# Parallel Computation of the Minimum Separation Distance of Bezier Curves and Surfaces

Software and s/w Development

Dec 1, 2013 (4 years and 5 months ago)

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Parallel Computation of the Minimum
Separation Distance of Bezier Curves
and Surfaces

Lauren Bissett,

lauren.bissett@uconn.edu

Nicholas Woodfield,
nicholas.woodfield@uconn.edu

REU Biogrid,
Summer 2009

University of
Connecticut

Storrs, CT 06269

Presentation Summary

Background

Overview

CUDA

Bezier Curves

Bezier Surfaces

Benefits/Future work

Background

Animation relies on continuous movement between
frames to give the illusion of motion.

When an unexpected change occurs between frames,
this is known as temporal aliasing.

Background

One method of dealing with temporal aliasing is
detecting self
-
intersections in geometric objects.

If self
-
intersections are unnoticed by an animator, it
could lead to the animation looking 'off'.

A scientist looking at a visual of a complicated
molecule likewise could not determine if there are
self
-
intersections.

Background

One method to detect self
-
intersections is finding a
geometric object's
minimum separation distance
.

The minimum separation distance is the smallest
doubly normal segment

between any two points on
the object.

A segment is doubly normal if both its endpoints are
normal to the curve/surface.

Background

The algorithm for finding the minimum separation
distance is time
-
consuming and not practical for use.

Therefore NVIDIA's CUDA parallel programming
langauge will be used to implement the algorithms
so that they run fast enough for practical use.

CUDA

CUDA was developed by NVIDIA for parallel
programming on the GPU.

Allows the launch of thousands of parallel threads of
execution.

Unlike most GPU programming, CUDA
development is relatively easy, because it extends C
(with additional bindings for C++ and Fortran).

It also allows for
general

programming, no prior
knowledge of graphics required

CUDA Memory Model

Threads

basic
element of
execution, has own
register and local
memory

Blocks

composed of
threads, has own
shared memory

Grid

composed
of blocks, contains
global, constant and
texture memory

Our Machine

The machine used for this research used a
QuadroFX 5800 GPU running Windows Vista.

512 threads per block

32 registers per thread

Minimum Separation Distance

To find the minimum separation distance, there are
three major steps:

Generate candidate double normal segments

Newton's method

Find segment of minimum segment length

Finding Candidate Segments

The method of finding potential double normal
segments is simple but time
-
consuming.

For a given sample size (
n
), take
n

sample points and
pair with all other points in the sample.

Then check if each pair is creates a double normal
segment.

Finding Candidate Segments

Newton's Method

The initial estimates are run on Newton's method
until they converge.

When finished, the double normal segment is
compared to determine if it is of smallest length.

Bezier Curves

Curve case implemented first

It's simple

only two parametric values

(s,t)

to
worry about

Bezier Curve Kernel Organization

Used a 1
-
Dim grid composed of 1
-
Dim blocks

Blocks divided into groups, one for each curve

Threads responsible for one s parametric value on
only one curve

Each thread tests its sample points vs all other points
on it's parent curve and all other curves

If the candidate segment passes the double normal
test, we run newton's method on it

Each thread ultimately returns the shortest segment
from its search

Bezier Curve Kernel Organization

What problems can arise?

Sample size > maximum thread allowance

Register usage

Must ensure consistency among multiple blocks

E.g. If we used a thread's built in index value,

and we used 2 blocks per curve, each block

would then only test values between 0 and .5!

This required equations to calculate which thread
belonged to whom

Bezier Curve Kernel Results

Double normals returned in a 2D array

Rows are blocks

Columns are threads in a block

Launched a second kernel to collapse the array into a
1
-
Dim array the size of the # of blocks

Iterated over those segments, and found the shortest
which is our minimum separation distance

Results

8x to 200x speedup

n

e1

e2

C code

CUDA code

512

0.4

0.4

28s

154.76ms

512

0.2

0.2

26s

154ms

200

0.4

0.4

4.3s

46.6ms

10

0.4

0.4

15ms

1.81ms

Bezier Surfaces

Next we moved onto surfaces

bicubic bezier
meshes.

Bezier Surfaces

Surfaces are just an extension of curves, but with
two parametric values

u and v.

Each thread in the kernel handles a u,v pair. It then
checks against all other u,v pairs on the surface.

Blocks were extended into 2 Dimensions, to
represent the unit square.

Bezier Surfaces

Similar to curves, the thread determines if the
segment is doubly normal, and if necessary, discards
an old one of greater length if it finds a short one.

And again, Newton's method is run on each doubly
normal segment.

Finally, a similar search through results to find the
minimum separation distance

Bezier Surfaces

The thread results are again searched for the smallest
segment, which is the minimum separation distance.

Results?

The mesh algorithm was largely completed in the
last few days

Curve case, although simple, took two weeks of
tweaking until the code was sastifactory

Mesh case still needs some tweaking

Still confident we'll observe similar results to the
curve case

However, we do not have comparable C code to
compare results

Benefits/Future work

How does this tie into bio
-
grid?

Nature of the problem:

Self
-
intersections in molecule simulations are of
interest

Leveraging the GPU for general programming

'Supercomputing for the masses'

Tremendous speedups for low costs

Useful for when supercomputing power is not
present nor available

Future work: Finish the mesh case!

Questions and Answers

Any Questions?

Images

CUDA Memory Model, Dr. Dobbs Supercomputing
for the Masses:
http://www.ddj.com/architect/208401741?pgno=3

Double normal segments & surfaces double normals,
Ed Moore's Ph.D. Thesis

Newton's method:
http://en.wikipedia.org/wiki/Newton%27s_method

Bezier surface w/control points:
http://www.cs.cf.ac.uk/Ralph/graphicspics/bez.GIF