Computational Geometry and Geometric Shape Matching

What is Computational Geometry?What is Computational Geometry? Algorithms for geometric objects

Convex HullConvex Hull Given a set of pins on a pinboard

And a rubber band around them

How does the rubber band look when it snaps tight?

Convex HullConvex Hull Given a set of pins on a pinboard

And a rubber band around them

How does the rubber band look when it snaps tight?

Voronoi DiagramVoronoi Diagram Given all post offices in San Antonio

Find a subdivision of San Antonio into cells such that points in a cell are all closest to one post office

Voronoi DiagramVoronoi Diagram Given all post offices in San Antonio

Find a subdivision of San Antonio into cells such that points in a cell are all closest to one post office

Security: Art GallerySecurity: Art Gallery Given an art gallery

How many guards do you need to guard the whole gallery? Where should they be located?

Data basesData bases Given a set of points (data sets) in high dimensional space

Preprocess them such that orthogonal range queries can be answered efficiently.

Geometric Shape MatchingGeometric Shape Matching Consider geometric shapes to be composed of a number of basic objects

Geometric Shape MatchingGeometric Shape Matching Consider geometric shapes to be composed of a number of basic objects such as points

Geometric Shape MatchingGeometric Shape Matching Consider geometric shapes to be composed of a number of basic objects such as points line segments

Geometric Shape MatchingGeometric Shape Matching Consider geometric shapes to be composed of a number of basic objects such as points line segments triangles

Geometric Shape MatchingGeometric Shape Matching Consider geometric shapes to be composed of a number of basic objects such as

How similar are two geometric shapes?

points line segments triangles

Geometric Shape MatchingGeometric Shape Matching Consider geometric shapes to be composed of a number of basic objects such as

How similar are two geometric shapes?

points line segments triangles

Choice of distance measure Full or partial matching Exact or approximate matching Transformations (translations, rotations, scalings)

Computer-Aided NeurosurgeryComputer-Aided Neurosurgery

FU Berlin, Functional Imaging Technologies GmbH and the medical school ‘Benjamin Franklin’ at FU Berlin

BackgroundBackground Computer assisted neuro surgery (esp. brain tumor surgery)

BackgroundBackground Computer assisted neuro surgery (esp. brain tumor surgery) Before Surgery:

Functional MR scan of the brain

3D model of the brain

BackgroundBackground Computer assisted neuro surgery (esp. brain tumor surgery) Before Surgery:

Functional MR scan of the brain

3D model of the brain

During Surgery:

BackgroundBackground Computer assisted neuro surgery (esp. brain tumor surgery) Before Surgery:

Functional MR scan of the brain

3D model of the brain

During Surgery:

Electromagnetic pointing device

Display positions in 3D model

BackgroundBackground Computer assisted neuro surgery (esp. brain tumor surgery) Before Surgery:

Functional MR scan of the brain

3D model of the brain

During Surgery:

Electromagnetic pointing device

Display positions in 3D model

Navigation aid mapping positions in the brain to a prerecorded 3D MR image of the brain

Landmark Registration Landmark Registration Set of markers attached to patient’s head

3D model during surgeryimage world

Small but very noisy point sets

Find nearly rigid motion that maps image markers to world markers

Rigid Point MatchingRigid Point Matching P={p1,p2,…,pn} Q={q1,q2,…,qm} point sets in R3

P Q

Rigid Point MatchingRigid Point Matching P={p1,p2,…,pn} Q={q1,q2,…,qm} point sets in R3

P Q

Rigid matching maps edges with same length onto each other

Rigid Point MatchingRigid Point Matching P={p1,p2,…,pn} Q={q1,q2,…,qm} point sets in R3

P Q

Rigid matching maps edges with same length onto each other

Nearly rigid matching maps edges with similar lengths onto each other

Scoring TableScoring Table

Edges with similar lengths indicate a possible matching of and or vice versa

For each pair of similar edges,

increase the score of all pairs of

involved endpoints

pi

pj

qu qv

Maintain score for each pair

indicating the “quality” of matching

those two points

p1 pn

q1

qm

• • •

• • •

Scoring TableScoring Table

Edges with similar lengths indicate a possible matching of and or vice versa

Maintain score for each pair

indicating the “quality” of matching

those two points

For each pair of similar edges,

increase the score of all pairs of

involved endpoints

pi

pj

qu qv

p1 pi pj pn

q1

qu

qv

qm

Finding a TransformationFinding a Transformation Extract combinatorial matching

from scoring table

Least-Squares Approximation:

Find affine transformation A that minimizes the sum of the squared distances between corresponding points

Test if A is nearly rigid (check determinant, unit vector images, etc.)

Computer-Aided Neurosurgery: Computer-Aided Neurosurgery: SummarySummary

Direct linear algebra approaches were numerically very unstable

Geometric approach of splitting the problem into - finding the combinatorial matching and then- computing the nearly rigid transformation

is very easy to implement and proved to be very robust.

The algorithm is integrated into a commercial product and used in practice.

Protein Gel MatchingProtein Gel Matching

FU Berlin, UofA, German Heart Center Berlin

2D Gel Electrophoresis2D Gel Electrophoresis

Two-dimensional Gel Electrophoresis (2DE) is

an important method in proteome research

a high resolution technique which is capable to separate thousands of proteins from a tissue sample

2D Gel Electrophoresis2D Gel Electrophoresis

2D Gel Electrophoresis2D Gel Electrophoresis

Proteins are concentrated in so called spots of (axis- parallel) elliptic shape

2D Gel Electrophoresis2D Gel Electrophoresis

Proteins are concentrated in so called spots of (axis- parallel) elliptic shape

Protein analysis by mass spectrometry (expensive)

2D Gel Electrophoresis2D Gel Electrophoresis

2D Gel Electrophoresis2D Gel Electrophoresis

Gel Matching

Protein identification by gel image comparison is faster and not expensive

The Algorithmic ApproachThe Algorithmic Approach

Spot detection

Assign to each spot the coordinates of its center point and its intensity

Make use of ideas and methods from Computational Geometry:

Point pattern matching

Consider a gel as a point pattern. Then the problem reduces to a partial approximate point pattern matching.

GPS Curve LocationGPS Curve Location

FU Berlin and UofA and UTSA

Given:• A geometric graph G

(embedded in R2 with line segments)

• A polygonal curve

Task:Find a path in G that

is the most similar to

Finding a Curve in a MapFinding a Curve in a Map

Given:• A geometric graph G

(embedded in R2 with line segments)

• A polygonal curve

Task:Find a path in G that

is the most similar to

Finding a Curve in a MapFinding a Curve in a Map

• Consider :– A given roadmap, and– a sequence of GPS positions obtained from a person

travelling on some of the roads while recording her positioning information using a GPS receiver

polygonal curve

• Problem: – The noise of the GPS receiver distorts the polygonal curve

inherently

• Task: – Find the roads in the roadmap that have been traveled

Application: Map ConstructionApplication: Map Construction