A Fast and Simple Surface Reconstruction Algorithm Siu-Wing Cheng, Jiongxin Jin and Man-Kit Lau Department of Computer Science and Engineering Hong Kong University of Science and Technology

Jun 17, 2012

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

1 / 18

Surface reconstruction

point cloud

Cheng, Jin and Lau (HKUST)

triangular mesh

Surface reconstruction

Jun 17, 2012

2 / 18

Surface reconstruction Given: a “dense” point set sampled from an unknown surface Σ ⊂ R3 Goal: to compute a surface mesh that is topologically equivalent - homeomorphic to Σ

geometrically faithful - small normal deviation - small Hausdorff distance

Applications: reverse engineering, medical imaging, computer graphics, . . .

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

3 / 18

ε-sample

Medial axis MΣ

closure of the set of points having at least two closest points in Σ

MΣ

Local feature size lfs(x) lfs(x) = d(x, MΣ )

lfs(x)

Σ

x

ε-sample P ∀x ∈ Σ, d(x, P) ≤ εlfs(x)

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

4 / 18

Previous results

Crust, PowerCrust, Cocone: O(n2 ), need to compute 3D DT/VD output-sensitive 3D VD algorithm: O((n + f ) log2 n) - uniform sample from generic smooth surface: f = O(n log n) Attali, Boissonnat and Lieutier [SCG ’03] √ - non-generic surface: f = Ω(n n) even for uniform sample Erickson [DCG ’03]

Funke and Ramos [SODA ’02]: O(n log n) (not practical)

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

5 / 18

FR algorithm 1

Extract a locally uniform O(ε)-sample S ⊆ P.

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

6 / 18

FR algorithm 1

Extract a locally uniform O(ε)-sample S ⊆ P.

2

Reconstruct a surface using S. O(n log n), Dey, Funke and Ramos [EuroCG ’01]

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

6 / 18

FR algorithm 1

Extract a locally uniform O(ε)-sample S ⊆ P.

2

Reconstruct a surface using S. O(n log n), Dey, Funke and Ramos [EuroCG ’01]

3

Add back P \ S. O(n log n)

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

6 / 18

FR algorithm 1

Extract a locally uniform O(ε)-sample S ⊆ P. O(n log n) well-separated pair decomposition approximate range searching approximate directional nearest neighbors

2

Reconstruct a surface using S. O(n log n), Dey, Funke and Ramos [EuroCG ’01]

3

Add back P \ S. O(n log n)

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

6 / 18

FR algorithm 1

Extract a locally uniform O(ε)-sample S ⊆ P. O(n log n) well-separated pair decomposition approximate range searching

⇒ Octrees

approximate directional nearest neighbors 2

Reconstruct a surface using S. O(n log n), Dey, Funke and Ramos [EuroCG ’01]

3

Add back P \ S. O(n log n)

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

6 / 18

Our results

Given an ε-sample P, compute a locally uniform O(ε)-sample S ⊆ P in O(n log n) time with a single octree.

surface reconstruction algorithm: O(n log n), optimal in the pointer machine model performance: - for non-uniform samples: 51% to 68% faster than Cocone - for locally uniform samples: still faster than Cocone

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

7 / 18

Locally uniform sample S

βrq rq

Σ

Bq q

S

Bq : largest empty ball with center in Σ and boundary through q ∀q ∈ S, |B(q, βrq ) ∩ S| = O(1)

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

8 / 18

Octree decomposition `C C

Root cell

1 8 `C

smallest bounding cube of P Splitting rule split a splittable leaf cell into eight children Balancing rule split a leaf cell C if it has a neighbor C 0 s.t. `C 0 < `C /2

non-splittable

C

Apply the two rules alternately until the tree stops growing. splittable Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

9 / 18

Example

splitting . . .

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

10 / 18

Example

splitting . . .

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

10 / 18

Example

splitting . . .

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

10 / 18

Example

splitting . . .

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

10 / 18

Example

splitting . . .

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

10 / 18

Example

splitting . . . balancing . . .

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

10 / 18

Example

splitting . . . balancing . . . splitting . . .

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

10 / 18

Example

splitting . . . balancing . . . splitting . . . balancing . . .

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

10 / 18

Example

splitting . . . balancing . . . splitting . . . balancing . . . done

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

10 / 18

Properties of the octree

O(n) size, O(n log n) construction time

balanced: side lengths of neighboring leaf cells differ by at most a factor 2

non-empty leaf cells have side lengths O(εlfs) (but can be much smaller than εlfs)

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

11 / 18

Overview 1

Trim the tree so that the sizes of non-empty leaf cells are “good” side lengths O(εlfs) the union of their const-factor expansions covers the surface

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

12 / 18

Overview 1

Trim the tree so that the sizes of non-empty leaf cells are “good” side lengths O(εlfs) the union of their const-factor expansions covers the surface

C

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

12 / 18

Overview 1

Trim the tree so that the sizes of non-empty leaf cells are “good” side lengths O(εlfs) the union of their const-factor expansions covers the surface

2

Smooth out the sizes of non-empty leaf cells no cell C 0 intersecting κ-factor expansion of C is smaller than half of C (κ = 2 in the example) can be done in linear time by processing non-empty leaf cells in decreasing order in their sizes

C

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

12 / 18

Overview 1

Trim the tree so that the sizes of non-empty leaf cells are “good” side lengths O(εlfs) the union of their const-factor expansions covers the surface

2

Smooth out the sizes of non-empty leaf cells no cell C 0 intersecting κ-factor expansion of C is smaller than half of C (κ = 2 in the example) can be done in linear time by processing non-empty leaf cells in decreasing order in their sizes

C

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

12 / 18

Overview 1

Trim the tree so that the sizes of non-empty leaf cells are “good” side lengths O(εlfs) the union of their const-factor expansions covers the surface

2

Smooth out the sizes of non-empty leaf cells no cell C 0 intersecting κ-factor expansion of C is smaller than half of C (κ = 2 in the example) can be done in linear time by processing non-empty leaf cells in decreasing order in their sizes

3

Pick one point from each non-empty leaf cell locally uniform O(ε)-sample

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

12 / 18

Overview 1

Trim the tree so that the sizes of non-empty leaf cells are “good” side lengths O(εlfs) the union of their const-factor expansions covers the surface

2

Smooth out the sizes of non-empty leaf cells no cell C 0 intersecting κ-factor expansion of C is smaller than half of C (κ = 2 in the example) can be done in linear time by processing non-empty leaf cells in decreasing order in their sizes

3

Pick one point from each non-empty leaf cell locally uniform O(ε)-sample

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

12 / 18

Tree trimming Trim the tree so that the sizes of non-empty leaf cells are “good”: 1

2

3

C

estimate surface normal at p ∈ C H: approximate tangent plane look for an empty cube with center in H and side length 18 `C in close neighborhood of C

p empty cube of size 18 `C

if normal estimation fails or an empty cube is found, then make parent(C ) a new leaf cell, and perform the checking on parent(C ) (`C = O(εlfs) in this case)

Cheng, Jin and Lau (HKUST)

˜p n

H

trim C

Surface reconstruction

Jun 17, 2012

13 / 18

Normal estimation

p1

Normal estimation at p ∈ C : 1 2 3

Pick pi ∈ P ∩ Ri , for i ∈ [1, 53 ]

Find pi , pj , s.t. ∠pi ppj ∈ [θ0 , π − θ0 ] ˜ p = npi ppj n

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Ri p C

5`C

p2

Jun 17, 2012

14 / 18

Locally uniform subsample

Cocone on P

Cheng, Jin and Lau (HKUST)

Cocone on S

Surface reconstruction

Jun 17, 2012

15 / 18

Locally uniform subsample

Cocone on P

Cheng, Jin and Lau (HKUST)

Cocone on S

Surface reconstruction

Jun 17, 2012

15 / 18

Performance (non-uniform input) sec

Cocone on P Extraction

Cocone on S

Insertion

300

200

100

0

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

16 / 18

Performance (uniform input) sec

Cocone on P Extraction

Cocone on S

Insertion

90

60

30

0

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

17 / 18

Extensions

multiple surfaces

handling noise

k-dimensional manifold in Rd

Cheng, Jin and Lau (HKUST)

Surface reconstruction

Jun 17, 2012

18 / 18

A Fast and Simple Surface Reconstruction Algorithm

Jun 17, 2012 - Octree decomposition. Root cell smallest bounding cube of P. Splitting rule split a splittable leaf cell into eight children. Balancing rule split a leaf cell C if it has a neighbor C/ s.t. lC < lC /2. Apply the two rules alternately until the tree stops growing. C splittable. C. lC. 1. 8. lC non-splittable. Cheng, Jin and Lau ...
Download PDF
5MB Sizes 0 Downloads 305 Views
Report

Recommend Documents

A Simple, Fast, and Effective Polygon Reduction Algorithm - Stan Melax
Special effects in your game modify the geometry of objects, bumping up your polygon count and requiring a method by which your engine can quickly reduce polygon counts at run time. G A M E D E V E L O P E R. NOVEMBER 1998 http://www.gdmag.com. 44. R
Schematic Surface Reconstruction - Semantic Scholar
multiple swept surfaces, of which the transport curves lie in horizontal planes. This section will introduce the basic reconstruction framework that initializes a set ...
Schematic Surface Reconstruction - Changchang Wu
This paper introduces a schematic representation for architectural scenes together with robust algorithms for reconstruction from sparse 3D point cloud data. The.
Bone Surface Reconstruction Using Localized ...
the case of long shaped bones such as the tibia, humerus, clavicle, ulna or .... as connected if their distance is less than 10 pixels (Fig. 2d). .... 987–1010, 2006.
A Fast String Searching Algorithm
number of characters actually inspected (on the aver- age) decreases ...... buffer area in virtual memory. .... One telephone number contact for those in- terested ...
A Fast String Searching Algorithm
An algorithm is presented that searches for the location, "i," of the first occurrence of a character string, "'pat,'" in another string, "string." During the search operation, the characters of pat are matched starting with the last character of pat
KinectFusion: real-time dynamic 3D surface reconstruction and ...
SIGGRAPH 2011, Vancouver, British Columbia, Canada, August 7 – 11, 2011. ... refinements of the 3D model, similar to the effect of image super- resolution.
A Fast and Efficient Algorithm for Low-rank ... - Semantic Scholar
The Johns Hopkins University [email protected]. Thong T. .... time O(Md + (n + m)d2) where M denotes the number of non-zero ...... Computer Science, pp. 143–152 ...
A Fast and Efficient Algorithm for Low-rank ... - Semantic Scholar
republish, to post on servers or to redistribute to lists, requires prior specific permission ..... For a fair comparison, we fix the transform matrix to be. Hardarmard and set .... The next theorem is dedicated for showing the bound of d upon which
A remote sensing surface energy balance algorithm for ...
(e.g. Sellers et al., 1996) can contribute to an improved future planning .... parameter is variable in the horizontal space domain with a resolution of one pixel.
A Simple, Fast, and Effective Polygon Reduction ...
method by which your engine can quickly reduce polygon counts at ..... search all neighboring edges for “least cost” edge ... ferent pieces to optimize for human.
Contour-Based Surface Reconstruction using MPU ...
fits a point-based implicit surface to the contour data, while allowing the user to .... visualization of point sets at interactive frame rates with good visual quality.
Road Surface 3D Reconstruction Based on Dense Subpixel ...
and computer vision have been increasingly applied in civil. Rui Fan is with the ... e.g., stereo vision, are more capable of reconstructing the 3D ..... Road Surface 3D Reconstruction Based on Dense Subpixel Disparity Map Estimation .pdf.
Implicit surface reconstruction from point clouds
... van de kwaliteit van een gereconstrueerd model is zowel objectief als subjectief. Objectief kan de kwaliteit van de reconstructie gemeten worden door de afs-.
A Fast Bit-Vector Algorithm for Approximate String ...
Mar 27, 1998 - algorithms compute a bit representation of the current state-set of the ... *Dept. of Computer Science, University of Arizona Tucson, AZ 85721 ...
A Fast Line Segment Based Dense Stereo Algorithm ...
Intitute of HCI and Media Integration, Key Lab of Pervasive Computing(MOE). 3-524, Fit building, Tsinghua University, Beijing 100084, P.R. China ... survey by Scharstern and Szeliski [1] and the one by Brown et al. [2]. ..... Two registers stores.
A Fast Algorithm for Mining Rare Itemsets
telecommunication equipment failures, linking cancer to medical tests, and ... rare itemsets and present a new algorithm, named Rarity, for discovering them in ...
A Fast Algorithm For Rate Optimized Motion Estimation
Abstract. Motion estimation is known to be the main bottleneck in real-time encoding applications, and the search for an effective motion estimation algorithm has ...
A Fast Bresenham Type Algorithm For Drawing Ellipses
We define a function which we call the which is an .... refer to the ellipse's center point coordinates and its horizontal and vertical radial values. So. \V+.3?= œ +.
A fast optimization transfer algorithm for image ...
Inpainting may be done in the pixel domain or in a transformed domain. In 2000 ... Here f and n are the original noise-free image and the Gaussian white noise ...... Step: δ t=0.08. Step: δ t=0.04. Step: Linear Search. Our Method. 0. 100. 200.
A fast convex conjugated algorithm for sparse recovery
of l1 minimization and run very fast on small dataset, they are still computationally expensive for large-scale ... quadratic constraint problem and make use of alternate minimiza- tion to solve it. At each iteration, we compute the ..... Windows XP
A Fast Greedy Algorithm for Generalized Column ...
In Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer. Science (FOCS'11), pages 305 –314, 2011. [3] C. Boutsidis, M. W. Mahoney, and P. Drineas. An improved approximation algorithm for the column subset selection problem. In P
A Fast Line Segment Based Dense Stereo Algorithm ...
problem stems from the pixel-based tree construction. Many edges in ... Stereo correspondence has been one of the most important problems in computer vision ...