Visual Perception,
Computation,
and Geometry
Jason Miller
Associate Professor of Mathematics
Truman State University
12 September 2009

Outline

Outline
• a bit about me

Outline
• a bit about me
• computers & sight

Outline
• a bit about me
• computers & sight
• medical imaging and medialness

Outline
• a bit about me
• computers & sight
• medical imaging and medialness
• relative critical sets

Outline
• a bit about me
• computers & sight
• medical imaging and medialness
• relative critical sets
• subsequent work

Me
• B.A. in math from small, private liberal arts
college
• Ph.D. in mathematics from University of
North Carolina
• area = differentiable topology & singularity
theory of René Thom
• “Relative Critical Sets in n-Space and their
application to Image Analysis.”

The miracle of appropriateness of the language of
mathematics for the formulation of the laws of [science] is a
wonderful gift which we neither understand nor deserve.
We should be grateful for it, and hope that it will remain
valid for future research, and that it will extend, for better
or for worse, to our pleasure even though perhaps also to
our bafflement, to wide branches of learning.
— Eugene Wigner, The Unreasonable
Effectiveness of Mathematics

Computers & Sight

Computers & Sight
Semi-Autonomous Vehicles

Computers & Sight
Semi-Autonomous Vehicles Descriptive and
Diagnostic Medicine

Computers & Sight
Semi-Autonomous Vehicles Descriptive and
Diagnostic Medicine
Automatic Annotation of
Digital Content

Computers & Sight
Semi-Autonomous Vehicles Descriptive and
Diagnostic Medicine
Automatic Annotation of Face Recognition,
Digital Content Motion Tracking, etc.

Computers & Sight
The secret is …

Computers & Sight
The secret is …
They Suck at it!

Computers & Sight
The secret is …
They Suck at it!
(they have no natural talent for sight)

Example: Captchas

Computers & Sight

Computers & Sight

Computers & Sight

Computers & Sight

Image Processing
• Challenges:
Segmentation and
Registration of Images
• Edge-based methods
• Medialness-based
methods

Medial Axis

Medial Axis

Medial Axis

Medial Axis

Medial Axis

Medial Axis

Medial Axis

Medial Axis

Medial Axis

Medial Axis
th
wid

Image Processing

Image Processing

Image Processing

Image Processing

Image Processing

Image Processing
• Digital images are
collections of pixels
• Each pixel has an
intensity
528 x 525 pixels
intensities: 0 ≤ I ≤ 255

Pixel intensity function

Pixel intensity function

Pixel intensity function

Pixel intensity function

Pixel intensity function

Pixel intensity function

Pixel intensity function
nsity values
Inte

Image
shapes

Image function
shapes geometry

Image
shapes
←→ function
geometry

Backstory: Why Me?
• high-powered computer science research group!
• they had algorithms computing medial axes of objects in
medical images
• dogged by some anomalous unexpected numerical
problems
• my advisor: “let’s figure out what should be happening”

Real Mathematical
World World
Assumptions Mathematical
about Phenomena Model
Logical Consequences
Real (Analyze Model)
Data

Real Mathematical
World World
translate
Assumptions Mathematical
about Phenomena Model
Logical Consequences
Real (Analyze Model)
Data

Real Mathematical
World World
translate
Assumptions Mathematical
about Phenomena Model
Logical Consequences
Real (Analyze Model)
Data

Real Mathematical
World World
translate
Assumptions Mathematical
about Phenomena Model
Logical Consequences
Real (Analyze Model)
Data compare

Real Mathematical
World World
translate
Assumptions Mathematical
about Phenomena Model
adjust assumptions
to improve
Logical Consequences
Real (Analyze Model)
Data compare

Relative Critical Sets
• They extended the concept of local extrema where
I=0
(vanishing derivative) to a higher dimensional set of
points.
• Let ei be the eigenvectors of the matrix of second
partials of I , and λi ≤ λi+1 be the eigenvalues.
I · ei = 0 for i < n
λn−1 < 0

Image
shapes

Image function
shapes geometry

Image
shapes
←→ function
geometry

Relative Critical Sets
• Used the following techniques to prove a
structure theorem for the CS’s group’s
medial axes
• wavelet theory (scale-space theory)
• Lie group actions
• transversality theorems
• semi-algebraic geometry
• combinatorics

Relative Critical Sets
• Used the following techniques to prove a
structure theorem for the CS’s group’s
medial axes
• wavelet theory (scale-space theory)
abstract
• Lie group actions mathematics in
service of
• transversality theorems
applied science
• semi-algebraic geometry
• combinatorics

Subsequent Work
• Undergraduate Research Project on
computing relative critical sets
• Applied wavelets to bat echolocation project
with Scott Burt (Biology)
• Use medialness methods in vascular network
project with Rob Baer (ATSU)

Subsequent Work
• Undergraduate Research Project on
computing relative critical sets ramming
Mathem atica prog
• Applied wavelets to bat echolocation project
with Scott Burt (Biology)
• Use medialness methods in vascular network
project with Rob Baer (ATSU)

Subsequent Work
• Undergraduate Research Project on
computing relative critical sets ramming
Mathem atica prog
• Applied wavelets to bat echolocation project
with Scott Burt (Biology) assific ation and
sta tistical cl ethods
cluster m
• Use medialness methods in vascular network
project with Rob Baer (ATSU)

Subsequent Work
• Undergraduate Research Project on
computing relative critical sets ramming
Mathem atica prog
• Applied wavelets to bat echolocation project
with Scott Burt (Biology) assific ation and
sta tistical cl ethods
cluster m
• Use medialness methods in vascular network
project with Rob Baer (ATSU)
grap h theor y
ramming
M atlab prog