Point-based Representation Point based graphics Implicit surfaces - - PDF document
Plan: Point-based Representation Point based graphics Implicit surfaces Defining implicit surfaces from point clouds Surface optimization Qsplat MLS surfaces Intensity image Laser scanning Range Image An image with
Point based graphics Implicit surfaces Defining implicit surfaces from point clouds Surface optimization Qsplat MLS surfaces
An image with depth
information in every pixel d Intensity image Range image
Point cloud Scattered data
(a) (c) (b)
7.5 meter scanner 800 kilo
1,080 man hours 22 people 480 scans 2G polygons 7000 Color images 32 Gigabyte
1M
Polygonal models are getting complex If we see 1M polygons, every polygon contributes to a single
pixel
Scanned models produce clouds of points Noisy scans - Denosing Ambiguities Oversampled – Simplification Goal – surface representation Parametric: polygon mesh, higher degree Implicit functions New types, direct methods
Render, hardware Store – compress / transmit Edit ÷ multi-resolution ÷ Constructive Solid Geometry (CSG) ÷ Blend, merge
Smooth, sharp features Manifold Local properties Interpolating / approximating Differential properties: normal… Noise handling Parameterization
Feature Noise Smooth Sharp
Parametric Polygonal meshes Splines Implicit functions Other, ad-hoc, MLS
Traditional computer graphics uses polygonal mesh as object representation
Well studied Hardware rendering Surface is well defined Geometry Topology Color and texture
Editing Hard to edit – self intersection CSG High resolution models –
Only piecewise smooth
F(x,y) = 0 F(x,y) = x2+y2-r2 Discrete form: image that
samples the function
Signed distance function:
F(x,y) = ||<x,y> ; F-1(0)|| For example r <0 >0 =0 r y x y x F
2
) , (
Pros Smooth CSG Blend Fill holes Differential properties Cons Direct rendering, real-time rendering Sharp features Hard to parameterize
Surface normal usually gradient of function
f(x,y,z) = (f/x, f/y, f/z)
Gradient not necessarily unit length Gradient points in direction of increasing f
Plane bounds half-space Specify plane with point p
and normal N
Points in plane x are perp.
to normal N
f is distance if ||N|| = 1
N p x
X’
f(x) = (x-p)N f < 0 f > 0
intersect union difference
Range Image Registration Meshing Intensity Image Texture to Geometry Texture maps
Model
Error Compensation Simplification
Extracts a polygonal iso-surface from a signed distance function
Use pattern rules
(configurations)
in in
in in
in
Volume is subdivided into unit cubes, which are
Threshold value marks points inside / outside
+
+
+
+
+ + +
+
+ +
+ + + + + +
+ +
+ +
+ +
In 3D there are more configurations and more ambiguities.
in
in in
in
(for a voxel), reduced to 15 (symmetry and rotations)
triangles being added to the isosurface
computed by:
Interpolation along edges
(according to pixel values) – better shading, smoother surfaces
Default - middle of the edge.
Given: points P sampled from
unknown surface U
S approximating U
Initial Surface Estimation Mesh Optimization Points {xi} Surface U Smooth subdivision
Input: set of points {xi} Goal: Contour tracing
(Marching cubes)
Estimate signed distance
function dU:DIR
Associate an oriented plane
with each point
Create a globally consistent
U {xi} D
Compute ni that minimizes
Where
Nbhd(xi) xi ni
p i i j
n x p
i j
x Nbhd p
Assumption: if xi and xk are close then
ni and nk are nearly parallel
If (nk· ni < 0) Flip nk
ni xi xk nk nk
du(p) is the distance to the
tangent plane of the closest point.
n
i
x
i
x
k
n
k
ni xi xk nk
du(p)
p
Determine the topological type
Improve mesh’s accuracy and conciseness Find piecewise smooth surface fitting on the data
Points Phase 1 Phase 2
Phase 1 (886 vertices) Phase2 (163 vertices)
Input: data points P, initial mesh Output: optimized mesh M, minimizing
initial
M
complexity distance
Optimize the energy
function:
Optimization rules use edge
collapse and expand
Sum of square distances (accurate) Number of vertices (sparse) Regulation term make vertices with
equidistance
K E K E V K E V K E
spring rep dist
, , ,
i dist
K x d V K E ) , ( , K c K E
r rep
k j k j s spring
v v c V K E
,
,
Repeat (K’,V’)=GenerateLegalMove(K,V) V’ = OptVertexPosition(K’,V’) if E(K’,V’) < E(K,V) (K,V) = (K’, V’) endif Until converges
Not everywhere smooth, but piecewise smooth!
Perform optimization for
the following:
Optimization steps uses
subdivision rules.
Input Phase-2 Phase-3
rep dist
E E E V L K E
,
K - mesh V - vertices L - sharp edges
Find a concise control mesh M: Same topological type as phase 2 defining a piecewise smooth subdivision
Subdivision is defined as:
“repeatedly refining a control mesh”
M=(K,V) M1=R(M) M2=R(R(M)) = R(M1)
Phase-3
2
C
v1 v5 v4 v3 v2 v0 w1
’
w2
’
w3
’
n n v v v n v
n r
1
n n v v v v wr
2 2 1 1
3 3
5 / 2 cos 2 3 8 5
2
a 5
Where
a n a n n
Not smooth everywhere! Creation of sharp features on surface: creases,
Corner Crease Dart Smooth
1
2
What is the surface Rendering the surface What other operations are supported
A point is infinitely small
Splatting Micro polygons Image-space hole filling
Back to front Sort points: from nearest to
farthest points
Z-Buffer Arbitrary order
Szymon Rusinkiewicz and Marc Levoy Stanford University
An interactive viewer for large models
Fast startup and progressive loading Maintains interactive frame rate Compact data structure Fast preprocessing Data structure: bounding sphere hierarchy Rendering algorithm: traverse tree and splat
Key observation: a single bounding sphere hierarchy
Hierarchical frustum and backface culling Level of detail control Splat rendering [Westover 89]
Start with a triangle mesh produced by aligning and
Place a sphere at each node, large enough to touch
Build up hierarchy
Position and radius encoded relative to
parent node
Hierarchical coding vs. delta coding
along a path for vertex positions
Number of children (0, 2, 3, or 4) – 2 bits Presence of grandchildren – 1 bit
Normal quantized to grid
Each node contains bounding cone of
children’s normals
Hierarchical backface culling [Kumar 96]
Per-vertex color is quantized 5-6-5 (R-G-B)
Traverse hierarchy recursively
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Feedback-driven frame rate control During motion: adjust recursion threshold based on
time to render previous frame
On mouse up: redraw with progressively smaller
thresholds
Consequence: frame rate may vary Alternative: Predictive control of detail [Funkhouser 93]
Tree layout: Breadth-first order in memory and on disk Working set management: Memory mapping disk file Consequence: lower detail for new geometry Alternative: Active working set management
with prefetching [Funkhouser 96, Aliaga 99]
3D scan of 2.7 meter statue at
102,868,637 points File size: 644 MB Preprocessing time: 1 hour
Marc Alexa Johannes Behr Daniel Cohen-Or Shachar Fleishman David Levin Claudio Silva
Smooth surface manifold
representation of point sets
Up-sample Down-sample Noise reduction Interactive rendering
Range Image Registration Meshing Intensity Image Texture to Geometry Texture maps
Model
Error Compensation Simplification
Concentrate on
Surface definition Measurement errors Simplification Rendering
Intensity Image Texture to Geometry Texture maps
Model
Range Image Registration Meshing Error Compensation Simplification
Input point r Compute a local reference
plane Hr=<q,n>
Compute a local polynomial
Project point r’=Gr(0)
r
Gr
Hr
q n
Plane equation: Hr=<q,n> Hr=min id(xi,Hr)(||q- xi||)
Non-linear optimization Where
{xi} neighbor points (x)=exp(-x2/h2) h reconstruction parameter,
depends on the spacing of points
Note: q != r
r Hr
q n
Gr=min i|G(xi,x)-xi,y|(||q- xi||)
Linear optimization
r’=Gr(0)
r
Gr
Hr
q n
Using high values of h Smoother surfaces Less detail
Create a local
Voronoi diagram
Add points at
Voronoi vertex with the largest radius
Stop when spacing is
small enough
Sp- Surface Remove Point pi if |Sp-SP-pi|< Speedup: Evaluate |pi-Project(pi, SP-pi)|
Upsample SR to conform with screen space resolution Adding points SR with the projection method is expensive Point sampling U(Gi) is faster Basic technique Sort points into bounding sphere hierarchy (QSplat) View frustrum + backface culling Traversal ends if screen space resolution is satisfied Points not in {ri} are sampled from a
regular grid over gr
Define the extent of a polynomial to be the spacing
between points:
Project points in radius h on the support plane The extent is the bounding rectangle “E”
E
(b), (c), (g) – Splatting (d),(h) - Polynoms
A traditional method
Align scans
1. Manual rough alignment 2. Local accurate ICP of pairs 3. Global alignment
Merge scans Geometry Pipeline Fill holes Mesh Map color on the mesh Color Pipeline Separate
1. Ambient 2. Diffuse 3. Specular
Merge scans
Geometry Pipeline
Fill holes Mesh Align