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Homework 1: Simplification and Subdivision Surfaces
The goal of this assignment is to try your hand at various mesh
processing tasks. A half-edge mesh adjacency data structure is
provided, along with a very simple .obj mesh parser. As you modify
the triangle mesh, it is critical that the manifold properties of the
input surface are maintained, and that the adjacency data structure
remains consistent. Throughout this assignment you are asked to
consider the efficiency of various mesh operations: is it quadratic,
linear, logarithmic, constant, etc? Since this assignment is rather
long, it's ok if you don't always implement the most efficient
strategy (which may require additional data structures), but please
discuss the tradeoffs in your README.txt file. Please use
the provided README.txt template.
Warning: This assignment has one small component (Gouraud shading)
& two larger components (simplification and subdivision). Start early,
and just get the basics working first. Spend a reasonable
amount of time coding and write about what you've learned in your
README.txt file. Also, to streamline grading, please
indicate which portions of the assignment are finished & bug free
(full credit), attempted (part credit) or not started (no credit) by
filling in the provided hw1_gradesheet.txt and submitting it with
your assignment. The TA will then check, edit as needed, and
finalize your assessment.
Download the provided source code and compile it as before. You
should be able to load simple .obj files and turn wireframe
visualization on & off by pressing "w".
NOTE: Grab the updated version the provided code for a better
wireframe visualization of the boundary and crease edges.
Try these command lines:
./mesher -input ../src/cube.obj -size 300
./mesher -input ../src/cube.obj -size 300 -wireframe
./mesher -input ../src/open_box.obj -size 300
./mesher -input ../src/open_box.obj -size 300 -wireframe
Notice that the adjacency data structure detects boundaries in the
mesh and draws them with thicker red lines in the wireframe visualization:
Now you're ready to start coding. First make use of the
adjacency data structure provided to implement gouraud
shading (we'll talk about this in Lecture 3). In gouraud
shading, average normals are computed for each vertex, these normals
are used for lighting, and the shaded values automatically
interpolated in GL. You can change the color and/or normal between
each specified vertex. This shading option should be enabled by
specifying -gouraud on the command line, or pressing "g".
./mesher -input ../src/bunny_1k.obj -size 500
./mesher -input ../src/bunny_1k.obj -size 500 -gouraud
Note that the average normals for each vertex can be efficiently
computed without any adjacency information at all, but that wouldn't
give you practice with the data structure! For extra credit, don't
average the normals across "sharp crease" edges (whose faces differ in
normal by more than a user-defined threshhold angle).
Comment on the efficiency & order notation of your implementation
in your README.txt.
Next, let's experiment with edge collapses, the key mesh
manipulation operation from the Progressive Mesh framework. When you
press the "d" key (for decimate, a.k.a. simplification) the number of
triangles in your mesh should be reduced by approximately 10 percent.
First implement the topology of an edge collapse. Identify a
target edge to collapse. For now, just pick a random edge (or any
edge) in the mesh. Using the adjacency data structure, identify which
two triangles will be removed permanently from the model, and which
other triangles will change. Choose a simple location for the
remaining vertex (e.g., it's old position, or a position averaged with
the deleted vertex). Note: in the provided adjacency data structure
it is recommended that you will delete and then re-create triangles
rather than modify them and the related edges. Initially you may want
to just collapse one edge each time the "d" key is pressed.
./mesher -input ../src/bunny_1k.obj -size 500 -wireframe
Even though a single random collapse works great, you will
probably quickly run into problems: 1) the mesh looks bad with small
triangles and big triangles that don't represent the high-resolution
model very well, 2) it creates self intersections of the surface (it
turns itself inside out and you can see the blue side), and 3)
(worst-of-all) it sometimes crashes with an error like:
assertion "opposite == NULL" failed: file "edge.h", line XXX
Think about what that error might indicate is wrong with your
implementation. (If you get a different error, of course, try to
explain that one.) Certain edges should not be collapsed
because it would cause the surface to become non-manifold. For
example, the "ring" of vertices surrounding the target edge should be
unique. In the diagram below, if vertices 1 and 4 refer to the same
vertex, then after the collapse of the solid black edge, four
triangles will meet at the same edge. This proposed edge collapse
will alter the topology of the model and yield a non-manifold
condition. Write code to check for these conditions. NOTE: This code
can be tricky. It's ok if you only just catch some of these
conditions and your program still crashes occasionally. Don't spend
too long trying to make it perfect. Discuss this in your
Now implement something to make a smarter choice for which edge to
collapse (this might also minimize the occurance of lingering bugs
from the previous part). Collapsing the shortest edge first (or
actually make that the shortest legal edge) will usually do a
very good job at preserving overall surface shape. Done naively,
repeatedly selecting the shortest edge for collapse can be an
expensive operation. That's fine for now, you can implement something
very inefficient. Discuss these performance issues in your
README.txt file and what common data structures would help.
Finally, implement the Loop Subdivision Surface scheme (we'll
talk about this in Lecture 4). When the "s" key is pressed, your
implementation should perform one iteration of subdivision. First you
want to get the topology correct. Add a vertex at the center of each
edge, then replace each triangle with 4 smaller triangles. Make sure
you share the new vertices across the edges and don't create multiple
copies of the vertex. Consider how to do this efficiently and discuss
various options in your README.txt file. The provided hash
table structure for tracking the "parents" of each newly added vertex
should be very helpful for this task. You can verify correct vertex
sharing by visualizing the wireframe and looking for the red boundary
edges. If you have extra red lines on your mesh (not at the
boundaries) you have not properly shared the vertices.
./mesher -input ../src/open_box.obj -size 300 -wireframe
Once you have the correct topology and vertex sharing, you can
adjust vertex positions according to the Loop Subdivision rules. To
handle general shapes you must implement the rules for regular
vertices (valence = 6, i.e., 6 edges meeting at a vertex),
extraordinary vertices (valence != 6), and boundary conditions. Here
are the Loop
Subdivision Rules from the SIGGRAPH
2000 course notes - Subdivision for Modeling and Animation (page 70).
Finish off your subdivision surface code by implementing the simple
binary (infinitely sharp or infinitely smooth) crease method described
in: "Piecewise Smooth Surface Reconstruction."
H. Hoppe, T. DeRose, T. Duchamp, M. Halstead, H. Jin, J. McDonald, J. Schweitzer, W. Stuetzle.
ACM SIGGRAPH 1994.
./mesher -input ../src/creased_cube.obj -size 300 -wireframe
Ideas for Extra Credit
Include a short paragraph in your README.txt file
describing your extensions and sample command lines to demo them.
Implement gouraud shading with a proper average weighted by
angle for the triangles meeting at each vertex (if you haven't done
this already) and a crease angle threshhold. Find or create a mesh
which illustrates these differences.
Implement a "better" edge weight selection method. First define
your goal for simplification and construct an edge weight function
that matches that goal. E.g., if your goal is to create equilateral
triangles than the shortest edge collapse is a very good choice.
Discuss in your README.txt.
Implement a better positioning method for the remaining vertex in
an edge collapse. Discuss in your README.txt.
Implement the geomorph option to smoothly transition
between two different resolutions within a progressive mesh.
Improve the efficiency of your implementation by using the right
data structures for various operations (hash tables, priority queues,
Implement the integer crease weight control described by "Subdivision
Surfaces in Character Animation", DeRose, Kass & Truong, SIGGRAPH
1998 to perform n iterations of crease subdivision
rules followed by infinite iterations of smooth subdivision.
Implement the floating point weight control by interpolating
between the nearest integer iterations.
Implement the butterfly interpolating subdivision surface scheme.
NOTE: If you need it... here is the original version of code,
before a few bug fixes & before the wireframe shader rendering was
NOTE: If you are using GLUT instead of GLFW/GLEW/GLM, see
last year's page & last year's code.
Basic Code & Compilation Files(CMakeLists.txt,
Similar to Homework 0.
An implementation of the half-edge data structure. The load
function will parse simple .obj files.
For efficient, constant time access of parts of the data
structure, we store edges and vertex ancestry in a hashtable. It is
used by the Mesh code to efficiently find pairings between opposite
edges at load time. Unfortunately, the hash map or
unordered map is not quite standard in STL, and depending on
your platform, you may need to tweak this file.
.obj files (cube.obj, creased_cube.obj, open_box.obj,
bunny_200.obj, bunny_1k.obj, bunny_40k.obj, cactus.obj,
knot.obj, and hypersheet.obj)
Some models from: "Piecewise Smooth Surface Reconstruction.", Hoppe et al, 1994
Files for submission (README.txt and hw1_gradesheet.txt)
Fill out these files before submitting your assignment.
Please read the Homework information page again before submitting.