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# Assignment 3 Lucas-Kanade Visual Tracker1

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COMP5421 Computer Vision
Homework Assignment 3
with the complete submission checklist at the end of this document.
• All tasks marked with a Q require a submission.
• Please stick to the provided function signatures, variable names, and file names.
• Start early! This homework cannot be completed within two hours!
• Verify your implementation as you proceed: otherwise you will risk having a huge mess
of malfunctioning code that can go wrong anywhere.
This homework consists of four sections. In the first section you will implement a simple
Lucas-Kanade (LK) tracker with one single template; in the second section, the tracker will be
generalized to accommodate for large appearance variance. The third section requires you to
implement a motion subtraction method for tracking moving pixels in a scene. In the final section
you shall study efficient tracking which includes inverse composition and correlation filters. Note
the first 3 sections are based on the Lucas-Kanade tracking framework; with the final section also
incorporating correlation filters. Other than the course slide decks, the following references may
1. Simon Baker, et al. CMU-RI-TR-02-16, Lucas-Kanade 20 Years On: A Unifying Framework:
Part 1, Robotics Institute, Carnegie Mellon University, 2002.
2. Simon Baker, et al. CMU-RI-TR-03-35, Lucas-Kanade 20 Years On: A Unifying Framework:
Part 2, Robotics Institute, Carnegie Mellon University, 2003.
Both are available at:
https://www.ri.cmu.edu/pub_files/pub3/baker_simon_2002_3/baker_simon_2002_3.pdf
In this section you will be implementing a simple Lucas & Kanade tracker with one single template.
In the scenario of two-dimensional tracking with a pure translation warp function,
W(x; p) = x + p. (1)
The problem can be described as follows: starting with a rectangular neighborhood of pixels
N ∈ {xd}
D
d=1 on frame It
, the Lucas-Kanade tracker aims to move it by an offset p = [px, py]
T
to
1Credit to CMU Simon Lucey, Nate Chodosh, Chengqian Che, Allie Chang, Akshita Mittel, Gaurav Mittal, Purna
Sowmya Munukutla
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obtain another rectangle on frame It+1, so that the pixel squared difference in the two rectangles
is minimized:
p
∗ = arg min
p
=
X
x∈N
||It+1(x + bp) − It(x)||2
(2)
=

It+1(x1 + p)
.
.
.
It+D(xD + p)

 −

It(x1)
.
.
.
It(xD)

2
2
(3)
Q1.1 (5 points) Starting with an initial guess of p (for instance, p = [0, 0]T
), we can compute the
optimal p

iteratively. In each iteration, the objective function is locally linearized by first-order
Taylor expansion,
It+1(x
0 + ∆p) ≈ It+1(x
0
) + ∂It+1(x
0
)
∂x0T
∂W(x; p)
∂pT ∆p (4)
where ∆p = [∆px, ∆py]
T
, is the template offset. Further, x
0 = W(x; p) = x + p and ∂I(x
0
)
∂x0T is a
vector of the x- and y- image gradients at pixel coordinate x
0
. In a similar manner to Equation 3
one can incorporate these linearized approximations into a vectorized form such that,
arg min
∆p
||A∆p − b||2
2
(5)
such that p ← p + ∆p at each iteration.
• What is ∂W(x;p)
∂pT ?
• What is A and b?
• What conditions must AT A meet so that a unique solution to ∆p can be found?
Q1.2 (15 points) Implement a function with the following signature
that computes the optimal local motion from frame It to frame It+1 that minimizes Equation 3.
Here It is the image frame It
, It1 is the image frame It+1, rect is the 4-by-1 vector that represents
a rectangle describing all the pixel coordinates within N within the image frame It
, and p0 is the
initial parameter guess (δx, δy). The four components of the rectangle are [x1, y1, x2, y2]
T
, where
[x1, y1]
T
is the top-left corner and [x2, y2]
T
is the bottom-right corner. The rectangle is inclusive,
i.e., in includes all the four corners. To deal with fractional movement of the template, you will need
to interpolate the image using the Scipy module ndimage.shift or something similar. You will also
need to iterate the estimation until the change in ||∆p||2
2
is below a threshold. In order to perform
interpolation you might find RectBivariateSpline from the scipy.interpolate package. Read
the documentation of defining the spline (RectBivariateSpline) as well as evaluating the spline
using RectBivariateSpline.ev carefully.
Q1.3 (10 points) Write a script testCarSequence.py that loads the video frames from carseq.npy,
and runs the Lucas-Kanade tracker that you have implemented in the previous task to track the
car. carseq.npy can be located in the data directory and it contains one single three-dimensional
matrix: the first two dimensions correspond to the height and width of the frames respectively,
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and the third dimension contain the indices of the frames (that is, the first frame can be visualized with imshow(frames[:, :, 0])). The rectangle in the first frame is [x1, y1, x2, y2]
T =
[59, 116, 145, 151]T
. Report your tracking performance (image + bounding rectangle) at frames 1,
100, 200, 300 and 400 in a format similar to Figure 1. Also, create a file called carseqrects.npy,
which contains one single n × 4 matrix, where each row stores the rect that you have obtained for
each frame, and n is the total number of frames.
Figure 1: Lucas-Kanade Tracking with One Single Template
Q1.4 (20 points) As you might have noticed, the image content we are tracking in the first
frame differs from the one in the last frame. This is understandable since we are updating the
template after processing each frame and the error can be accumulating. This problem is known
as template drifting. There are several ways to mitigate this problem. Iain Matthews et al.
(2003), https://www.ri.cmu.edu/publication_view.html?pub_id=4433) suggested one possible approach. Write a script testCarSequenceWithTemplateCorrection.py with a similar functionality to Q1.3, but with a template correction routine incorporated. Save the resulting rects
as carseqrects-wcrt.npy, and also report the performance at those frames. An example is given
in Figure 2.
Figure 2: Lucas-Kanade Tracking with Template Correction
Here the green rectangles are created with the baseline tracker in Q1.3, the yellow ones with
the tracker in Q1.4. The tracking performance has been improved non-trivially. Note that you
do not necessarily have to draw two rectangles in each frame, but make sure that the performance
improvement can be easily visually inspected.
2 Lucas-Kanade Tracking with Appearance Basis
The tracker we have implemented in the first section, with or without template drifting correction,
may suffice if the object being tracked is not subject to drastic appearance variance. However, in
real life, this can hardly be the case. We have prepared another sequence sylvseq.npy (the initial
rectangle is [101, 61, 155, 107]), with exactly the same format as carseq.mat, on which you can
test the baseline implementation and see what would happen. In this section, you will implement
a variant of the Lucas-Kanade tracker (see Section 3.4 in ), to model linear appearance variation
in the tracking.
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2.1 Appearance Basis
One way to address this issue is to use eigen-space approach (aka, principal component analysis,
or PCA). The idea is to analyze the historic data we have collected on the object, and produce a
few bases, whose linear combination would most likely to constitute the appearance of the object
in the new frame. This is actually similar to the idea of having a lot of templates, but looking for
too many templates may be expensive, so we only worry about the principal templates.
Mathematically, suppose we are given a set of k image bases {Bk}
K
k=1 of the same size. We
can approximate the appearance variation of the new frame It+1 as a linear combination of the
previous frame It and the bases weighted by w = [w1, · · · , wK]
T
, such that
It+1(x) = It(x) +X
K
k=1
wkBk(x) (6)
Q2.1 (5 points) Express w as a function of It+1, It
, and {Bk}
K
k=1, given Equation 6. Note that
since the Bk’s are orthobases, they are orthogonal to each other.
2.2 Tracking
Given K bases, {Bk}
K
k=1, our goal is then to simultaneously find the translation p = [px, py]
T and
the weights w = [w1, · · · , wK]
T
that minimizes the following objective function:
min
p,w
=
X
x∈N
||It+1(x + p) − It(x) −
X
K
k=1
wkBk(x)||2
2
. (7)
Again, starting with an initial guess of p (for instance, p = [0, 0]T
), one can linearize It+1(x +
p+∆p) with respect to ∆p. In a similar manner to Equation 5 one can incorporate these linearized
approximations into a vectorized form such that,
arg min
∆p,w
||A∆p − b − Bw||2
2
. (8)
As discussed in Section 3.4 of  (ignore the inverse compositional discussion) this can be
simplified down to
arg min
∆p
||B
⊥(Ap − b)||2
2
(9)
where B⊥ spans the null space of B. Note that ||B⊥z||2
2 = ||z−BBT z||2
2 when B is an orthobasis.
Q2.2 (15 points) Implement a function with the following signature
where bases is a three-dimensional matrix that contains the bases. It has the same format as
frames as is described earlier and can be found in sylvbases.npy.
Q2.3 (15 points) Write a script testSylvSequence.py that loads the video frames from sylvseq.npy
and runs the new Lucas-Kanade tracker to track the sylv (the toy). The bases are available
in sylvbases.npy in the data directory. The rectangle in the first frame is [x1, y1, x2, y2]
T =
[101, 61, 155, 107]T
. Please report the performance of this tracker at frames 1, 200, 300, 350
and 400 (the frame + bounding box), in comparison to that of the tracker in the first section.
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That is, there should be two rectangles for each frame, as exemplified in Figure 3. Also, create a sylvseqrects.npy for all the rects you have obtained for each frame. It should contain
one single N × 4 matrix named rects, where N is the number of frames, and each row contains
[x1, y1, x2, y2]
T
, where [x1, y1]
T
is the coordinate of the top-left corner of the tracking box, and
[x2, y2]
T
the bottom-right corner.
Figure 3: Lucas-Kanade Tracking with Appearance Basis
3 Affine Motion Subtraction
In this section, you will implement a tracker for estimating dominant affine motion in a sequence
of images and subsequently identify pixels corresponding to moving objects in the scene. You will
be using the images in the file aerialseq.npy, which consists aerial views of moving vehicles from
a non-stationary camera.
3.1 Dominant Motion Estimation
In the first section of this homework we assumed the the motion is limited to pure translation. In
this section you shall implement a tracker for affine motion using a planar affine warp function. To
estimate dominant motion, the entire image It will serve as the template to be tracked in image
It+1 , that is, It+1 is assumed to be approximately an affine warped version of It
. This approach is
reasonable under the assumption that a majority of the pixels correspond to the stationary objects
in the scene whose depth variation is small relative to their distance from the camera.
Using a planar affine warp function you can recover the vector ∆p = [p1, …, p6]
T
,
x
0 = W(x; p) = 
1 + p1 p2
p4 1 + p5
  x
y

+

p3
p6

(10)
One can represent this affine warp in homogeneous coordinates as,
x
0 = Mx (11)
where,
M =

1 + p1 p2 p3
p4 1 + p5 p6
0 0 1

 (12)
Note that M will differ between successive image pairs. Starting with an initial guess of p = 0
(i.e. M = I) you will need to solve a sequence of least-squares problem to determine ∆p such
that p → p + ∆p at each iteration. Note that unlike previous examples where the template to
be tracked is usually small in comparison with the size of the image, image It will almost always
not be contained fully in the warped version It+1. Hence, one must only consider pixels lying in
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the region common to It and the warped version of It+1 when forming the linear system at each
iteration.
Q3.1 (15 points) Write a function with the following signature
which returns the affine transformation matrix M, and It and It1 are It and It+1 respectively.
probably also find scipy.ndimage.affine transform helpful).
3.2 Moving Object Detection
Once you are able to compute the transformation matrix M relating an image pair It and It+1, a
naive way for determining pixels lying on moving objects is as follows: warp the image It using M
so that it is registered to It+1 and subtract it from It+1; the locations where the absolute difference
exceeds a threshold can then be declared as corresponding to locations of moving objects. To obtain
better results, you can check out the following scipy.morphology functions: binary erosion, and
binary dilation.
Q3.2 (10 points) Using the function you have developed for dominant motion estimation, write
a function with the following signature
SubtractDominantMotion(image1, image2)
where image1 and image2 form the input image pair, and the return value mask is a binary image
of the same size that dictates which pixels are considered to be corresponding to moving objects.
You should invoke LucasKanadeAffine in this function to derive the transformation matrix M,
and produce the aforementioned binary mask accordingly.
Q3.3 (10 points) Write a script testAerialSequence.py that loads the image sequence from
aerialseq.npy and run the motion detection routine you have developed to detect the moving
objects. Report the performance at frames 30, 60, 90 and 120 with the corresponding binary masks
superimposed, as exemplified in Figure 4. Feel free to visualize the motion detection performance
in a way that you would prefer, but please make sure it can be visually inspected without undue
effort.
Figure 4: Lucas-Kanade Tracking with Appearance Basis
4 Efficient Tracking
4.1 Inverse Composition
The inverse compositional extension of the Lucas-Kanade algorithm (see ) has been used in
literature to great effect for the task of efficient tracking. When utilized within tracking it attempts
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to linearize the current frame as,
It(W(x; 0 + ∆p)) ≈ It(x) + ∂It(x)
∂xT
∂W(x; 0)
∂pT ∆p (13)
In a similar manner to the conventional Lucas-Kanade algorithm one can incorporate these
linearized approximations into a vectorized form such that,
arg min
∆p
kA0∆p − b
0
k
2
2
(14)
for the specific case of an affine warp where p ← M and ∆p ← M this results in the update
M ← M(∆M)
−1
.
InverseCompositionAffine(It,It1) using the inverse compositional method. In your own words
please describe why the inverse compositional approach is more computationally efficient than the
classical approach?
4.2 Correlation Filters
Enter the directory Corr-Filters. Run the script file example.py, and observe a visual depiction
of the extraction of sub-images from the Lena image (Figure 1) as well as the desired output response
from our linear discriminant (Figure 2). Inspecting example.py you can see that it extracts a set
of sub-images X = [x1, · · · , xN ] from within the Lena image. These sub-images are stored in vector
form so that xn ∈ R
D (in the example code all the sub-images are 29 × 45 therefore D = 1305).
Associated with these images are desired output labels y = [y1, · · · , yN ] where yn lies between zero
and one. For this example we have made D = N on purpose.
Q4.2 (15 points) A linear least-squares discriminant can be estimated by solving
arg min
g
X
N
n=1
1
2
kyn − x
T
n gk
2
2
we can simplify this objective in vector form and include an additional penalty term
arg min
g
1
2
ky − XT gk
2
2 +
λ
2
kgk
2
2
(15)
Please write down the solution to Equation 15 in terms of the matrices S = XXT
, X and y.
Q4.3 (15 points) Add your solution to Equation 15 in the example.py code. Visualize the
resultant linear discriminant weight vector g for the penalty values λ = 0 and λ = 1 (remember to
use the reshape function to convert g back into a 2D array). Apply the filter to the entire Lena
image using the scipy.ndimage.correlate function. Visualize the responses for both values of λ.
Include these visualizations in your written response document. Can you comment on which value
Q4.4 (15 points) Visualize the response you get if you attempt to use the 2D convolution function scipy.ndimage.convolve. Why does this get a different response to the one obtained using
correlate? How could you use the numpy indexing operations to get a response more similar to
the one obtained using correlate? Give the answer in your written response.
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5 Deliverables
The assignment should be submitted to canvas. The writeup should be submitted as a pdf named
<ustlogin-id>.pdf. The code should be submitted as a zip named <ustlogin-id>.zip. The zip
when uncompressed should produce the following files.
• SubtractDominantMotion.py
• InverseCompositionAffine.py
• testCarSequence.py
• testSylvSequence.py
• testCarSequenceWithTemplateCorrection.py
• testAerialSequence.py
• carseqrects.npy
• carseqrects-wcrt.npy
• sylvseqrects.npy
• aerialseqrects.npy
Do not include the data directory in your submission.
8 Assignment 3 Lucas-Kanade Visual Tracker1
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