Homework 5
Computer Vision, Fall 2021
Due Date: 10AM November 15, 2021
Total Points: 18
This homework contains one written assignment, one programming walkthrough
and one programming challenge. All submissions are due at 10:00 a.m. on
November 15, 2021. Follow the submission instructions in the document titled
“Guidelines for Homework Assignments.”Note that there is no credit for late
submissions. So please start working on the homework early.
Written Assignments
Problem 1: A Lambertian surface is illuminated simultaneously by two distant point
sources with equal intensity in the direction s1 and s2. Show that for all normals on
the surface that are visible to both sources, illumination can be viewed as coming
from a single“effective”direction s3. How is s3 related to s1 and s2? Now, if the two
distant sources have unequal intensities I1 and I2, respectively, what is the direction
and intensity of the“effective”source? (4 Points)
Problem 2: The reflectance map can be parameterized in various ways. In class we
have concentrated on using the gradient (p, q) as a means of specifying surface
orientation. In some cases, the Gaussian sphere is more suitable for this purpose.
Each point on the Gaussian sphere corresponds to a particular direction, from the
center of the sphere to that point. The orientation of a surface patch can be specified
by giving the direction of its surface normal. Thus a given surface orientation can be
identified with a particular point on the Gaussian sphere. The reflectance map is
merely a means of associating brightness with orientation.
a. What are the contours of constant brightness on the Gaussian sphere in the
case of a Lambertian surface illuminated by a point source? Hint: See Figure
1a. (2 Points)
b. Show that there are at most two surface orientations that give rise to a given
pair of brightness values when the photometric stereo method is applied to a
Lambertian surface. Assume that two different light sources are used. Hint:
1
See Figure 1b. (2 Points)
Figure 1: The reflectance map can be plotted on the Gaussian sphere. (a) The contours of constant
brightness are particularly simple for a Lambertian surface illuminated by a point source. (b) This
makes it easy to prove that there are at most two solutions to the two-source photometric stereo
problem in this case.
Programming Assignments
This programming assignment has one challenge (with its own subset of milestones
or unit tests). Instructions and summary corresponding to these are given below.
runHw5.m 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
runHw5(‘all’) with no errors.
MATLAB is optimized for operations involving matrices and vectors. Avoid
using loops (e.g., for, while) in MATLAB whenever possible—looping can result
in long running code. Instead, you should“vectorize [1]”loops to optimize your
code for performance. In many cases, vectorization also results in more
compact code (fewer lines to write!). If you are new to MATLAB, refer to these
articles [1] [2] on techniques to optimize MATLAB code.
2
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:
● 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).
3
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.
You may use the Image Processing Toolbox functions im2bw and
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
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
right-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_cell)
Center and radius are the resulted computed in (a). img_cell contains the
5 images of the sphere. The resulting light_dirs_5x3 is a 5×3 matrix. Row i
4
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_cell)
The img_cell 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_cell,
mask)
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)
5
At this point you can use the outputs of your program to reconstruct the
surface of the object. demoSurfaceReconstruction and
reconstructSurf.m demonstrate how to use the Frankot-Chellappa
algorithm to compute the 3D shape from surface normals. You can use the
MATLAB function surf to visualize the reconstructed surface. Surface
reconstruction is provided as a demo–no submission is required