5.
Outline
• a bit about me
• computers & sight
• medical imaging and medialness
6.
Outline
• a bit about me
• computers & sight
• medical imaging and medialness
• relative critical sets
7.
Outline
• a bit about me
• computers & sight
• medical imaging and medialness
• relative critical sets
• subsequent work
8.
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.”
9.
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
12.
Computers & Sight
Semi-Autonomous Vehicles Descriptive and
Diagnostic Medicine
13.
Computers & Sight
Semi-Autonomous Vehicles Descriptive and
Diagnostic Medicine
Automatic Annotation of
Digital Content
14.
Computers & Sight
Semi-Autonomous Vehicles Descriptive and
Diagnostic Medicine
Automatic Annotation of Face Recognition,
Digital Content Motion Tracking, etc.
50.
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”
51.
Real Mathematical
World World
Assumptions Mathematical
about Phenomena Model
Logical Consequences
Real (Analyze Model)
Data
52.
Real Mathematical
World World
translate
Assumptions Mathematical
about Phenomena Model
Logical Consequences
Real (Analyze Model)
Data
53.
Real Mathematical
World World
translate
Assumptions Mathematical
about Phenomena Model
Logical Consequences
Real (Analyze Model)
Data
54.
Real Mathematical
World World
translate
Assumptions Mathematical
about Phenomena Model
Logical Consequences
Real (Analyze Model)
Data compare
55.
Real Mathematical
World World
translate
Assumptions Mathematical
about Phenomena Model
adjust assumptions
to improve
Logical Consequences
Real (Analyze Model)
Data compare
56.
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
60.
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
61.
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
62.
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)
63.
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)
64.
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)
65.
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
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