Problem Set 5: Optic Flow


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CSx495 — Computer Vision
Problem Set 5: Optic Flow
In class we discussed optic flow as the problem of computing a dense flow field where a flow field
is a vector field < u(x, y), v(x, y) >. We discussed a standard method — Hierarchical Lucas and
Kanade — for computing this field. The basic idea is to compute a single translational vector over
a window centered about a point that best predicts the change in intensities over that window by
looking a the image gradients.
For this problem set you will implement the necessary elements to compute the LK optic flow field.
This will include the necessary functions to create a Gaussian pyramid.
As a reminder as to what you hand in: A Zip file that has
1. Images (either as JPG or PNG or some other easy to recognize format) clearly labeled using
the convention PS¡number¿–¡question number¿–¡question sub¿–counter.jpg
2. Code you used for each question. It should be clear how to run your code to generate the
results. Code should be in different folders for each main part with names like PS1-1-code.
For some parts – especially if using Matlab – the entire code might be just a few lines.
3. Finally a PDF file that shows all the results you need for the problem set. This will include
the images appropriately labeled so it is clear which section they are for and the small number
of written responses necessary to answer some of the questions. Also, for each main section,
if it is not obvious how to run your code please provide brief but clear instructions. If there
is no Zip file you will lose at least 50%!
This project uses files stored in the directory
ProblemSets/PS5 There are a few sequences there. First, there is a test sequence called TestSeq
that just translates a textured rectangle over a textured background. In that directory you’ll find
the images Shift0, ShiftR2, ShiftR10, ShiftR20, ShiftR40 and ShiftR5U5. You’ll use these
images to test your code and to see the effect of the hierarchy. DataSeq1 has 3 frames of the
so-called ”Yosemite” sequence and DataSeq2 has larger displacements. There are also two more
intereting sequences: taxi and Juggle.
1 Gaussian and Laplacian Pyramids
In class we described the Gaussian pyramid constructed using the REDUCE operator. On the class
website’s Problem Sets main page there is a link to the original Burt and Adelson Laplacian Pyramid
paper that defines the REDUCE and EXPAND operators. And it’s in the main directory given above.
1.1 Write a function to implement REDUCE. Use this to produce a Gaussian Pyramid of 4 levels
(0-3). Demonstrate using the first frame of the DataSeq1 sequence.
Output: The code, and the 4 images that make up the Gaussian Pyramid
Although the Lucas and Kanade method does not use the Laplacian Pyramid, you do need to
expand the warped coarser levels (more on this in a minute). Therefore you will need to implement
the EXPAND operator. Once you have that the Laplacian Pyramid is just some subtractions.
1.2 Write the function EXPAND. Using it and the above REDUCE, create the 4 level Laplacian pyramid
of DataSeq1 (which has 1 Gaussian image and the 3 Laplacian images).
Output: The code and the Laplacian pyramid images (3 Laplacian images and 1 Gaussian
image) of the first image of DataSeq1.
2 Lucas Kanade optic flow
Next you need to implement the basic LK step. Given the two operators above and code to create
gradient images (you did those last problem set) you can implement the Lucas and Kanade optic
flow algorithm. Recall that we compute the gradients Ix and Iy and then over a window centered
around each pixel we solve the following:




Remember a weighted sum could be computed by just filtering the gradient image (or the gradient
squared or product of the two gradients) by a funciton like a 5×5 box filter or a 5×5 smoothing
filter (wide Gaussian) instead of actually looping. Convolution is just a normalized sum.
2.1 Write the code to do the LK optic flow estimation. For each pixel you solve the equation above.
You can display the recovered values a variety of ways. The easiest is to make two new images
U and V that are the x and y displacements [u(x, y) and v(x, y)]. These are displacement
images. You can also use the MATLAB quiver function which draws little arrows – though
you may have to scale the values to see the arrows. TestSeq has images of a smoothed image
texture with the different texture center rectangle displaced by a set number of pixels. Shift0
is the “base” image; images listed as ShiftR2 have the center portion shifted to the right by
2 pixels; ShiftR5U5 have the center portion shifted to the right by 2 pixels and up 5, etc.
When you try your code on the images TestSeq you should get a simple translating rectangle
in the middle and everything else zero. Try your LK on the base image and the ShiftR2 and
between the base image and ShiftR5U5. Remember LK only works for small displacements
with respect to the gradients so you might have to smooth your images a little to get it to
work; try to find a blur amount that works for both cases but keep it as little as possible.
Output: The code and the images showing the x and y displacements either as images (make
sure you scale them so you can see the values) or as arrows when computing motion between
(1) the base and ShiftR2 and (2) the base and ShiftR5U5. If you blur (smooth) the images
say how much you did.
2.2 Now try the code comparing the base image Shift0 with the remaining images of ShiftR10,
ShiftR20 and ShiftR40. Use the same amount of blurring as you did in the previous
section. Does it still work? Does it fall apart on any of the pairs?
Output: The code and the images showing the x and y displacements either as images or as
arrows when applied to ShiftR10, ShiftR20 and ShiftR40. Describe your results.
Next you’ll try it on the two Data sequences. You’ll need to determine a level of the pyramid where
it seems to work. To test your results you should use the recovered motion field to warp the second
image back to the first (or the first to the second).
The challenge in this is to create a Warp function and then use it correctly. This is going to be
somewhat tricky. I suggest you try and use the test sequence or some simple motion sequence
you create where it’s clear that a block is moving in a specific direction. The first question is are
you recovering the amount you need to move to bring the second image to the first or the first
to the second. If you look carefully at the slides you’ll see we’re solving for the amount that is
the change from I1 to I2 Consider a case the image moves 2 pixels to the right. This means that
I2(5, 7) = I1(3, 7) where I am indexing by x, y and not by row and column. So to warp I2 back to
I1 to create a new image W, would set W(x, y) to the value of I2(x + 2, y). The W would align
with I1.
MATLAB has a function to do this interpolation: INTERP2. To use this, you’ll need to understand
the function MESHGRID – which tends to think about matrices as X and Y not rows and columns.
So the call:
creates a matrix called x which is just repeated columns of the x value – the column index, and y
is the row index. In this case, if M – the number of rows – is 4 and N is 3, then x and y are 4×3.
This can be confusing. Another way to think about it is that these are the x and y values(column
and row) of the (i,j) locations. So if you want to get a value from somewhere near by in the image,
you would add the displacement to this value. This can be seen in the following very good warp
function (courtesy Yair Weiss when still a student):
function [warpI2]=warp(i2,vx,vy)
% warp i2 according to flow field in vx vy
% this is a “backwards” warp: if vx,vy are correct then warpI2==i1
warpI3=interp2(x,y,i2,x+vx,y+vy,’*nearest’); % use Matlab interpolation routine
warpI2=interp2(x,y,i2,x+vx,y+vy,’*linear’); % use Matlab interpolation routine
In OpenCV there is the function remap which behaves similarly.
2.3 Apply your single-level LK code to theDataSeq1 sequence (from 1 to 2 and 2 to 3). Because
LK only works for small displacements, find a Gaussian pyramid level that works for these.
To show that it works, you will show the output flow fields as above and you’ll show a warped
version of image 2 to the coordinate system of image 1. That is, you’ll warp Image 2 back
into alignment with image 1. If you flicker (rapidly show them back and forth) the warped
image 2 and the original image 1, you should see almost no motion: the second image pixels
have been ”moved back” to the location they came from in the first image. Same is true for
image 2 to 3. Next try this for DataSeq2 where the displacements are greater. You will likely
need to use a coarser level in the pyramid (more blurring) to work for this one.
Note: for this question you are only comparing between images at some chosen
level of the pyramid! In the next section you’ll do the hierarchy.
Output: The code and the images showing the x and y displacements for DataSeq1 and
DataSeq2 either as images or as arrows. Then, for each sequence, show the ***difference***
image between the warped image 2 and the original image 1.
3 Hierarchical LK optic flow
Recall that the basic steps of the hierarchy:
1. Given input images L and R. Initialize k = n where n is the max level.
2. REDUCE both input images to level k. Call these images Lk and Rk.
3. If k = n initialize U and V to be zero images the size of Lk; otherwise expand the flow field
and double (why?) to get to the next level: U = 2*EXPAND(U), V =2* EXPAND(V).
4. Warp Lk using U and V to form Wk.
5. Perform LK on Wk and Rk to yield two incremental flow fields Dx and Dy.
6. Add these to original flow: U = U + Dx and V = V + Dy.
7. If k > 0 let k = k − 1 and goto (2).
8. Return U and V .
When developing this code you should try it on TestSeq between the base and the larger shifts,
or you might try and create some test sequences of your own. Take a textured image and displace
a center square by a fixed translation amount. Vary the amount and make sure your hierarchical
method does the right thing. In principle, for a displacement of δ pixels, you’ll need to set n to (at
least) log2
3.1 Write the function to compute the hierarchical LK (surprised, huh?). First apply to the
TestSeq for the displacements of 10, 20 and 40 pixels. Then apply to both DataSeq1 and
Output: The code, the displacement images and the difference image between the warped I2
and the original I1 for each of the cases.
4 The Juggle Sequence
The sequence Juggle has significant displacement between frames — the juggled balls move significantly. Try your hierarchical LK on that sequence and see if you can warp frame 2 back to frame
4.1 Apply your hierarchical LK to the Juggle sequence.
Output: Displacement image (as image or quiver diagram) and the difference image between
the warped I2 and the original I1.
5 The Taxi Sequence
The sequence taxi has distinct moving objects. Use the output of hierarchical LK to create a
segmentation of the scene. A segmentation would be a set of image regions where the motion all
belongs to the same parametric flow, like translation (all pixels with same velocity) or similarity
(pixels movement is determined by 4 coefficients and the < x, y > value of the pixel.
5.1 Apply your hierarchical LK to the taxi sequence and then generate a segmentation map based
upon consistent motion in the regions. State your model (translation, similarity, affine, or

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