Description
1 Lucas-Kanade Tracking
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
1
p = [px, py]
T
to obtain another rectangle on frame It+1, so that the pixel squared difference
in the two rectangles is minimized:
p
∗ = arg minp =
X
x∈N
||It+1(x + p) − It(x)||2
2
(2)
=
It+1(x1 + p)
.
.
.
It+1(xD + p)
−
It(x1)
.
.
.
It(xD)
2
2
(3)
Q1.1 (5 points) (Talha) 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) (Rishi) Implement a function with the following signature
LucasKanade(It, It1, rect, p0 = np.zeros(2))
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., it 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) (Rishi) 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
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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, 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) (Talha) 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 with a similar functionality to Q1.3
named testCarSequenceWithTemplateCorrection.py, 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 blue rectangles are created with the baseline tracker in Q1.3, the red 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 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 of aerial views
of moving vehicles from a non-stationary camera.
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2.1 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
. (6)
One can represent this affine warp in homogeneous coordinates as,
x˜
0 = Mx˜ (7)
where,
M =
1 + p1 p2 p3
p4 1 + p5 p6
0 0 1
. (8)
Here M represents W(x; p) in homogeneous coordinates as described in [1]. Also 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 the region common to It and the warped version of It+1 when
forming the linear system at each iteration.
Q2.1 (15 points) (Harsh) Write a function with the following signature
LucasKanadeAffine(It, It1)
which returns the affine transformation matrix M, and It and It1 are It and It+1 respectively.
LucasKanadeAffine should be relatively similar to LucasKanade from the first section (you
will probably also find scipy.ndimage.affine transform helpful).
2.2 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.
Q2.2 (10 points) (Harsh) Using the function you have developed for dominant motion
estimation, write a function with the following signature
4
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.
Q2.3 (10 points) (Shumian) 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 3. 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 3: Lucas-Kanade Tracking with Motion Detection
3 Efficient Tracking
3.1 Inverse Composition
The inverse compositional extension of the Lucas-Kanade algorithm (see [1]) has been used
in literature to great effect for the task of efficient tracking. When utilized within tracking
it attempts to linearize the current frame as,
It(W(x; 0 + ∆p) ≈ It(x) + ∂It(x)
∂xT
∂W(x; 0)
∂pT ∆p . (9)
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
||A0∆p − b
0
||2
2
(10)
for the specific case of an affine warp where p ← M and ∆p ← ∆M this results in
the update M = M(∆M)
−1
. Just to clarify, the notation M(∆M)
−1
corresponds to
W(W(x; ∆p)
−1
; p) in Section 2.2 from [2].
Q3.1 (15 points) (Shumian) Reimplement the function LucasKanadeAffine(It,It1)
as
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?
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4 Deliverables
The assignment (code and writeup) should be submitted to canvas. The writeup should also
be submitted to Gradescope named hw6.pdf. The code should be submitted
as a zip named .zip. The zip when uncompressed should produce the following
files.
• LucasKanade.py
• LucasKanadeAffine.py
• SubtractDominantMotion.py
• InverseCompositionAffine.py
• testCarSequence.py
• testCarSequenceWithTemplateCorrection.py
• testAerialSequence.py
• carseqrects.npy
• carseqrects-wcrt.npy
Do not include the data directory in your submission.
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