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TRANSCRIPT
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Geometric Modeling Summer Semester 2012
Introduction Motivation · Topics · Basic Modeling Techniques
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Today...
Topics:
• Formalities & Organization
• Introduction: Geometric Modeling
Motivation
Overview: Topics
Basic modeling techniques
• Mathematical Background
Function Spaces
Differential Geometry
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Today...
Topics:
• Formalities & Organization
• Introduction: Geometric Modeling
Motivation
Overview: Topics
Basic modeling techniques
• Mathematical Background
Function Spaces
Differential Geometry
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Motivation
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Motivation
This lecture covers two related areas:
• Classic geometric modeling
• Geometry processing
Common techniques (math, models, terminology), but different goals
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Geometric Modeling
Geometric Modeling:
• You start with a blank screen, design a geometric model
• Typical techniques:
Triangle meshes
Constructive Solid Geometry (CSG)
Spline curves & surfaces
Subdivision surfaces
• Goal is interactive modeling
• Mathematical tools are designed with the user in mind
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Geometry Processing
Geometry Processing
• You already have a geometric model Typically: 3D scanner (read: not nice)
You need to process & edit the geometry
Complex, unstructured data
• Typical problems
Noise removal, filtering
Surface reconstruction
Registration
Statistical analysis (features, symmetry, hole-filling etc...)
Structure retrieval / data interpretation
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Two fields
Goals
• Geometric modeling
Create nice images
Build stuff
Long term goal: model anything
• Geometry processing
Data processing (think “Photoshop”)
Reverse engineering
Long term goal: shape understanding machines make sense of the world
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Examples:
Geometric Modeling
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The Modern World... designed on a computer
(the building)
designed on a computer as well (the cars)
fortunately, not (yet) designed on a computer
(the trees)
[c.f. Danny Hillis, Siggraph 2001 keynote]
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Impact of Geometric Modeling
We live in a world designed using CAD
• Almost any man-made structure designed w/computers
Architecture
Commodities
Your car (bikes as well)
Spline curves invented in automotive industry
Typesetting
• <advertising> Our abilities in geometric modeling shapes the world we live in each day. </advertising>
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Different Modeling Tasks
CAD / CAM
• Precision Guarantees
• Handle geometric constraints exactly (e.g. exact circles)
• Modeling guided by rules and constraints
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Different Modeling Tasks
Photorealistic Rendering
• Has to “look” good
• Ad-hoc techniques are ok
• Using textures & shaders to “fake” details
• More complexity, but less rigorous
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Examples:
Geometry Processing
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Geometry Processing
A rather new area
• Motivation: 3D scanning
3D scanners
Clouds of millions of measurement points
• Sources of spatial data:
Science: CT, [F]MRI, ET, Cryo-EM, ...
3D movie making
Game / movie industry: Servers with GBs of “polygon soup”
Crawl the internet
• Need to process the geometry further
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Photoshopping Geometry
Geometry Processing:
• Cleanup:
Remove inconsistencies
Make watertight (well defined inside/outside, for 3D printers)
Simplify – keep only the main “structure”
Remove noise, small holes, etc...
• Touch-up /Edit:
Texturing, painting, carving
Deformation
Stitch together pieces
• Lots of other stuff – similar to image processing
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Scan Registration
[data set: Stanford 3D Scanning Repository]
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Feature Tracking
Fully Automatic:
[Implementation: Martin Bokeloh (Diploma thesis)]
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Example
Example: The Stanford “Digital Michelangelo Project” [Levoy et al.: The Digital Michelangelo Project, Siggraph 2000]
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Scanning the World....
Example: The “Wägele” Laser scanners (2D sheets of distance measurments)
[Biber et al. 2005]
A pull-through measurement
device – can acquire complete
buildings in a few hours
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This is what you get...
Corridor – CS Building University of Tübingen (6.5 GB)
...lots of artifacts (the scanner does not really like windows)
CS Building, Outside (nicer colors...)
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Automatic Processing
Example: Automatic Outlier Removal
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Think Big
More Problems:
• Occluded areas, shiny / transparent objects holes (lots of holes, actually)
• Huge amounts of data (really huge)
City Scanning
• There are big companies trying to scan large areas
• Think Google Earth in full resolution
• How about a virtual online walk through New York, Tokyo, Saarbrücken?
• Lots of open research problems to get there
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[data set: Institute for Cartography, Leibnitz University Hannover]
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HUGE Data Sets
The Largest Data Set Currently I have On My Hard-Drive...
Data set: Outdoor Scan (structure from video) of a part of the UNC campus (2.2·109 pts / 63.5 GB), courtesy of J.-M. Frahm, University of North Carolina
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Geometry Processing
Examples of Our Own Research
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Symmetry Detection
[data sets: C. Brenner, IKG, Universität Hannover]
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Symmetry Detection
[data sets: C. Brenner, IKG, Universität Hannover]
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Reconstruction by Symmetry
overlay of
16 parts [data sets: C. Brenner, IKG, Universität Hannover]
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Results
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Results
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Regularity Aware Deformation
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Algebraic Resizing
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Scanning Moving Geometry
Real-time 3D scanners:
• Acquire geometry at video rates
• Capture 3D movies: “performance capture”
• Not done yet – highly active research area
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Kinect Example Data
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Animation Reconstruction
Problems
• Noisy data
• Incomplete data (acquisition holes)
• No correspondences
• Just point clouds
noise
holes
missing correspondences
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Animation Reconstruction
Remove noise, outliers
Fill-in holes (from all frames)
Dense correspondences
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Lecture Overview Topics
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Geometric Modeling 2012
Mathematical Background
• Function spaces
• Differential geometry
Geometric Modeling
• Smooth curves: polynomial interpolation & approximation, Bezier curves, B-Splines, NURBS
• Smooth surfaces: spline surfaces, implicit functions, variational modeling
• Meshes: meshes, multi-resolution, subdivision
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Geometric Modeling 2012
Geometry Processing
• 3D Scanner data
• Registration
• Surface reconstruction
• Point cloud processing
Preliminary List:
• Subject to change & reordering
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Overview
Modeling Techniques
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Geometric Modeling
What do we want to do?
empty space (typically 3)
geometric object
B 3
B
d
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Fundamental Problem
The Problem:
B
d
infinite number of points my computer: 4GB of memory
We need to encode a continuous model with a finite amount of information
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Modeling Approaches
Two Basic Approaches
• Discrete representations
“Pixels”
Fixed discrete bins
• “Continuous” representations
“Vector graphics”
Mathematical description
Evaluate continuously
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Discrete Representations
Discrete represenations
• Fixed Grid of values:
(i1, ..., ids) ds (x1, ..., xdt
) dt
• Typical scenarios:
ds = 2, dt = 3: Bitmap images
ds = 3, dt = 1: Volume data
ds = 2, dt = 1: Depth maps
• PDEs / Numerics “Finite Differences” models
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Modeling Approaches
Two Basic Approaches
• Discrete representations
“Pixels”
Fixed discrete bins
• “Continuous” representations
“Vector graphics”
Mathematical description
Evaluate continuously Designing Functions
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Continuous Models
Basic principle: procedural modeling
finite set of Shape Parameters
determines the object shape
Algorithm(s)
determines the class of objects that can
be represented
Query Parameters (a finite set of numbers from
a continuous set)
Answer
𝑓
𝐩
𝐱
𝑓𝐩(𝐱)
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Example: Continuous Model
Example: Sphere
• Shape Parameters: center, radius (4 numbers)
• Algorithms:
Ray Intersection (e.g. for display) – Input: Ray (angle, position: 5 numbers) – Output: {true, false}
Inside/outside test (e.g. for rasterization) – Input: Position (3 numbers) – Output: {true, false}
Parametrization (e.g. for display) – Input: longitude, latitude (, ) – Output: position (3 numbers)
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Example: Continuous Model
Example: Sphere
• Shape Parameters: center, radius (4 numbers)
• Algorithms:
Ray Intersection (e.g. for display) – Input: Ray (angle, position: 5 numbers) – Output: {true, false}
Inside/outside test (e.g. for rasterization) – Input: Position (3 numbers) – Output: {true, false}
Parametrization (e.g. for display) – Input: longitude, latitude (, ) – Output: position (3 numbers)
𝑓:ℝ5 → [0,1]
𝑓:ℝ3 → [0,1]
𝑓:ℝ2 → ℝ3
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So Many Questions...
Several algorithms for the same representation: • Parametrization – compute surface points according to
continuous parameters
• (Signed) distance computation – distance to surface of points in space, inside/outside test
• Intersection – with rays (rendering), other objects (collision detection)
• Conversion – into other representations.
• Many more...
And: algorithms to construct and alter models
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Continuous, Procedural Models
“Continuous” representations • Algorithm (math: function) describes the shape
Definition: finite number of continuous parameters
Query: finite number of continuous parameters
• Characteristics: More involved (have to ask for information)
Potentially “infinite” resolution (continuous model)
• Structural complexity limited by algorithm
This lecture: focus on these represenations
• Mathematically, we study function design
• Mostly linear design approaches
Just find a basis to a linear vector space, that’s all the magic
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Classes of Models
(Main) classes of models in this lecture:
• Primitive meshes
• Parametric models
• Implicit models
• Particle / point-based models
Remarks
• Most models are hybrid (combine several of these)
• Representations can be converted (may be approximate)
• Some questions are much easier to answer for certain representations
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Modeling Zoo
Parametric Models Primitive Meshes
Implicit Models Particle Models
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Modeling Zoo
Parametric Models Primitive Meshes
Implicit Models Particle Models
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Parametric Models
Parametric Models
• Function f maps from parameter domain to target space
• Evaluation of f gives one point on the model
u
v
(u, v)
f (u, v) f
ds S dt
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output: 1D output: 2D output: 3D in
pu
t: 3
D
inp
ut:
2D
in
pu
t: 1
D
u
f(t) t
function graph
x
t
plane curve
t
space curve
plane warp surface
space warp
y
x
y
z
u
v
x
y u
v
x
y
z
u v
y
z
w
x
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output: 1D output: 2D output: 3D in
pu
t: 3
D
inp
ut:
2D
in
pu
t: 1
D
u
f(t) t
function graph
x
t
plane curve
t
space curve
plane warp surface
space warp
y
x
y
z
u
v
x
y u
v
x
y
z
u v
y
z
w
x
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Modeling Zoo
Parametric Models Primitive Meshes
Implicit Models Particle Models
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Primitive Meshes
Primitive Meshes
• Collection of geometric primitives
Triangles
Quadrilaterals
More general primitives (spline patches)
• Typically, the primitives are parametric surfaces
• Composite model:
Mesh encodes topology, rough shape
Primitive parameter encode local geometry
• Triangle meshes rule the world (“triangle soup”)
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Primitive Meshes
Complex Topology for Parametric Models
• Mesh of parameter domains attached in a mesh
• Domain can have complex shape (“trimmed patches”)
• Separate mapping function f for each part (typically of the same class)
1
2
3
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Meshes are Great
Advantages of mesh-based modeling:
• Compact representation (usually)
• Can represent arbitrary topology
• Important objects can be represented exactly
When using the right parametric parts
E.g. NURBS: circles, cylinders, spheres CAD/CAM
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Meshes are not so great
Problem with Meshes:
• Need to specify a mesh first, then edit geometry
• Problems for larger changes
Mesh structure and shape need to be adjusted
Mesh encodes object topology Changing object topology is painful
• Sometimes difficult to use
Rule of thumb: drastic topology changes ⇒ meshes are hard to use
Extreme example: fluid simulation (splashing water)
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Modeling Zoo
Parametric Models Primitive Meshes
Implicit Models Particle Models
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Implicit Modeling
General Formulation:
• Curve / Surface S = {x | f (x) = 0}
• x d (d = 2,3), f (x)
• S is (usually) a d-1 dimensional object
This means...:
• Surface = set of points where f vanishes ( f (x) = 0)
• Also known as “level-set methods”
• Alternative notation: S = f -1(0)
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Implicit Modeling
Example:
• Circle: x2 + y2 = r2 fr(x,y) = x2 + y2 - r2 = 0
• Sphere: x2 + y2 + z2 = r2
Special Case:
• Signed distance field
• Function value is signed distance to surface
• Negative means inside, positive means outside
x2
y2 r2
||)(),( 222222 ryxryxyx signf
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Implicit Modeling: Pros & Cons
Advantages:
• More general than parametric techniques
• Topology can be changed easily
• Standard technique for simulations with free boundaries.
Example: Fluid simulation (evolving water-air interface)
Geometric modeling: Surface reconstruction, “blobby surfaces”
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Implicit Modeling: Pros & Cons
Disadvantages:
• Need to solve inversion: S = f -1(0)
• Many algorithms more difficult
Difficult: display, surface sampling, conversion (larger run-time costs as well)
Easy: inside/outside, Boolean operations
• Memory: Often more costly
In particular: sharp boundaries, flat surfaces
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Modeling Zoo
Parametric Models Primitive Meshes
Implicit Models Particle Models
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Particle Representations
Particle / Point-based Representations
• Geometry is represented as a set of points / particles
• Irregular sample of geometry
• Need additional information to deal with “the empty space around the particles”
Reconstruction for processing
Also a type of implicit representation
additional assumptions
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Particle Representations
Helpful Information
• Each particle may carries a set of attributes
Must have: position
Additional:
– Particle density (sample spacing)
– Surface normals
– Color
– physical quantities (mass, pressure, temperature), ...
• Additional information might facilitate reconstruction
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The Wrath of Khan
Why Star Trek is at fault...
• Particle methods first used in computer graphics to represent fuzzy phenomena (fire, clouds, smoke)
• “Particle Systems—a Technique for Modeling a Class of Fuzzy Objects” [Reeves 1984]
• Probably most well-known example: Genesis sequence
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Genesis Sequence [Reeves 1983]
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Non-Fire Objects
Particle Traces for Modeling Plants (also from [Reeves 1983])
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Geometric Modeling
How became the geometric modeling crowd interested in this?
3D Scanners
• 3D scanners yield point clouds
• Have to deal with the problem anyway
• Need algorithms to directly work on “point clouds”
(this is the geometry name for particle system)
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Geometric Modeling
How became the geometric modeling crowd interested in this?
Other Reasons:
• Similar advantages as implicit techniques
• Topology does not matter (for the good and for the bad)
Topology is easy to change
Topology might be hard to determine
Multi-scale representations are easy to do (more details on multi-resolution techniques later)
• Often easier to use than implicit or parametric techniques
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Multi-Scale Geometry w/Points
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Summary
Summary
• Lots of different representations
• No silver bullet
• In theory, everything always works, but might be just too complicated/expensive
• Best choice depends on the application
• We will look on all of this...
Focus on parametric techniques though
Most common approach