Schrodinger
Before moving ahead, it is very much useful to investigate the different
correspondence matching scheme which had proposed until now and to learn
about the current state-of-the art techniques in this area of research. This
chapter starts with providing extensive information about the earlier work
addressing correspondence matching techniques. Then we give a brief expla-
nation on why we have chosen this approach.
Correspondence Point is basically disparity between two images. It is very
important factor in Image registration and in 3D reconstruction. Correspon-
dence Problem is arises when we take the images from different scene, different
view,at different time, and by using different sensor, it may create some dis-
parity between two images. In correspondence problem we just try to find
out set of points in one image which would be identical to the some points in
7
other image.
As we have different kind of images that we have to registered and due to
different types of degradations (geometric degradation, noise corruption and
radiometric deformation) in the images, it is impossible to design a universal
method applicable to all registration tasks. Generally there are two basic ways
to find out the correspondence between two images
1. Feature Based Correspondence Matching
2. Area Based Correspondence Matching
2.1 Feature Based Correspondence Matching
This approach is based on the extraction of distinct features in the images.
These features include significant regions (fields, lands ,sea etc), lines or points
( corners, line intersections, centre of any object). These features should be
distinct, have better density in the image and easy to find in both images.
For better feature matching, number of features should be large and image
geometry, noise and different degradation parameter should not affect them.
Feature based correspondence matching consist of following stages:
1. Feature Detection
2. Feature Matching
3. Feature Mapping
4. Image Re-sampling and Transformation
2.1.1 Feature Detection
Selection of proper feature from the images plays very important role in corre-
spondence matching. The most important property of feature is that it should
8
be unique. There are different kind of features which can be calculated.
Edges
Object boundaries or edges generally generate significant changes in image
intensities. Edge detection is used to find these changes. generally edge
detection consist of three stages:Filtering, Differentiation and Detection[4].
There are different kind of edge detectors used in recent days, main difference
between edge detectors are, they have different filtering scheme and different
threshold for detection[1]. Basic edge detectors are Robert‘s Operator, Pre-
wit‘s Operator, Sobel‘s Operator and Laplacian Operator [5][6] . Most impor-
tant property of edges feature is, they are robust against illumination changes
compared to color features. Because of its simplicity and accuracy, the most
popular edge detection approach is the Canny Edge detector[6].
Region
This is a complementary approach to the edge detection. Every object in the
image occupied some regions.In the region based segmentation method we are
just finding that occupies region.
Let‘s consider an image Ipx, yq consisting of sub images R0, R1, R2, ... (as
shown in figure) satisfying the following constraints.
Figure 2.1: Different Image Segments
9
ď
Ri “ Ipx, yq (2.1)
And
Ri XRj » φ, i ‰ j (2.2)
In any sub image, change in pixel intensity should not exceed from the thresh-
old value and standard deviation should be less[4].There are different kind of
segmentation techniques we use to find the region.
Most basic and simple segmentation technique is Threshold Based Segmen-
tation. In this technique, we define some threshold value and compare it with
pixel intensity. If pixel intensity is greater than threshold then replace that
value with one otherwise keep it zero.Sometimes we select multiple threshold
for proper segmentation[7].
Another technique is K-Means Algorithm Segmentation. It is an unsuper-
vised clustering algorithm that classifies the input data points into different
classes based on means and distance from each other. In K-Means, we basi-
cally divide object in to different K clusters and then compute the means of
each cluster[8]. And after that find out the distance of each point from each
cluster by computing its distance from the corresponding cluster mean[9].
Some other techniques are also used for segmentation purpose like Region
Growing, Region Split and Merging, Phagocyte Algorithm etc. Every tech-
nique use different ways to get those regions. In region spilt and merging
technique, first we divide whole image in to four parts then find out simi-
larity if similar then merge, otherwise, again spilt till final segmentation[10].
Phagocyte Algorithm is a boundary melting technique. It detects the weak
boundary between two adjacent region and melts that boundary. The strength
of boundary between two regions can be calculated by absolute difference of
gray levels[4].
10
Segments
Segments represent geometric boundary of regions and segments in the image.
Hough Transform is generally used to define the shape of any plane curves like
line, circle and parabolas etc. Hough Transform and Least Square fit, both are
inter related with each other, one belongs to over constraints system (Least
Square fit) i.e. number of unknowns are less than number of equations and
other one belongs to the under constraints system (Hough Transform) where
number of unknowns are more than the equations[4]. Generally there are two
kind of Hough transform used for shape identification[11]:
1. Standard Hough Transform
2. Probabilistic Hough Transform
Standard Hough Transform is robust against the noise for line detection com-
parison to the Probabilistic Hough Transform.The accuracy of the segmen-
tation technique can significantly affect the final registration. Accuracy of
Probabilistic Hough Transform is lower than the Standard Hough Transform
but it is effective.
Points
The point features comprises of centroid of any object, Corners, crossing point
of two lines, points having higher variance etc. there are different kinds of op-
erator used for these point of interest[1].
The most commonly used technique for corner detection is Harris Detector.
This technique uses concept of Hessian Matrix. And Hessian matrix generated
by using second derivative of image intensities. Hessian matrix around the
image Ipx, yq is defined as:
Hppq “
»
–
BI2
Bx2BI2
BxBy
BI2
BxByBI2
By2
fi
fl (2.3)
11
For Harris corner, we count auto correlation matrix of the second derivative of
the images over a window around every point. Then we find two eigenvalues
for each corner because at each corner there will be two edges, orthogonal
with each other[3] .
For detection of point feature, most basic and use technique is Moravec‘s
Interest Operator, which is most widely used in object detection and tracking
another different application. In this technique we select 4 ˆ 4 window and
find the variance in horizontal direction, vertical direction, diagonal and anti
diagonal direction. And then find the smallest of them and decide the interest
point[12].
2.1.2 Feature matching
This is the second stage of Feature based image matching where we will find
out the correspondence between the detected feature from sensed image and
reference image. There are different kind of feature descriptors and similar-
ities used for this purpose. The main problem is incorrect feature detection
that results image degradation. So, that we should select feature descriptor
and similarity measure considering the degradation fact. And these descrip-
tors and measures should be invariant to different imaging conditions. The
matching algorithm should be robust and efficient. We can match the sensed
and reference image by using the image intensity without detecting any other
feature. Some methods combines feature correspondence and estimate the
parameters of mapping functions simultaneously[1].
2.1.3 Feature Mapping
The main task is to choose the type of the mapping function and its parameter
estimation. Mapping function should correspond to the assumed geometric
12
deformation of the sensed image, to the method of image acquisition and to
the required accuracy of the registration.
According to the different image data and different features we can model
the mapping function in two ways:
1. Global Mapping model
2. Local Mapping Model
Global Mapping model uses all Correspondence points to find out the parame-
ters for mapping functions and those parameters will be valid for entire image.
Whereas ,in the local mapping functions, the parameters totally depend on
the location in the image.
2.1.4 Re-sampling and Transformation
Image Transformation can be done by using Mapping function. Intensities
of image at non integer value can be computed by proper interpolation tech-
nique. Generally interpolation technique is used to find out the missing point
between two consecutive pixels[13].
We can interpolate the image by performing convolution between image
and interpolation kernel. A suitable interpolation kernel is sinc function ker-
nel. This kernel has the infinite sequence up to last extent and this sequence
is periodic[14].
Feature based matching is preferred when less information lies in image in-
tensities in comparison to the local structure of image. feature based matching
technique is robust against the illumination change i.e. accuracy does not de-
pend on the illumination changes. It is faster than the correlation method[15].
The main drawback of feature based method is that feature should distinct
13
and suitable. Generally, in an image we have less distinct features so disparity
map is not so dense, it is sparse. In case of remote sensing and computer vision,
we have distinct and detectable features so feature based matching is appro-
priate for these fields. In case of medical imaging, images don‘t have enough
features so area based method method is suitable for medical imaging[1].
2.2 Area Based Correspondence Matching
Area based image matching is used when we don‘t have distinct and detectable
salient features. It is also known as template matching technique or correla-
tion method.In this method we select a window template, and slides that
window template for computing the similarity between two images[1]. The
main drawbacks of this method is that it is slower than the feature based
matching. Sometimes, window template has smooth surface or we can say
SNR is less so in that case we can get some false detection of correspondence
point. Another drawback of area based matching is that we cant choose any
other shape of window except rectangular window[16].
There are different area based correspondence matching techniques we use:
2.2.1 Normalized Cross Correlation
This is the most basic and classical approach of area based matching tech-
niques. In this technique, we select a small rectangular window template and
find out the correlation coefficient between template image and reference im-
age by using given formula[4]:
CC “
ř
pW ´ EpW qqpIpi, jq ´ EpIpi, jqqqb
ř
pW ´ EpW qq2b
ř
pIpi, jq ´ EpIpi, jqqq2(2.4)
where W is window and EpW q shows expectation value of window tem-
plate and Ipi, jq show reference image.
14
After calculating the correlation coefficient for whole image, we find out
the maximum of the correlation coefficient and that point shows the match-
ing point or correspondence point[17]. For sub pixel accuracy, we have to use
the concept of interpolation with normalized cross correlation. There are two
main drawbacks of this method, first the flatness of the maxima function and
other is the high computational cost.
2.2.2 Sequential Similarity Detection Algorithm
As we had discussed, Normalized Cross Correlation has higher computational
complexity. To overcome this problem, we use Sequential Similarity Detection
Algorithm (SSDA). It reduces the computational complexity but it has lower
estimation accuracy in comparison to the normalized cross correlation method.
Generally there are two kinds of algorithm used belonging to SSDA[1]:
1. Sum of Absolute Difference
2. Sum of Squared Difference
Sum of Absolute Difference (SAD) simply computes the distance and store the
SAD values in terms of image intensity after applying some threshold value, if
the SAD value is more than the threshold then take it in to the consideration
otherwise check for next value of SAD[18]. We can calculate the SAD values
from the given equation:
SAD “ÿ
|I1pi, jq ´ I2pi` d, jq| (2.5)
Where d is the disparity between two images. And for Sum of Squared Dif-
ference (SSD) we can use another equation which is given below:
SSD “ÿ
pI1pi, jq ´ I2pi` d, jqq2 (2.6)
15
In SSD we calculate for minimum value of SSD whereas in NCC (Normalized
Cross Correlation), we calculate for maximum value of correlation coefficient.
We can use SSDA Algorithms where accuracy does not matter too much but
computational time is important[1].
2.2.3 Fourier Method
Techniques, we had explained in area based matching, are sensitive to fre-
quency dependent and correlated noise. NCC have average estimation accu-
racy but computational complexity is more whereas SSDA algorithms have
less computational complexity but they have less accuracy.
Fourier Method uses the concept Fourier analysis and provides robustness
against the correlated and frequency dependent noise because the final ex-
pression has only phase term and that is not too much affected by the noise.
This is also known as the Phase only correlation as it has only phase term.
It provides better accuracy than Normalized cross correlation and Sequential
Similarity Detection Algorithm. And has less computational time in compar-
ison to the Normalized cross correlation but more in comparison to SSDA[19].
16
Chapter 3
1-D Band Limited Phase Only
Correlation
“No problem can be solved from
the same level of consciousness
that created it”
-Albert Einstein
This chapter will explain algorithm we implemented for calculate the dis-
parity between two images. This algorithm, we will see follows a 2D POC
approach and uses Fourier Method. This algorithm has several stages for
complete implementation so this chapter also explains some basic back-
ground related to each stages.
As we had discussed in the literature part, corresponding point matching
divided into two parts. In feature based matching, implementation of each
registration step has its own typical problems.Firstly, we have to decide the
proper features for the given task. The features should be able to distinguish
the objects, and those features are often spread over the images and are easily
detectable[1]. The detected feature sets in the reference and sensed images
must have enough common elements, even in situations when the images do
not cover exactly the same scene or when there is object occlusion or other
17
unexpected changes.The detected features in the reference and sensed images
can matched by image intensity values in their close neighbourhoods, the fea-
ture spatial distribution, or the feature symbolic description. The detection
methods should have good localization accuracy and should not sensitive to
the assumed image degradation. In an ideal case, the algorithm should able
to detect the same features in all projections of the scene regardless of the
particular image deformation[20].
So feature-based methods is generally used if the images contain enough
different and easily detectable objects. This is usually the case of applications
in remote sensing and computer vision. Usually, images contain a lot of de-
tails and on the basis of those details we can find out the different features[21].
The algorithm on which we worked belongs to second part of correspond-
ing point matching i.e. Area based matching. In area based method we had
discussed different technique which are generally used for image matching
purpose. Most classical approach is Normalized Cross Correlation which is
simpler and easy to implement but sensitive to illumination change and also
affected by noise interference.It also has high computational complexity[1].
Another approach is Sequential Similarity Detection Algorithm(SSDA) in
which we use sequential search approach and compute simpler distant mea-
sure. It consist of Sum of Absolute Difference(SAD)[18], Sum of Squared
difference(SSD) etc. It is computationally faster than Correlation method
but less accurate[4].
Fourier Methods or Phase only Correlation Method are preferred than cor-
relation method because it provides robustness against frequency dependent
noise, Illumination variations and also having less computational cost provides
better Accuracy than correlation like method. it is also known as 2-D Phase
only correlation (2DPOC)[22].
18
Even this method shows strong robustness against the correlated and fre-
quency dependent noise and non-uniform, time varying illumination distur-
bances.The computational time savings are more significant if the images,
which are to be matched, are large.But if image is not large then it takes
large computational time. So for reducing that computational time we are
using here 1D approach of Phase only Correlation(POC).Which is known as
1 Dimensional Band Limited Phase Only Correlation(1D BLPOC)[23].
This approach makes possible significant reduction in computational time
and has reconstruction accuracy similar to 2D POC.Resulting Reconstruc-
tion accuracy is much higher than Sequential Similarity Detection Algorithm
(SSDA)[24].
This algorithm consist of following stages:
1. Image Rectification
2. Frequency Domain Transformation
3. Windowing Technique
4. Cross Phase Power Spectrum
5. Spectral Weighting
6. Interpolation
7. Time Domain POC
3.1 Image Rectification
We are using here 1D BLPOC so first of all, convert the image pair in to
1D or we have to rectify the image pairs. Image rectification is process in
which we project image pairs in common image plane.And after image rectifi-
cation of image pairs their epi-polar line coincide and become parallel to the
x-axis of the image . Image Rectification is widely used in Computer vision
19
and Geographic Information System.In figure 2, consider an example of Com-
puter Stereo Vision , in which first image is without rectification and taken
by Stereo camera pair and 2nd image shows the image after rectification[25].
In most of the applications, finding corresponding points requires a search in
1.jpg
Figure 3.1: Epipolar Geometry and Image Rectification Image Source:
Wikipedia
two dimensions. And because of that computation time increases. It restricts
the search domain for each match to a line parallel to the X-axis.
Let’s consider two images I and J which we have to be rectified. A refer-
ence point p in the image I is given and we have to find out the corresponding
point q in the image J . Firstly, rectify the image I centered at point p along
the epipolar line and extarct the 1D image signal fpnq. Similarly, in the im-
age J extract 1D image signal centered at q1 a random and initial estimation
for corresponding point q. As shown in figure 3 both points q and q1 lie on
the same epipolar line. figure:3paqshows without rectification and 3pbq after
rectification[23].
Rectification required if correspondence between two images is in pixel level.
but if shift between two images is very tiny so disparity between images will
be in sub-pixel and for that both images have same epi-polar line. In our
we are shifting image in micrometer range so no need of image rectification
because both having same epi-polar line.
20
Figure 3.2: Image Rectification. Image Source:IEEE Xplore
3.2 Frequency Domain Transformation
Phase only correlation is also known as the Fourier method in which we ex-
ploit Fourier presentation of images in frequency domain.There are a lot of
Fourier methods used for different applications and different methods having
their different attribute[26]. For fixed length signal, most suitable Fourier
method is Discrete Fourier Transform. Here we have 1D image signal and for
that one dimensional Discrete Fourier Transform (DFT)is applicable[27].This
is a computational approach and implemented by Cooley and Tukey in 1965.
Let’s assume our 1D image signal having fixed length N . And discrete spatial
distribution index for both 1D imagesfpnq and gpnq are:
n P p´M,´pM ´ 1q....0....pM ´ 1q,Mq (3.1)
So from here we can say:
N “ 2M ` 1 (3.2)
We are considering here sign symmetric range p´M, ., .,Mq. This assumption
provides a better mathematical simplicity. Let’s consider F pkq and Gpkq are
Discrete Fourier Transform of 1D image signal fpnq and gpnq then:
F pkq “Mÿ
n“´M
fpnqWNkn (3.3)
And
Gpkq “Mÿ
n“´M
gpnqWNkn (3.4)
21
Where
WN “ e´j2πN (3.5)
Key point regarding significant reduction in computational time is to convert-
ing the 2D image into 1D image signal and then applying 1D Discrete Fourier
Transform.So to do this, first we create a fftw array and then copy all row
elements inside this array and then perform 1D DFT on that array.For finding
single correspondence point,if we are using 1DBLPOC then it reduces com-
putational cost near about to 1{8th for addition, for substraction near about
to 1{6th and for Division near about to 1{3th, in comparison to 2DPOC[5].
3.3 The Windowing Technique
As we are using DFT for frequency domain transformation because it’s suit-
able for fixed length signal. But the main drawback of DFT is Wrap Around [28].
This problem arises due periodicity nature of DFT in both time and frequency
domain. Because of this Wrap Around affect, an identical image appear on
each sides of the image. But here we are using 1D image signal so periodic
repetition occurs along the X-axis means, left and right edges are effectively
next to each other and can interfere . And this affect is mainly known as
Wrap Around Error [29]. The wrap around effect applies in both real space
and reciprocal space and is responsible for unpredictable distribution of spa-
tial frequencies in DFT. For that reason, we must perform truncation.The
window function reduces the ringing effects at the band edge and does result
in lower side lobes at the cost of an increase in the width of the transition
band of the filter. For truncation purpose we are using Windowing Technique
.
Let’s consider our desired frequency response specification Hdpwq regarding
unit sample response hdpnq:
hdpnq “1
2π
ż π
´π
Hdpwqejwn dx (3.6)
22
Where
Hdpwq “8ÿ
n“´8
hdpnqe´jwn (3.7)
Now hdpnq must truncate at some fixed point to yield a FIR filter. Let’s say
that fixed point is n “M ´ 1 i.e. p0, ..M ´ 1q so length of filter would be M .
So for truncation of hdpnq we can use the most basic Rectangular Window and
that is defined as:
wpnq “
$
&
%
1 0 ă w ď pM ´ 1q
0 otherwise(3.8)
Ripples and Smoothness in frequency domain both depend on the length of
window function. If length of window increases, main lobe width reduced
which reduces the width of the transition band, but this also introduces more
ripple in the frequency response and vice versa.Leakage is another problem
in windowing technique but we can reduce it by selecting proper window[30].
Leakage problem can be completely removed if the signal periodicity is equal
to window length. The different Windowing techniques we generally use:
1. Bartlett Triangular Window
2. Generalized Cosine Windows
3. Kaiser Windows
3.3.1 Bartlett Triangular Window
We can get triangular window function by convolving two rectangular win-
dow.The spectrum of a triangular window has a sinc2 characteristics, i.e. it
decays asymptotically at twice the rate of the spectrum of a rectangular[29].
Bartlett triangular Window function defined as:
wpnq “
$
’
’
’
&
’
’
’
%
2pn`1qN`1 0 ă n ď pN´1q
2
2´ 2pn`1qN`1
pN´1q2 ă n ď pN ´ 1q
0 Otherwise
(3.9)
23
The width of the main lobe and side lobes are twice that in the spectrum
of a rectangular window (because of the “halfing” in the time domain).The
Bartlett window having quiet smoother behaviour in the designed filter and
having less abrupt change but spreads the transition region considerably.Figure
4 shows the spectrum of Bartlett window.
Figure 3.3: Bartlett Window
3.3.2 Generalized Raised Cosine Windows
Raised Cosine Filters are derived basically from Nyquist Filter[31]. Raised
cosine filter is defined as:
W pnq “
$
&
%
a´ b cosp2ppn` 1q{pN ` 1qq ` c cosp4ppn` 1q{pN ` 1qq 0 ă n ď pN ´ 1q
0 Otherwise
(3.10)
On the basis of three parameters a,b and c, we can derive different window
filter which are given below:
1. Rectangular Window (a “ 1,b “ c “ 0)
2. Hanning Window (a “ 0.5,b “ 0.5,c “ 0)
3. Hamming Window (a “ 0.54,b “ 0.46,c “ 0)
4. Blackman Window (a “ 0.42,b “ 0.5,c “ 0.08)
24
We can derive Hanning Window spectrum from spectrum of rectangular win-
dow. It is sum three spectrum of rectangular windows which are shifted with
respect to each other. .It has a wider Main Lobe than Rectangular window
also, it has the much lower side lobes compare to the Rectangular window and
taper off very smoothly to zero[29].
Hamming Window also, it has similar spectrum i.e. wider main lobe because
of the same reason. Only difference between Hanning Window and Hamming
Window is their Roll off Factor.The side lobes, although lower than that of
the rectangular window, decays very slowly. This is due to the discontinuities
at the two edges of the window. Blackman Window has wider main lobe in
Figure 3.4: Different Window Functions
spectrum compare to Hanning/hamming window because of the additional
cosine term in the windowing function (c “ 0.08). Side lobes are also lower
because of smoother transition to zero.
Figure 5 shows different window functions. The Hanning, Hamming and
Blackman windows use progressively more complicated cosine functions to
provide a smooth truncation of the ideal impulse response and a frequency
response that looks better[28].
25
3.3.3 Kaiser Window
Windows which we explained here can be approximated or derived by using
Kaiser Window function.It is also known as the Kaiser-Bessel Window. It is
one parameter family of window functions. Kaiser window has linear phase re-
lation, The best window results probably come from using the Kaiser window,
which has a parameter that allows adjustment of the compromise between the
overshoot reduction and transition region width spreading[28].
The main advantages of using window filter method is that it is quiet sim-
ple to design and easy to use. Also, well defined mathematical equations are
available to find out the filter coefficients, which has made it quit successful.
In this work, we are using Hanning Window because it has wider main
lobe in the spectrum in comparison to rectangular and similar to Hamming
Window and also lower side lobes[32]. This windowing function tapers off to
zero very smoothly, less spectral leakage and implementation of this window is
simpler in comparison to other windowing technique[29]. For 1D image signal
Hanning Window function is defined as:
W pnq “1` cospπn
Mq
2(3.11)
3.4 Phase Only Correlation
We use Phase Only Correlation to find out relative translation movement
between two images corrupted by frequency dependent noise and correlated
noise. This is robust against the noise and external interference.It is a method
of image matching and exploit frequency domain analysis to estimate disparity
between two similar images[1].
Let’s consider frequency domain signal of 1D image signal fpnq and gpnq.
F pkq “Mÿ
n“´M
fpnqWNkn (3.12)
26
We can write in polar form i.e. in Amplitude and Phase term.
F pkq “ AF pkqejθF pkq (3.13)
Similarly we can write for second image signal,
Gpkq “Mÿ
n“´M
gpnqWNkn (3.14)
In Polar form,
Gpkq “ AGpkqejθGpkq (3.15)
The goal of image correlation is, for given two images,find the displacement
that maximises their similarity[19].Let’s assume that our Cross Power Spec-
trum is denoted by Rpkq. Then Normalized Cross Power Spectrum is defined
as:
Rpkq “F pkqGpkq
|F pkqGpkq|(3.16)
So Normalized Cross Power Spectrum in polar form:
Rpkq “ ejpθF pkq´θGpkqq (3.17)
From the above equation it is clear that final expression of Normalized Cross
Power Spectrum does not contain any Amplitude term and it depends on
phase term only. Because of this Phase Only Correlation, it is robust against
correlated and frequency dependent noise[33].
3.5 Spectral Weighting
If we consider the general Natural image or the typical images around us
so we can easily identify that those images do not have too much higher
frequency component means most of the energy of the image resides in the
low frequency component[5]. So if talk about low frequency components then
they have higher Signal to Noise Ratio (SNR) but high frequency components
have Low SNR. Low SNR is not desirable because it reduces the estimation
27
accuracy and increases the false detection of peaks in correlation function[34].
To improve the estimation accuracy and reduce the false detection we have to
restrict the high frequency component from image i.e. filtered them out. For
that purpose we employ spectral weighting concept in image matching. We
can use following Weighting functions:
1. Rectangular Low Pass Weighting Function
2. Triangular Weighting Function
3. Gaussian Weighting Function
Weighting functions mentioned above are basically Low Pass Filter. They
restrict the high Frequency component and pass the lower frequency compo-
nent.
3.5.1 Rectangular Low Pass Weighting Function
Rectangular low pass filter or brick wall filter is most basic and an idealized
electronic filter. It has full transmission in pass band and complete attenuation
in pass band, with abrupt transition[28]. transfer function of brick wall is
defined as:
Hpωq “
$
&
%
1 |k| ď ωL
0 Otherwise(3.18)
Transfer function is depicted in Figure 6: Rectangular Weighting function
Figure 3.5: Rectangular Weighting Filter
provides considerable wider main lobe but peak value is lower. It also has
higher side lobes compared to other weighting function. So generally it is not
used in image matching.
28
3.5.2 Triangular Weighting Function
Triangular Weighting Function is also popular weighting function in image
matching. It has less wider main lobe in comparison to rectangular weighting
function but having peak value larger that rectangular weighting function. It
has lower side lobes than Rectangular filter[29].
3.5.3 Gaussian Weighting Function
This is one of the important filter in signal and image processing. Gaussian
Filter is a filter whose frequency domain transformation is also Gaussian func-
tion. It does not provides overshoot to step function input while minimizing
the rise and fall time.
1D Gaussian Weighting function is defined as:
Hpkq “ e´2π2σ2k2 (3.19)
where σ is a parameter that controls the function width.Shape of Gaussian
Weighting Function shown in figure 7. It provides best result compared to
Figure 3.6: Shape of Gaussian Weighting Filter
other weighting function. It has lowest side lobes and has significant width of
29
main lobe with high peak value[35].
In this work, we are using Gaussian weighting function with different values
of variance. we have used others weighting technique also but this one pro-
vides better estimation accuracy and is robust against noise compare to those
technique. Here we multiply this Gaussian weighting function with Cross
power spectrum and then take inverse and look for peak[36].
3.6 Interpolation
Till now we did not talk about sub pixel accuracy. All procedures which are
explained above provide disparity at pixel level. But here we are working on
the sub-pixel accuracy.So it may possible that Peak of cross phase spectrum
occurs between the pixels and to find out that exact place we are using In-
terpolation Concept.
Image Interpolation is widely used in different application of Computer Vi-
sion . It is basically Re-Sampling method used to transform discrete samples
in to a continuous signal. Re-sampling is necessary for discrete signal or image
processing, such as geometric adjustment and registration, to improve image
quality on different display devices or in the field of lossy image compression
wherein some pixels or some frames are discarded during the encoding process
and must be regenerated from the remaining information for decoding[28]. In-
terpolation is basically used to find out the some unknown or missing points
or samples from the signal. It uses weighted average of a number of known
samples at the neighbourhood points.
The main goal of Interpolation is to provide high-fidelity reconstruction of
unknown or missing samples of signal[37].
There are different types of interpolation we use:
1. Nearest Neighbour
2. Linear Interpolation
3. Cubic Interpolation
30
4. Bilinear Interpolation
3.6.1 Digital Interpolation
Digital interpolation is widely used in the field of image and signal processing.
Digital interpolation is used for sampling rate conversion in multi rate commu-
nication systems and up-sampling for improved graphical representation[28].
Let’s consider our 1D image signal fpnq and we are going to interpolate it by
a factor of I. Means we are going to insert I ´ 1 zeros between two sample
of image signal. It is also known as Zero Padding, because we are inserting
number of zeros between two points of image[28][13] .
So let’s assume our Zero padded signal is denoted by fzpnq and defined as:
fzpnq “
$
&
%
fpnIq n “ 0,˘I,˘2I...
0 Otherwise(3.20)
Figure 8 shows an example of interpolation.
Figure 3.7: Interpolation technique implementation
Nearest Neighbour is the most basic interpolation technique for image sig-
nal.With the most basic nearest neighbour interpolation, just copy the exact
same pixel values over to the filler pixel closest to the pixel. As the actual
pixels are proportionally copied to their new locations, their position in rela-
tion to one another remains the same[13].
This technique is also known as point shift algorithm and pixel replication.
31
The interpolation kernel for the nearest neighbour algorithm is defined as
hpxq “
$
&
%
1 0 ď |x| ă 0.5
0 0.5 ď |x|(3.21)
3.6.2 Linear Interpolation
Linear Interpolation is also another basic interpolation technique but better
in comparison to Nearest Neighbour.It takes into account the gradual transi-
tion of pixel values. By finding the means between two pixel values, the filler
pixel is better suited for overall image enhancement. In other words, it just
looks plain better. It is a first degree method that passes a straight line
through every two consecutive points of the input signal[38].
In the spatial domain, linear interpolation kernel is defined as:
hpxq “
$
&
%
1´ |x| 0 ď |x| ă 1
0 1 ď |x|(3.22)
It has triangular kernel, and is also called Triangular filter,Roof function or
Bartlett Window.
3.6.3 Bilinear Interpolation
Bilinear Interpolation is a Re-Sampling method that uses the distance
weighted average of the four nearest pixel values to estimate a new pixel
value. We can say ,it is extension of linear interpolation and interpolation is
quadratic rather than linear. Here in our case we have 1D image signal so it
uses two nearest pixel to predict the new pixel[13].
32
3.6.4 Cubic Interpolation
Cubic Interpolation is quit better interpolation algorithm and provides better
approximation compare to other algorithms. It is a third degree interpolation
algorithm and approximates signal optimum sinc interpolation function. The
kernel is composed of piecewise cubic polynomials[38].
Cubic Interpolation kernel is defined as:
hpxq “
$
’
’
’
&
’
’
’
%
1´ a|x|2 ` b|x|3 0 ď |x| ă 1
c´ d|x| ` e|x|2 ´ f |x|3 1 ď |x| ă 2
0 Otherwise
(3.23)
So here we explained some basic interpolation technique for image and 1D
signal.
In This Work, Interpolation factor is 100 here in our experiment means
we are adding 99 zeros between two consecutive pixels[22]. This helps us to
find out sub-pixel accuracy because by using this we are finding the exact
place where the Cross Phase Function has peak value. Without Interpolation
it may possible that peak value of Cross Phase Function can have peak value
any where between two pixels but it shows at pixel level. But if we use inter-
polation technique, so we put some zeros between the pixels and then its easy
to find accuracy at Sub-Pixel accuracy [39].
Accuracy of measurement is directly proportional to Interpolation Factor
i.e. if use higher Interpolation factor it provides high accuracy at sub pixel
level[1]. But computational time is inversely proportional to Interpolation
Factor means higher computational cost for higher interpolation factor. So
this is a kind of Trade-off between Accuracy and Computational Time. Be-
cause of this here Interpolation factor is 100 and this value provides optimal
solution.
33
3.7 Time Domain POC
Time Domain POC is basically 1D inverse DFT of Cross Phase Spectrum
Rpkq. Inverse DFT (IDFT) of Cross Phase power Spectrum is given by[26]:
rKpnq “1
L
Kÿ
k“´K
RpkqWL´kn (3.24)
Where
L “ 2K ` 1 (3.25)
Here K is an Important parameter because it limits the frequency bandwidth
of the images. Basically, it is a control parameter for limit the bandwidth and
because of this it is known as Band Limited also. This parameter eliminates
unwanted high frequency component and noise also[23]. It improves matching
accuracy between two images.
Now if we solve this equation for a pair of images displaced by very small
distance, then it will look like Sinc function having a peak value[32]. So
analytical peak value model of this algorithm will be given by:
rKpnq “ αsin pπpn` δ L
Nqq
πpn` δLLq
(3.26)
Here α is a multiplicative constant and generally its value is 1 or in most of
case α ď 1 .This equation shows shape of peak in 1D BLPOC. And δ is a
real number and shows the shift between two images fpnq and gpnq. If δ “ 0
expression would look like Kronecker Delta Function[23].
But this shows general expression for Time domain Phase Only Correlation.
This expression basically depends on Spectral Weighting Function. So before
taking inverse DFT of Cross Phase Spectrum Rpkq, it is multiplied by any of
Spectral Weighting Function Hpkq and after that we perform IDFT.
Let‘s consider another function P pkq defined as:
P pkq “ RpkqHpkq (3.27)
Now we can perform Inverse Discrete Fourier Transform on P pkq shown in
following equation:
rKpnq “1
L
Kÿ
k“´K
P pkqWL´kn (3.28)
34
Put the value of P pkq in the given equation:
rKpnq “1
L
Kÿ
k“´K
RpkqHpkqWL´kn (3.29)
In Polar form Cross Phase Spectrum is given by:
Rpkq “ ejpθF pkq´θGpkqq (3.30)
We can write it like:
Rpkq “ e´j2πNkδ (3.31)
Now putting this value of Cross Phase Spectrum into IDFT equation.
rKpnq “1
L
Kÿ
k“´K
e´j2πNkδHpkqWL
´kn (3.32)
Now apply the DFT shifting Theorem:
rKpnq “ hpn` δq (3.33)
So here final expression shows shifted version of spatial domain of Spectral
Weighting Function. But as we had explained about different Spectral weight-
ing function and proper spectral weighting is also important for proper esti-
mation accuracy.
Let‘s consider H1pkq is Rectangular Weighting Function. So we can derive
other Weighting function by using convolution operation.
H2pkq “ H1pkq bH1pkq (3.34)
Here H2pkq is Triangular Weighting Function for that we will get different 1D
BLPOC expression:
rKpnq “1
L
Kÿ
k“´K
e´j2πNkδH2pkqWL
´kn (3.35)
Put the value of H2pkq in the above equation:
rKpnq “1
L
Kÿ
k“´K
e´j2πNkδH1pkq bH1pkqWL
´kn (3.36)
35
By using the property of DFT:
rKpnq “
ˆ
αsin pπpn`δ LN qq
πpn`δLL q
˙2
(3.37)
Another Spectral weighting factor can be obtained by performing the convo-
lution between H1pkq and H2pkq :
H3pkq “ H2pkq bH1pkq (3.38)
Now this shows another kind of spectral weighting function and peak value
model of 1D BLPOC by using this spectral weighting function can be calcu-
lated similarly and defined as[34]:
rKpnq “
ˆ
αsin pπpn`δ LN qq
πpn`δLL q
˙3
(3.39)
Most important spectral weighting function is Gaussian Spectral Weighting
Function . Kernel for 1D Gaussian Spectral Weighting factor is defined as[35]:
Hpkq « e´2π2σ2k2 (3.40)
In This Work we are using Gaussian Spectral Weighting. Gaussian filter
provides better smoothing and preserve the information compare to other
weighting functions. Gaussian filters are rotationally symmetric and filter
weights decrease monotonically from central peak, giving most weight to cen-
tral pixels. It provides simple and intuitively relationship between size of σ
and the smoothing[22].
In this case our peak model for 1D BLPOC will be:
rKpnq «1
2πσ2e
´n2
2σ2 (3.41)
So equation 3.41 shows the peak model of algorithm and we are plotting this
function for find out the shift between two images.
3.8 Flow Chart
Here we are giving the summary of algorithm which is briefly explained in
this chapter. For a short explanation we are using a flow diagram.In the flow
36
we started from reading the input image of pairs. Then we apply normal 2D
Phase Only Correlation and calculate disparity at pixel level[33][19][36]. Af-
ter that, those values (Disparity at pixel level) are used for reference purpose.
And then we go for Sub-Pixel Accuracy on the basis of results from normal
POC result. To do that, first we choose small window from the image of width
N and Height H and then copy all row component in to a fftw Array [22]. As
we have explained before, this is the key point to reduce the computational
time. Then we apply frequency Domain Transformation by using 1D Discrete
Fourier Transform (DFT)[26]. After Frequency Domain transformation we
are finding out cross power spectrum and then normalized its value. Finally,
as Cross Power Spectrum has no Amplitude term and it consist only phase
term that‘s why this method is also known as Phase Only Correlation[27].
Noise(Frequency dependent and Correlated) is more sensitive towards Ampli-
tude term in comparison to the phase so this method is robust against white
noise, frequency dependent and correlated noise[1].
Before calculation of Cross Power Spectrum, we have to apply windowing
technique to the 1D image signal because of the periodic nature of DFT in
both Time and Frequency domain, and this periodicity creates problem of
”Wrap Around”[26]. In this problem, there is repetition of signal at the edges
and that is not allowed in real world problem. So to overcome this problem,
we use Hanning Window to truncate the 1D image signal and it removes the
Wrap Around problem[30].
Upto calculation of Cross Phase Spectrum we are not considering the effect
of noise and general frequency distribution of natural images. Generally in
natural images, most of the energy resides in low frequency domain and low
energy in high frequency i.e. we can say a natural image having more low fre-
quency component compared to the high frequency term. So Signal to Noise
Ratio (SNR) for high frequency component is less than that for low frequency
term. And this low SNR reduces the estimation accuracy[35].
To improve estimation accuracy we have to enhance SNR and for that
37
Figure 3.8: Flow Diagram
we are using Spectral Weighting Function. This is a filtering concept and it
blocks high frequency components which have low SNR and allows Low fre-
quency component. We are using Gaussian Spectral Weighting Function and
this provides better estimation accuracy in comparison to the other Spectral
Weighting Function because its weights decreases monotonically from central
peak very smoothly so, it blocks high frequency term as well as preserve more
details[35].
The most important stage is Interpolation because this is responsible for
Sub-Pixel Accuracy [13]. It is possible that the peak value of peak model func-
38
tion lies between two pixels and it shows that disparity, at pixel level so to
achieve proper Sub-Pixel Accuracy we are using Interpolation Concept. We
are providing zero padding between two pixels of image and then interpolate
that 1D image signal[25]. Interpolation increases the Computational Time but
provides better accuracy. We are using digital interpolation and interpolation
factor is 100.
39
Chapter 4
Experimental Set-Up
“It doesn’t matter how beautiful
your theory is, it doesn’t mat-
ter how smart you are. If it
doesn’t agree with experiment,
it’s wrong.”
-Richard P. Feynman
This section describes experimental set up for estimating translational im-
age displacements using the proposed technique. As it explained in Problem
Statement that we are investigating the limit of accuracy of the algorithm i.e.
we are finding the minimum shift between two images at Sub-Pixel Accuracy.
To provide some fix and minor shift we need some experimental set up i.e
some bench on which we can put some object and can take the picture of that
object.
Whole Experimental set up consist of following articles:
1. Translational Stage
2. Camera
3. Wooden Cube
4. Complete Set Up
40
4.1 Translational Stage
We are working on 1D BLPOC for calculating translation displacement be-
tween two images. So for collecting that kind of data i.e. images which are
displaced only in X-Direction, we need some translational stage.
For that purpose we are using a Micrometer Translational Stage TCS42-
05A. This is a product of Laser Components GmbH, Germany . Translational
Stage dimension is 42ˆ 42mm and has Travel range up to 13mmp˘6.5mmq .
It is made up of Aluminium and has load capacity of 5 kg. It has repeatability
of 0.003mm. It is shown in figure We fixed this micrometer translational stage
Figure 4.1: Micrometer Translational Stage TCS42
on the table with wooden block as shown in figure.
Figure 4.2: Micrometer Translational Stage TCS42 with Wooden Cube
41
4.2 Camera
We are using Canon EOS 600D for taking the pictures of object (Wooden
Cube). It is an 18 Megapixel digital single-lens reflex camera. This camera has
CMOS APS-C 22.3ˆ14.9mm and provides 5184ˆ3456 maximum resolution.
In our experiment we fix the camera on the table and it is not movable. We
are not touching camera, we are operating it by remote as shown in figure 12.
Figure 4.3: Canon EOS 600D Camera with Remote
4.3 Wooden Cube
We are using a wooden cube as an object. And putting this object on the
translational stage to take the picture of this wooden block as input to algo-
rithm. Size of Wooden Cube is 18 ˆ 18 ˆ 18cm. Each face of wooden block
having different texture and we can rotate the wooden block on the translation
stage to click picture of different texture.As it is shown in figure 11.
42
4.4 Complete Set Up
Figure 13 shows complete set up for experiment. In this set-up, we put camera
at 2 meter distance from the object. And camera is fixed on the table whereas
object is movable on the translation stage. We are moving the object in
micrometer range by using the knob of micrometer translation stage.
Figure 4.4: Complete Experimental Set Up
43
Chapter 5
Results
“A work of art is the unique re-
sult of a unique temperament.”
-Oscar Wilde
This chapter provides the results and benchmarks done using the 1D
BLPOC on the large input images shifted by some distance. First section
gives result information regarding 1D BLPOC with alot of input image pairs.
Then we compare with two techniques, Normalized Cross Correlation and 2D
Phase Only correlation. .
In this section, we describe a set of experiments, performed by using explained
experimental set up shown in figure 4.4, for estimating translational displace-
ments using the proposed technique and also evaluating the accuracy of the
proposed algorithm.We have estimated the displacements between two images
taken by the camera (Canon EOS 600D with CMOS APS-C sensors). The
target object is a wooden cube with the size of 18cmˆ 18cmˆ 18cm, which is
mounted on a micro stage that allows precise alignment of the cube position.
In the first experiment, we have two images for matching and they are shifted
randomly under optimal conditions.As shown in figure 5.1.
Figure 5.2 shows peak model of 1D BLPOC algorithm. From figure, its con-
firm that image signal is influenced by individual noise and the correlation
function shapes deviate from the theoretical graph i.e. from the pure sinc
44
Figure 5.1: Input Image Pair
graph. But still the peak location can be identified with proper accuracy.
And in this case it shows near about 27.08 pixel shifting between both of
images. Here we are plotting disparities occurring along an image row in the
direction of epipolar line and in the figure we can see the noise influence in
the direction of epipolar line also.
Figure 5.3 shows the deviation of matching accuracy from the straight regres-
sion line. Dotted line show the deviation and this deviation arises because
of noise influence. We can reduce it by providing different value of σ in the
Gaussian spectral weighting function. But more filtering provides more flat
peak of the peak model and then it becomes quiet difficult to say that where
is the exact peak value with proper accuracy.
Figure 5.4 shows the disparities in the form of gray level of image. This im-
age provides a corresponding impression, with disparities linearly increasing
from left to right. Image is not looking so smooth and its just because of
noise effect. Smoothness of the disparity image depends on different Spec-
tral weighting function and window size. As shown in figure 5.5, two output
images with rectangular spectral weighting function. Smoother image having
window size larger in first image.
In figure 5.6, the output images are with Gaussian Spectral weighting Func-
tion and these images are smoother than images with Rectangular Spectral
weighting Function . From figure 5.6, we can easily say that second image is
smoother than first image. In second image window size is 101 ˆ 21 whereas
45
in first image window size is 101ˆ 11.
As the size of window increases, information inside that window also increases
and SNR for that window is also increased because CCD noise is common for
entire image. For larger window we have better information compare to the
a window having smaller size. Higher the SNR, higher estimation accuracy
and matching accuracy does not deviate too much from the straight regres-
sion line. And we can easily verify it from second image of figure 5.6 having
window size 101ˆ 21.
Input image pair shown in figure 5.1 is taken randomly from camera and shift
is not precise because we want to show the effect of different spectral weight-
ing factor, and effect of variance for Gaussian Spectral weighting Function and
different window size.
Now we are going to use experimental set up shown in figure 4.4. Image taken
by this set up is shown in figure 5.7. In that image, we just paste some other
texture image on the wooden block and then shift the block by 10 micrometer
and then took another image. From that image, we can easily say that whole
image is not region of interest because only wooden block part is moving and
rest part is stationary. So first of all we have to find out or crop the ROI
(region of interest). figure 5.8 shows the Region of Interest of size 250ˆ 200.
We took images first shifted by 10 micrometer then 20 micrometer and so
on up to 600 micrometer with step size of 10 micrometer.Result shows in term
of sub-pixel shift. At each step size we took 4 images and apply to algorithm
for final result. we took average of all the result to reduce human error and
also surrounding noise effect. figure 5.9 shows the final result for the image
pair shown in figure 5.8.
Figure 5.9 shows the variation in pixels of the image when we shifted the
image by 10 micrometer i.e. variation along X-direction has fixed step size of
10 micrometer. Maximum shift between two images we used is 600 microme-
ter. Y-direction shows shift in image in terms of pixels.
Image we used in 1st experiment and shown in figure 5.8 having standard
46
Figure 5.2: Time Domain POC
deviation of 40.3561 means this image has proper Texture. In this figure we
have two data, one regarding the window size 101ˆ 11 and another is regard-
ing window size 101 ˆ 31. As we had explained earlier higher window size
has high signal to noise ratio (SNR) and provides better estimation accuracy
that we can see from the figure, the data regarding higher window size has
the smoother behaviour compared to smaller window size.
But its difficult to say how much sub pixel shift is occurring when we are
shifting the image by 10 micrometer from micrometer translational stage. So
for accurate measurement of shifting we plotting the graph with step size of
20 micrometer as shown in figure 5.10. In this figure behaviour of each data
is quiet smooth. And from this figure we concluded that, if we shift the image
by 20 micrometer from the translational stage, the change in the image is
1/14th of a Pixel.
Now consider the another input image pair shown in figure 5.11, having stan-
dard deviation 19.3581 i.e. this image pair does not have proper texture like
the previous image pair but still has good texture. Figure 5.11 shows the vari-
ation in pixels of the image when we are shifting the image by 10 micrometer
47
Figure 5.3: Deviations between Disparity and Regression Line
i.e. variation along X-direction has fixed step size of 20 micrometer. Maxi-
mum shift between two images we used is 600 micrometer. Y-direction shows
shift in image in terms of pixels.
And from this figure, we find out the sub pixel shift is 1/12th of pixel if we
shift the image by 20 micrometer from translational stage.
5.1 Comparison with Normalized Cross Cor-
relation
Normalized Cross Correlation comes in the category of Correlation like meth-
ods. This is the classical Area Based Method for correspondence matching.
The main problems of the Normalized cross correlation is the flatness of the
peak model which provides us maxima and high computational complexity.
As the Normalized Cross Correlation is most classical and basic method
for similarity measurement, so we want compare the performance our of algo-
48
Figure 5.4: Disparities as gray level image
rithm with NCC(Normalized Cross Correlation).
So for comparison, we are using the input image pair shown in figure 5.8.
We computed Normalized Cross Correlation coefficient between two shifted
images i.e. one image is shifted with very small distance compared to other
and we had done shifting in same manner like previous experiment i.e. by 10
micrometer ,20 micrometer and so on. And we calculated the NCC coefficient.
In the figure 5.13, we have drawn the NCC coefficient value with respect to
shift in image in micrometers.
From the figure, we have analysed that, if the shift between the images is
10 micrometer then there is no change in NCC Coefficient, same for 20 mi-
crometer means no change occur in NCC coefficient. So we can say if
shift in image is more then 20 micrometer then it is detectable.
Whereas in the case 1D BLPOC, for shift of 20 micrometer in the images
provides proper and stable shift in pixel and that is 1/14th of pixel . And
Computational time is more than 2D POC .
49
Figure 5.5: Disparity image with Rectangular Weighting Factor and windows
size are 101ˆ 11 and 101ˆ 21
5.2 Comparison with 2D POC
2D POC is also known as Fourier Series method. And most basic approach
in the field of Phase Only Correlation. It provides similar accuracy as the 1D
BLPOC but computational time is higher in comparison to the 1D BLPOC.
We used same image pair given in the figure 5.8 for comparison purpose. We
have applied 2D POC with windows size 101ˆ31 and 101ˆ21. So from figure
it is a clear that it has a bit smoother behaviour than the 1D BLPOC and
accuracy is also good. Accuracy for 2D POC is near about 1/16th of Pixel.
So in term of accuracy it is also better than the 1D BLPOC.
But the main drawback of this technique is High Computational time.And to
overcome this problem we used the 1D BLPOC that reduced the computa-
tional time to 1/3rd of 2D POC.
50
Figure 5.6: Disparity image with Gaussian Weighting Factor and windows
size are 101ˆ 11 and 101ˆ 21
Figure 5.7: Input Image Pair
Figure 5.8: Input Image Pair (Cropped)
51
Figure 5.9: Shift in pixels in the image Vs shift in micrometer stage with step
size 20 micrometer
Figure 5.10: Shift in pixels in the image Vs shift in micrometer stage with
step size 20 micrometer
52
Figure 5.11: Input Image Pair (Cropped)
Figure 5.12: Shift in pixels in the image Vs shift in micrometer stage with
step size 20 micrometer
53
Figure 5.13: Normalized Cross Correlation Coefficient Vs Shift in the image
Figure 5.14: 2D POC Vs shift in image
54
Chapter 6
Summary
“There will come a time when
you believe everything is fin-
ished. Yet that will be the be-
ginning”
-Louis L’Amour
This chapter completes the work giving a conclusion for the Accuracy of
the 1D BLPOC algorithm using different data input. A later section will
give future directions and ideas on how the accuracy of 1D BLPOC can be
improved and we will discuss about the applications of algorithm.
6.1 Conclusion
This work includes implementation of 1D BLPOC and comparison of this
algorithm with two other techniques.Other techniques are Normalized Cross
Correlation Method and 2D POC.
1D BLPOC concept is initially given by shihabara. But he did not use
the concept of interpolation, he used the concept of fitting equation. Here
55
we used concept of interpolation for estimate Sub-Pixel Accuracy with high
precision. We exploit the concept of basic Fourier analysis. We used different
weighting functions that also affects the accuracy of estimation as shown in
previous section and found out that Gaussian Spectral Weighting Factor pro-
vides better accuracy than others.
Next parameter which affects the accuracy of algorithm is Window Size.
Larger window size provides higher estimation accuracy compare to smaller
window size. As we had explained in the previous sections that larger window
has higher SNR and smaller window has lower SNR because larger window
has more information compare to the smaller ones.
Now we are going to compare 1D BLPOC with most classical area based
method of correspondence matching, Normalized Cross Correlation Method.
As it shown in figure 5.13, if shift in the image is more than 20 micrometer
then NCC method is able to detect otherwise there is no change in NCC co-
efficient. On the other hand 1DBLPOC gives proper accuracy for the shift
of 20 micrometer. If we shift the image by 20 micrometer then we will get
1/14th of Pixel shift. 1D BLPOC algorithm is computationally efficient
than NCC method.
Finally in conclusion subsection, we are comparing 2D POC and 1D
BLPOC. As it shown in figure 5.14, 2D POC has a bit better accuracy than
1D BLPOC or we can say both have similar accuracy.But the main drawback
of 2D POC is high computational cost because of 2D correspondence match-
ing whereas 1D BLPOC has 1D correspondence search along the epipolar
line. So 1D BLPOC is two or three times computationally efficient than 2D
POC.
Finally some points for conclusion.
1. For 20 micrometer shift, in pixel would be 1/12th to 1/14th of a
pixel.
56
2. Accuracy of explained algorithm is depends upon following factor:
(a) Spectral Weighting Function
(b) Window Size
(c) Texture of Image
3. Accuracy is similar to the 2D POC
4. Computationally efficient in comparison to the 2D POC and Normalized
Cross Correlation Method
6.2 Applications
This algorithm provides high accuracy measurement for calculating shift be-
tween the two images. So we can apply this algorithm where high accuracy
measurement is necessary. Some of them are mentioned below.
1. High Accuracy 3D Measurement
2. Civil Engineering and surveillance
3. Geo-informatics
4. Weather Forecasting etc.
The most important application of algorithm in 3D Reconstruction. As we
know for 3D reconstruction of any object by using stereo pair image we need
two important parameters, Base line width and corresponding point. And 3D
measurement accuracy depends mainly on accuracy of correspondence point.
By using this algorithm we can calculate correspondence point with high ac-
curacy so accuracy of 3D measurement would be enhanced.
Another application is in the field of surveillance and Civil Engineering.
Let‘s consider we want to measure deformation in buildings. Deformation in
the building occurs in the micrometer range so we can‘t see it with bare eyes.
57
If we can take two images at different time, like between months and then we
can apply explained algorithm to find the small changes.
Some other applications also like satellite imaging, Geo-informatics, medi-
cal imaging and all other fields where high accuracy measurement is required.
6.3 Future Work
we can extend this work in different senses like
1. Rotational Measurement
2. 3D Measurement and compare the accuracy
3. Apply Pyramidal concept for further reduction of computational time.
In our work we are considering only translation displacement and calculating
shift in pixels according to translational displacement only. We can extend
it to Rotational Measurement also. Initially 2D POC is also proposed only
for translational displacement but after some time it extended to rotational
measurement also[40]. Similarly we can extend this one also for rotational
measurement and can check minimum rotational shift between images.
Another extension in the work, we can think about 3D Measurement by
using the explained algorithm and then compare the accuracy of reconstructed
3D object with some existed techniques of 3D measurement.
We had already explained that 1D BLPOC is more computationally ef-
ficient than 2D POC and Normalized Cross Correlation Method. But still
this algorithm has considerable computational time and because of that we
can not use it in Real Time Applications. So we can think to use Pyramidal
Concept for calculating the disparity between two images and that can reduce
58
the computational time so that we can use it for real time applications also
like Object Tracking etc.
59
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