【论文阅读】7-Discovering Structural Regularity in 3D Geometry

1、Input:
2、Processing pipeline
- 2.1、Transform analysis
- - 2.1.0、Pre
  - 2.1.2、Sample
  - 2.1.3、Similarity sets
  - 2.1.4、Compute transformation
  - 2.1.5、Transformation mapping
- 2.2 Model estimation
- - 2.2.1、Clusters
  - 2.2.2、Energy(min)：
- 2.3、Aggregation
- - 2.3.1、Local region growing
  - 2.3.2、Geometry registration (simultaneous registration)
3、Limits:
4、Future work :

1、Input:

points/mesh output: representative element transformation

2、Processing pipeline

2.1、Transform analysis

yields characteristic lattice patterns for shapes containing regular structures.

2.1.0、Pre

Transformation && regularity

Similarity transformation：相似变换，保向性，与regularity 相关

Regular structure : ( P, G) -----(P0, Ti, ni)

P0: representative element
Ti: transformation
Ni: regularity paramaters
G: 可采用two steps 实现： translation – rotation、reflection、symmetrics
Regular 描述： 在相似变换T下，Pi—Pj(Pij为P中元素，即P关于相似变换自相关)
Ni 越大，similarity set(见下)应该越接近surface

结构规律性的复合与层次：

1-parameter regularity
Commutative 2-p( translation \ rotation \ scale )
……
Commutative k-p(K>=3, only translation)

2.1.2、Sample

采样处理的原理

Observation :
If is the representative P0 element of (P,G),
then any subset of P0 will also generate a regular (P,G) structure
with transformation group G.

2.1.3、Similarity sets

surface
采样points（represent small local patches）
计算descriptor :invariant similarity transformation

采用Mean && Gaussican curvatures
一般性的相似变换：H^2/K
无缩放的相似变换：H,k

基于descriptor得到多个Similarity sets

原理：Only sample points with similar descriptor value are potential candidates for a regular structure

Ignore : Umbilical point —transformation 不唯一

ps: umbilical point

In the differential geometry of surfaces in three dimensions,
umbilics or umbilical points are points on a surface that are locally spherical.
At such points the normal curvatures in all directions are equal, hence,
both principal curvatures are equal,
and every tangent vector is a principal direction.
The name "umbilic" comes from the Latin umbilicus - navel.

2.1.4、Compute transformation

Translation and rotation ：

can be derived by aligning the local frames computed from
the surface normal and principal curvature directions(主曲率方向).

The uniform scaling factor：

can be calculated from the ratio of corresponding mean curvatures Hi/Hj （曲率均值之比）

ICP—在计算transformation时，用于去除outliers

Serves as tool to prune unnecessary sample pairs.

Sampling the similarity set
原因如下：

PS: Affine transformation space characterize: ( regular srtucture)

Through origin 2D-plane — search sub-space through origin
clusters that form a uniformly spaced grid

应用： similarity set 采样
（sub-sample，cluster remain,减少underlying transformation,即减小变换空间的点数）；

2.1.5、Transformation mapping

原因：

In general, transformations cluster at points on a curved manifold, which is significantly more difficult to detect.
Cluster 并非完全理想情况下的在2D plane，采用mapping ，得到transformation 的二维映射

Affine transformation space – 2D plane

H(相似变换矩阵) -----invariant ( the choice of origin ----例如：旋转角度、平移方向…)
根据H具体表示的变换选择mapping 的invariants

具体如下图：

2/k- parameter transformation mapping : 对应组合相加：

2.2 Model estimation

a set of regular structures at the scale of the initial local surface patches

To reliably detect regular structures we therefore need a grid fitting approach that is robust to outliers and holes.操作对象：set of cluster centers C = ck （mapping 2D space）

2.2.1、Clusters

Mean-shift （解决 wrong cluster\ weak cluster）

2.2.2、Energy(min)：

estimate gi,gj（gird elements）,aij,bi ( weight, 与holes \ outliers 有关)

Energy—与gird location（网格角点） & cluster location 有关，几何意义：是的cluster为最终得到的grid 的网格角点

迭代求解--采用RANSAC
初值:
gi,gj（gird elements）: dominant lines through origin
aij,bi  1

（迭代求解过程示意图）

2.3、Aggregation

aggregate spatially adjacent patches to build larger repetitive elements
using a novel simultaneous registration method that optimizes the generating transformations
in the spatial domain.

以下两种方法交错进行:

2.3.1、Local region growing

aggregate spatially adjacent patches of regular structures with compatible group structure.

2.3.2、Geometry registration (simultaneous registration)

(F = ∑Qij)_min

F-度量distance (变换前后): point-plane ; point- point// 用于对regular的一种描述
（sampling patches matches—similarity sets）

aggregation && refine Transformation

After solving the resulting linear system we have to update the
group generator by a true similarity or true congruence. T

3、Limits:

Wrap sequential repetitive

4、Future work :

Extend Dimension

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【论文阅读】7-Discovering Structural Regularity in 3D Geometry

【论文阅读】7-Discovering Structural Regularity in 3D Geometry

1、Input:

2、Processing pipeline

2.1、Transform analysis

2.1.0、Pre

2.1.2、Sample

2.1.3、Similarity sets

2.1.4、Compute transformation

2.1.5、Transformation mapping

2.2 Model estimation

2.2.1、Clusters

2.2.2、Energy(min)：

2.3、Aggregation

2.3.1、Local region growing

2.3.2、Geometry registration (simultaneous registration)

3、Limits:

4、Future work :

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