Download presentation

Presentation is loading. Please wait.

Published byKayden Woodmancy Modified over 2 years ago

1
CSE554Cell ComplexesSlide 1 CSE 554 Lecture 3: Shape Analysis (Part II) Fall 2014

2
CSE554Cell ComplexesSlide 2 Review Skeletons – Centered curves/surfaces Approximations of medial axes – Useful for shape analysis 2D skeletons 3D surface and curve skeletons

3
CSE554Cell ComplexesSlide 3 Review Thinning on binary pictures – Removable pixels (voxels) Whose removal does not alter the object’s shape or topology – Strategies Serial thinning or parallel thinning Removal pixels Thinning

4
CSE554Cell ComplexesSlide 4 Review Issues (with thinning on a binary picture) – Difficult to write a 3D thinning algorithm E.g., simple voxel criteria, surface-end voxel criteria – Skeletons can be noisy Requires pruning x

5
CSE554Cell ComplexesSlide 5 This lecture… Thinning on a cell complex – One algorithm that works for shapes in any dimensions (2D, 3D, etc.) – Integrates pruning with thinning

6
CSE554Cell ComplexesSlide 6 Cells Geometric elements with simple topology – k-cell: an element at dimension k 0-cell: point 1-cell: line segment, curve segment, … 2-cell: triangle, quad, … 3-cell: cube, tetrahedra, … – Formally: A k-cell has the topology of a closed k-dimensional ball 0-D ball: point 1-D ball: edge 2-D ball: disk 3-D ball: sphere 0-cell1-cell 2-cell3-cell Not a 2-cell Not a 3-cell

7
CSE554Cell ComplexesSlide 7 Cells The boundary of a k-cell (k>0) has dimension k-1 – Examples: A 1-cell is bounded by two 0-D points A 2-cell is bounded by a 1-D curve A 3-cell is bounded by a 2-D surface 0-cell1-cell 2-cell3-cell (boundary is colored blue)

8
CSE554Cell ComplexesSlide 8 Cell Complex Union of cells and their boundaries – The formal name is CW (closure-finite, weak-topology) Complex Precise definition can be found in algebraic topology books 000 0 11 11 2

9
CSE554Cell ComplexesSlide 9 Cell Complex Are these cell complexes? (this edge is not bounded) (the triangle is not bounded)

10
CSE554Cell ComplexesSlide 10 Example Cell Complexes 2D – Polyline – Triangulated polygon 3D – Triangulated surface – Tetrahedral volume 0-,1-cells0-,1-, 2-cells 0-,1-, 2-, 3-cells

11
CSE554Cell ComplexesSlide 11 Cell Complex from Binary Pic Representing the object as a cell complex – Approach 1: create a 2-cell (3-cell) for each object pixel (voxel), and add all boundary cells Reproducing 8-connectivity in 2D and 26-connectivity in 3D

12
CSE554Cell ComplexesSlide 12 Representing the object as a cell complex – Approach 2: create a 0-cell for each object pixel (voxel), and connect them to form higher dimensional cells. Reproducing 4-connectivity in 2D and 6-connectivity in 3D Cell Complex from Binary Pic

13
CSE554Cell ComplexesSlide 13 Algorithm: Approach 1 2D: – For each object pixel, create a 2-cell (square), four 1-cells (edges), and four 0- cells (points). 3D : – For each object voxel, create a 3-cell (cube), six 2-cells (squares), twelve 1-cells (edges), and eight 0-cells (points).

14
CSE554Cell ComplexesSlide 14 Algorithm: Approach 1 Handling cells shared by adjacent pixels (voxels) – Create an array of integers that keeps track of whether a cell is created (e.g., at a pixel, at a pixel’s edge, or at a pixel’s point), and if it is created, the index of the cell. – Look up this array before creating a new cell 0 0000 0 000 0 000 0 0 000 0 0 0 0 0 00 0 00 00 0 00 0 0 Index of 2-cell Index of 1-cell Index of 0-cell

15
CSE554Cell ComplexesSlide 15 Algorithm: Approach 1 0 0000 0 000 0 000 0 0 000 0 0 0 0 0 00 0 00 00 0 00 0 0 Index of 2-cell Index of 1-cell Index of 0-cell 0 0 0000 0 000 10 000 0 0 000 0 040 30 120 30 12 0 4 0 0 0000 0 000 12 000 0 0 000 0 047 36 125 346 125

16
CSE554Cell ComplexesSlide 16 Algorithm: Approach 2 2D: – Create a 0-cell at each object pixel, a 1-cell for two object pixels sharing a common edge, and a 2-cell for four object pixels sharing a common point 3D: – Create a 0-cell at each object voxel, a 1-cell for two object voxels sharing a common face, a 2-cell for four object voxels sharing a common edge, and a 3- cell for eight object voxels sharing a common point Follow the note in Approach 1 for storing cell indices

17
CSE554Cell ComplexesSlide 17 Thinning on Binary Pictures Remove simple pixels (voxels) – Whose removal does not affect topology Protect end pixels (voxels) of skeleton curves and surfaces – To prevent shrinking of skeleton

18
CSE554Cell ComplexesSlide 18 Thinning on Cell Complexes Remove simple pairs – Whose removal does not affect topology Protect medial cells – Which are important for describing shape Advantages: – Easy to detect in 2D and 3D (same code) – Medial cells are robust to boundary noise

19
CSE554Cell ComplexesSlide 19 Simple Pairs How can we remove cells from a complex without changing its topology?

20
CSE554Cell ComplexesSlide 20 Simple Pairs Definition – A pair {x, y} such that y is on the boundary of x, and there is no other cell in the complex with y on its boundary. x y {x, y} is a simple pair y {x,y} is not a simple pair x

21
CSE554Cell ComplexesSlide 21 Simple Pairs Definition – A pair {x, y} such that y is on the boundary of x, and there is no other cell in the complex with y on its boundary. x y {x, y} is a simple pair x y

22
CSE554Cell ComplexesSlide 22 Simple Pairs Definition – A pair {x, y} such that y is on the boundary of x, and there is no other cell in the complex with y on its boundary. – In a simple pair, x is called a simple cell, and y is called the witness of x. A simple cell can pair up with different witnesses x y {x, y} is a simple pair y y x y

23
CSE554Cell ComplexesSlide 23 Simple Pairs Removing a simple pair does not change topology – True even when multiple simple pairs are removed together As long as the pairs are disjoint x y x y x y x y x1x1 y1y1 x2x2 y2y2

24
CSE554Cell ComplexesSlide 24 Exhaustive Thinning Removing all simple pairs in parallel at each iteration – Only the topology of the cell complex is preserved – If a simple cell has multiple witnesses, an arbitrary choice is made // Exhaustive thinning on a cell complex C 1.Repeat: 1.Let S be all simple pairs in C 2.If S is empty, Break. 3.Remove all cells in S from C 2.Output C

25
CSE554Cell ComplexesSlide 25 Exhaustive Thinning Removing all simple pairs in parallel at each iteration – Only the topology of the cell complex is preserved 2D example

26
CSE554Cell ComplexesSlide 26 Exhaustive Thinning Removing all simple pairs in parallel at each iteration – Only the topology of the cell complex is preserved 3D example

27
CSE554Cell ComplexesSlide 27 Exhaustive Thinning Removing all simple pairs in parallel at each iteration – Only the topology of the cell complex is preserved A more interesting 2D shape

28
CSE554Cell ComplexesSlide 28 Exhaustive Thinning Removing all simple pairs in parallel at each iteration – Only the topology of the cell complex is preserved A 3D shape

29
CSE554Cell ComplexesSlide 29 Isolated cells A cell x is isolated if it is not on the boundary of other cells – A k-dimensional skeleton is made up of isolated k-cells Isolated cells are colored

30
CSE554Cell ComplexesSlide 30 Medial Cells (2D) “Meaningful” skeleton edges survive longer during thinning

31
CSE554Cell ComplexesSlide 31 Medial Cells (2D) Isolation iteration (I(x)): # thinning iterations before cell is isolated Removal iteration (R(x)): # thinning iterations before cell is removed Medial cells: isolated cells whose difference between R and I is greater than a threshold – Absolute difference: R(x) – I(x) – Relative difference: (R(x) – I(x)) / R(x) I R

32
CSE554Cell ComplexesSlide 32 Medial Cells (2D) Isolation iteration (I(x)): # thinning iterations before cell is isolated Removal iteration (R(x)): # thinning iterations before cell is removed Medial cells: isolated cells whose difference between R and I is greater than a threshold – Absolute difference: R(x) – I(x) – Relative difference: (R(x) – I(x)) / R(x) R-I 1-I/R

33
CSE554Cell ComplexesSlide 33 Medial Cells (2D) For a 1-cell x that is isolated during thinning: – I(x), R(x) measures the “thickness” and “length” of shape – A greater difference means the local shape is more “elongated” I R R-I 1-I/R

34
CSE554Cell ComplexesSlide 34 Medial Cells (3D) For a 2-cell x that is isolated during thinning: – I(x), R(x) measures the “thickness” and “width” of shape – A greater difference means the local shape is more “plate-like” I R R-I: face size 1-I/R: face color

35
CSE554Cell ComplexesSlide 35 Medial Cells (3D) For a 1-cell x that is isolated during thinning: – I(x), R(x) measures the “width” and “length” of shape – A greater difference means the local shape is more “tubular” R-I: edge length 1-I/R: edge color I R

36
CSE554Cell ComplexesSlide 36 Medial Cells and Thinning A cell x is a medial cell if it is isolated and the difference between R(x) and I(x) exceeds given thresholds – A pair of absolute/relative difference thresholds is needed for medial cells at each dimension 2D: thresholds for medial 1-cells – t1 abs, t1 rel 3D: thresholds for both medial 1-cells and 2-cells – t1 abs, t1 rel – t2 abs, t2 rel Thinning: removing simple pairs that are not medial cells – Note: only need to check the simple cell in a pair (the witness is never isolated)

37
CSE554Cell ComplexesSlide 37 Thinning Algorithm (2D) // Thinning on a 2D cell complex C // Thresholds t1 abs and t1 rel for medial 1-cells 1.k = 1 2.For all x in C, set I(x) be 0 if x is isolated, NULL otherwise 3.Repeat and increment k: 1.Let S be all simple pairs in C 2.Repeat for each pair {x,y} in S: 1.If x is 1-cell and ( k-I(x)>t1 abs and 1-I(x)/k>t1 rel ), exclude {x,y} from S. 3.If S is empty, Break. 4.Remove all cells in S from C 5.Set I(x) be k for newly isolated cells x in C 4.Output C Current iteration

38
CSE554Cell ComplexesSlide 38 Thinning Algorithm (3D) // Thinning on a 3D cell complex C // Thresholds t1 abs and t1 rel for medial 1-cells // Thresholds t2 abs and t2 rel for medial 2-cells 1.k = 1 2.For all x in C, set I(x) be 0 if x is isolated, NULL otherwise 3.Repeat and increment k: 1.Let S be all simple pairs in C 2.Repeat for each pair {x,y} in S: 1.If x is 1-cell and ( k-I(x)>t1 abs and 1-I(x)/k>t1 rel ), exclude {x,y} from S. 2.If x is 2-cell and ( k-I(x)>t2 abs and 1-I(x)/k>t2 rel ), exclude {x,y} from S. 3.If S is empty, Break. 4.Remove all cells in S from C 5.Set I(x) be k for newly isolated cells x in C 4.Output C Current iteration

39
CSE554Cell ComplexesSlide 39 Choosing Thresholds Higher thresholds result in smaller skeleton – Threshold the absolute difference at ∞ will generally purge all cells at that dimension Except those for keeping the topology – Absolute threshold has more impact on features at small scales (e.g., noise) – Relative threshold has more impact on rounded features (e.g., blunt corners) t1 abs = ∞ Skeletons computed at threshold

40
CSE554Cell ComplexesSlide 40 t1 rel = 0.5t1 rel = 0.6t1 rel = 0.7t1 rel = 0.8 T1 abs = 1 T1 abs = 3 T1 abs = 5 T1 abs = 7

41
CSE554Cell ComplexesSlide 41 More Examples 2D t1 abs = ∞ t1 rel = 1 t1 abs = 4 t1 rel =.4

42
CSE554Cell ComplexesSlide 42 More Examples 3D t1 abs =∞ t1 rel =1 t2 abs =∞ t2 rel =1 t1 abs =4 t1 rel =.4 t2 abs =∞ t2 rel =1 t1 abs =∞ t1 rel =1 t2 abs =4 t2 rel =.4 t1 abs =4 t1 rel =.4 t2 abs =4 t2 rel =.4

43
CSE554Cell ComplexesSlide 43 More Examples 3D t1 abs =∞ t1 rel =1 t2 abs =∞ t2 rel =1 t1 abs =∞ t1 rel =1 t2 abs =5 t2 rel =.4 t1 abs =5 t1 rel =.4 t2 abs =∞ t2 rel =1 t1 abs =5 t1 rel =.4 t2 abs =5 t2 rel =.4

Similar presentations

Presentation is loading. Please wait....

OK

Focus on 2011-2012. Can you spell each word correctly ?

Focus on 2011-2012. Can you spell each word correctly ?

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on hot and cold places Ppt on leadership in nursing Ppt on science and technology advantages and disadvantages Ppt on information technology act 2008 Ppt on computer software download Ppt on 4g technology Ppt on basics of ms word 2007 Ppt on fast food retailing in india Ppt on viruses and bacteria worksheets Elements of one act play ppt on apple