Description
1. Vanishing Points and Vanishing Lines [10 pts]1
. Consider a plane defined by NT X = d, that is undergoing
perspective projection with focal length f. Show that the vanishing points of lines on this plane lie on the
vanishing line of this plane.
1Adapted from Jitendra Malik.
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2. Sphere Under Perspective Projection [30 pts].2
(a) [20 pts] Under typical conditions, the silhouette of a sphere of radius r with center (X, 0, Z) under planar
perspective projection is an ellipse. Show that the eccentricity of this ellipse is X
√
X2 + Z2 − r
2
. Recall
that, under perspective projection a point (X, Y, Z) in 3D space maps to
f
X
Z
, f Y
Z
in the image, where
f is the distance of the image plane from the pinhole.
Hint: There are different ways you can solve this. One line of attack would be to compute the lengths of
major and minor axes of the projected ellipse, and compute eccentricity via e =
r
1 −
length of minor axis
length of major axis2
,
but there could be other simpler alternatives as well.
2Adapted from Jitendra Malik.
3
(b) [10 pts] Are there circumstances under which the projection could be a parabola or hyperbola? If yes,
write down the conditions on X, Z, r and f, for parabola and hyperbola respectively; if no, explain why.
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3. Shape from Shading [50 pts].
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The goal of this problem is to implement shape from shading as described in Lecture 4 and to start becoming
comfortable with python image and matrix processing and display functions.
You will need the following python libraries: numpy, matplotlib, jupyter, Pillow.
Download the data (http://saurabhg.web.illinois.edu/teaching/ece549/sp2020/mp/mp1-data.tgz), and use this
ipython notebook (http://saurabhg.web.illinois.edu/teaching/ece549/sp2020/mp/mp1.ipynb). The data consists
of 64 images each of four subjects from the Yale Face database. The light source directions are encoded in the
file names. We have provided utilities to load the data and display the output. Your task will be to implement
the functions preprocess, photometric_stereo and get_surface in the ipython notebook, as explained below.
(a) For each subject (subdirectory in croppedyale), read in the images and light source directions. This is
accomplished by the function LoadFaceImages which returns the images for the 64 light source directions
and an ambient image (i.e., image taken with all the light sources turned off).
(b) Preprocess the data: subtract the ambient image from each image in the light source stack, set any negative
values to zero, rescale the resulting intensities to between 0 and 1 by dividing 255 (they are originally
between 0 and 255).
Hint: These operations can be done without using any loops. You may want to look into the concept of
array broadcasting in numpy.
(c) Estimate the albedo and surface normals. For this, you need to fill in code in photometric_stereo, which
is a function taking as input the image stack corresponding to the different light source directions and the
matrix of the light source directions, and returning an albedo image and surface normal estimates. The
latter should be stored in a three-dimensional matrix. That is, if your original image dimensions are h×w,
the surface normal matrix should be h × w × 3, where the third dimension corresponds to the x-, y-, and
z-components of the surface normals. To solve for the albedo and the normals, you will need to set up a
linear system as shown in slide 19 of Lecture 4.
[15 pts] For each of the four subjects, display your estimated albedo maps and surface normals using
plot_albedo_and_surface_normals. When inserting results images into your report, you should resize/-
compress them appropriately to keep the file size manageable – but make sure that the correctness and
quality of your output can be clearly and easily judged. For each subject, you should also report the mean
residual of the least square fitting (a single number defined as the squared reconstruction error averaged
over all pixels and images).
Hint: To get the least-squares solution of a linear system, use numpy.linalg.lstsq function. If you
directly implement the formulation of slide 19 of the lecture, you will have to loop over every image pixel
and separately solve a linear system in each iteration. There is a way to get all the solutions at once by
stacking the unknown g vectors for every pixel into a 3 × npix matrix and getting all the solutions with
3Adapted from Svetlana Lazebnik.
5
a single call to numpy solver. You will most likely need to reshape your data in various ways before and
after solving the linear system. Useful numpy functions for this include reshape, expand_dims and stack.
(d) Compute the surface height map by integration. The method is shown in slide 22 of Lecture 4, except
that instead of continuous integration of the partial derivatives over a path, you will simply be summing
their discrete values. Your code implementing the integration should go in the get_surface function. As
stated in the slide, to get the best results, you should compute integrals over multiple paths and average the
results. You should implement the following variants of integration:
• Integrating first the rows, then the columns. That is, your path first goes along the same row as the
pixel along the top, and then goes vertically down to the pixel. It is possible to implement this without
nested loops using the cumsum function.
• Integrating first along the columns, then the rows.
• Average of the first two options.
• Average of multiple random paths. For this, it is fine to use nested loops. You should determine the
number of paths experimentally.
i. [10 pts] Discuss the differences between the different integration methods. Specifically, you should
choose one subject, display the outputs for all of four variants above with display_3d (be sure to
choose viewpoints that make the differences especially visible), and discuss which method produces
the best results and why. You should also compare the running times of the different approaches.
ii. [10 pts] For the remaining subjects, it is sufficient to simply show the output of your best method, and
it is not necessary to give running times. Display the 3D screenshots of height maps using display_3d.
Be sure to choose a viewpoint that makes the structure as clear as possible (and/or feel free to include
screenshots from multiple viewpoints).
(e) [15 pts] Discuss how the Yale Face data violate the assumptions of the shape-from-shading method covered
in the slides. What features of the data can contribute to errors in the results? Feel free to include specific
input images to illustrate your points. Choose one subject and attempt to select a subset of all viewpoints
that better match the assumptions of the method. Show your results for that subset and discuss whether
you were able to get any improvement over a reconstruction computed from all the viewpoints.
(f) [Upto 10 pts] Extra Credit
On this assignment, there are not too many opportunities for easy extra credit. This said, here are some
ideas for exploration:
• Generate synthetic input data using a 3D model and a graphics renderer and run your method on this
data. Do you get better results than on the face data? How close do you get to the ground truth (i.e.,
the true surface shape and albedo)?
• Investigate more advanced methods for shape from shading or surface reconstruction from normal
fields.
• Try to detect and/or correct misalignment problems in the initial images and see if you can improve
the solution.
• Using your initial solution, try to detect areas of the original images that do not meet the assumptions
of the method (shadows, specularities, etc.). Then try to recompute the solution without that data and
see if you can improve the quality of the solution.
In your report describe the improvements you tried along with relevant implementation details. Describe
the results you obtained using images, visualizations, and supporting quantitative metrics, as necessary.
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