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Homework 5
Computer Vision, Spring 2024
Due Date: March 22, 2024
Total Points: 10
This homework contains one programming challenge. All submissions are due at
midnight on March 22, 2024, and should be submitted according to the
instructions in the document “Guidelines for Programming Assignments.pdf”.
runHw5.py will be your main interface for executing and testing your code.
Parameters for the different programs or unit tests can also be set in that file.
Before submission, make sure you can run all your programs with the command
python runHw5.py with no errors.
The numpy package is optimized for operations involving matrices and
vectors. Avoid using loops (e.g., for, while) whenever possible—looping can
result in long running code. Instead, you should “vectorize” loops to optimize
your code for performance. In many cases, vectorization also results in more
compact code (fewer lines to write!).
Challenge 1: Your task is to develop a vision system that recovers the shape,
surface normal and reflectance of an object. For this purpose you will use
photometric stereo.
You will be given 5 images of an object taken using five different light sources. Your
task is to compute the surface normal and albedo for the object. For this purpose,
however, you will need to know the directions and intensities of the five light
sources. Thus, in the first part of the assignment, you will compute the light
sources directions and intensities from 5 images of a sphere and use this
information in the second part to recover the orientation and reflectance.
The 11 images, sphere0…sphere5, and vase1…vase5 are provided to you.
Before you begin, pay attention to the following assumptions you can make about
the capture settings:
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• The surfaces of all objects (including the sphere) are Lambertian. This means
there are only diffuse peaks in the reflectance maps (no specular
components).
• For the images, assume orthographic projections.
• Image files with the same indices are taken using the same light source. For
example, sphere1 and vase1 are taken using light source number 1 only.
• The objects maintain the same position, orientation and scale through the
different images – the only difference is the light source. For example, the
sphere in sphere0…sphere5 has the same coordinates and the same radius.
• The light sources are not in singular configuration, i.e., the S-matrix that you
will compute should not be singular.
• You may NOT assume that the light sources are of equal intensities. This
means that you need to recover not only the directions of the light sources
but also their intensities.
• The background in the image is black (0 pixel value) in all images.
The task is divided into four parts, each corresponding to a program you need to
write and submit.
a. First you need to find the locations of the sphere and its radius. For this
purpose you will use the image sphere0, which is taken using many light
sources (so that the entire front hemisphere is visible).
Write a program findSphere that locates the sphere in an image and
computes its center and radius.
center, radius = findSphere(input_img)
Assuming an orthographic project, the sphere projects into a circle on the
image plane. Find the location of the circle by computing its centroid. Also
estimate the area of the circle and from this, compute the radius of the circle.
Report your results in README file.
You may use skimage.filters.threshold_otsu and
skimage.measure.regionprops.
(1 points)
b. Now you need to compute the directions and intensities of the light sources.
For this purpose you should use the images sphere1…sphere5.
Derive a formula to compute the normal vector to the sphere’s surface at a
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given point, knowing the point’s coordinates (in the image coordinate frame),
and the center and radius of the sphere’s projection onto the image plane
(again, assume an orthographic projection). This formula should give you the
resulting normal vector in a 3-D coordinate system, originating at the
sphere’s center, having its x-axis and y-axis parallel respectively to the x-axis
and the y-axis of the image, and z-axis chosen such as to form an orthonormal
left-hand coordinate system. Don’t forget to include your formula in your
README file.
Write a program computeLightDirections that uses this formula, along
with the parameters computed in (a), to find the normal to the brightest
surface spot on the sphere in each of the 5 images. Assume that this is the
direction of the corresponding light source (Why is it safe to assume this?
State this in the README).
Finally, for the intensity of the light source, use the magnitude (brightness) of
the brightness pixel found in the corresponding image. Scale the direction
vector so that its length equals this value.
light_dirs_5x3 = computeLightDirections(center, radius,
img_list)
Center and radius are the resulted computed in (a). img_list contains the
5 images of the sphere. The resulting light_dirs_5x3 is a 5x3 matrix. Row i
of light_dirs_5x3 contains the x-, y-, and z-components of the vector
computed for light source i. (2 points)
c. Write a program computeMask to compute a binary foreground mask for the
object. A pixel in the mask has a value 1 if it belongs to the object and 0 if it
belongs to the background. Distinguishing between the foreground and
background is simple: if a pixel is zero in all 5 images, then it is background.
Otherwise, it is foreground.
mask = computeMask(img_list)
The img_list contains the 5 images of an object and mask is the binary
image mask. (1 points)
d. Write a program computeNormals that, given 5 images of an object,
computes the normals and albedo to that object’s surface.
normal, albedo_img = computeNormals(light_dirs, img_list,
mask)
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You may want to use the formula given in the class lecture notes. Be careful
here! Make sure to take into account the different intensities of the light
source.
Photometric stereo requires the object to be lit by at least 3 light sources.
However, in our case, we have a total of 5 light sources. The lighting has been
arranged in such a way that all visible surface points on an object are lit by at
least 3 light sources. Therefore, while computing the surface normal at a
point, choose the 3 light sources for which the point appears brightest. Be
careful – choosing the wrong light sources will result in erroneous surface
normal. (You may also decide to choose more than 3 light sources to compute
the surface normal. This results in an over-determined linear system and can
provide robust estimates. However, such a computation is not mandatory.)
Do not compute the surface normal for the background. You can assume that
the surface normal in this region is looking toward the camera. Use the mask
generated in the previous program to identify whether a given pixel
corresponds to the object or the background.
Scale the albedo up or down to fit in the range 0…1 and show them in the
output image. Thus each pixel in the output image should be the pixel’s
albedo scaled by a constant factor. (6 points)
At this point you can use the outputs of your program to reconstruct the
surface of the object. demoSurfaceReconstruction and
reconstructSurf demonstrate how to use the Frankot-Chellappa
algorithm to compute the 3D shape from surface normals. Surface
reconstruction is provided as a demo--no submission is required.