cse 554 lecture 3: shape analysis (part ii) -...
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CSE554 Cell Complexes Slide 1
CSE 554
Lecture 3: Shape Analysis
(Part II)
Fall 2016
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CSE554 Cell Complexes Slide 2
Review
• Skeletons
– Centered curves/surfaces
• Approximations of medial axes
– Useful for shape analysis
2D skeletons
3D surface and curve skeletons
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CSE554 Cell Complexes Slide 3
Review
• Thinning on binary pictures
– Removable pixels (voxels)
• Whose removal does not alter the object’s
shape or topology
• Border, Simple, and not curve-end
– Strategies
• Parallel thinning: topology is lost
• Serial thinning: topology is preserved
– But result depends on pixel order
Removal pixels
Thinning
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CSE554 Cell Complexes Slide 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
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CSE554 Cell Complexes Slide 5
This lecture…
• Thinning on a cell complex
– One algorithm that works for shapes in any dimensions (2D, 3D, etc.)
– Integrates pruning with thinning
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CSE554 Cell Complexes Slide 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 can be continuously deformed to a k-D ball
• 0-D ball: point
• 1-D ball: edge
• 2-D ball: disk
• 3-D ball: sphere
0-cell 1-cell
2-cell 3-cell
Not a 2-cell
Not a 3-cell
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CSE554 Cell Complexes Slide 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-cell 1-cell
2-cell 3-cell
(boundary is colored blue)
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CSE554 Cell Complexes Slide 8
Cell Complex
• Union of cells and cells on their boundaries
– The formal name is CW (closure-finite, weak-topology) Complex
• Precise definition can be found in algebraic topology books
0 0 0
0
1 1
1 12
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CSE554 Cell Complexes Slide 9
Cell Complex
• Are these cell complexes?
(this edge is not bounded) (the triangle is not bounded)
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CSE554 Cell Complexes Slide 10
Example Cell Complexes
Polyline
(0-,1-cells)Triangulated polygon
(0-,1-, 2-cells)
Triangular mesh
(0-,1-, 2-cells)
Tetrahedral volume
(0-,1-, 2-, 3-cells)
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CSE554 Cell Complexes Slide 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
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CSE554 Cell Complexes Slide 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
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CSE554 Cell Complexes Slide 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).
• Challenge: avoid creating duplicated cells
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CSE554 Cell Complexes Slide 14
Algorithm: Approach 1
• Avoid duplicates
– Use a data structure to keep track of the index of cells (initially zero).
– Look up this array before creating a new cell, and update the array if a
new cell is created
0
0 0 0 0
0
000
00 0
Index of 2-cell0 0 0
0
0
000
0
0
0
0
0
0 0
0
0
Index of 1-cell0
0 0 0
0 0
Index of 0-cell
All indices occupy a
(2n-1)*(2n-1) grid!
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CSE554 Cell Complexes Slide 15
Algorithm: Approach 1
0
0 0 0 0
0
000
0
0 0 0
0
0
000
0
0
0
0
0
0 0 0
0 0
0 0
0
0 0
0
0
Index of 2-cell
Index of 1-cell
Index of 0-cell
0
0
0 0 0 0
0
000
1 0
0 0 0
0
0
000
0
0 4 0
3 0
1 2 0
3 0
1 2 0
4
0
0
0 0 0 0
0
000
1 2
0 0 0
0
0
000
0
0 4 7
3 6
1 2 5
3 4 6
1 2 5
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CSE554 Cell Complexes Slide 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
• Same strategy as in Approach 1 for storing cell
indices
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CSE554 Cell Complexes Slide 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
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CSE554 Cell Complexes Slide 18
Thinning on Cell Complexes
• Remove simple pairs
– Whose removal does not affect topology
• Protect medial cells
– To prevent shrinking of skeleton
– To prune noise
• Advantages:
– Easy to detect in 2D and 3D (same code)
– Robust to noise
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CSE554 Cell Complexes Slide 19
Simple Pairs
• How can we remove cells from a complex without changing
its topology?
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CSE554 Cell Complexes Slide 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
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CSE554 Cell Complexes Slide 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
xy
{x,y} is a simple pair
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CSE554 Cell Complexes Slide 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
xy
{x,y} is a simple pair
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CSE554 Cell Complexes Slide 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
• “Almost” parallel thinning
xy
xy
x
y
x y
x1
y1
x2y2
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CSE554 Cell Complexes Slide 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 disjoint simple pairs in C
2. If S is empty, Break.
3. Remove all cells in S from C
2. Output C
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CSE554 Cell Complexes Slide 25
Exhaustive Thinning
• Removing all simple pairs in parallel at each iteration
– Only the topology of the cell complex is preserved
2D example
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CSE554 Cell Complexes Slide 26
Exhaustive Thinning
• Removing all simple pairs in parallel at each iteration
– Only the topology of the cell complex is preserved
3D example
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CSE554 Cell Complexes Slide 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
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CSE554 Cell Complexes Slide 28
Exhaustive Thinning
• Removing all simple pairs in parallel at each iteration
– Only the topology of the cell complex is preserved
A 3D shape
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CSE554 Cell Complexes Slide 29
Medial Cells (2D)
“Meaningful” skeleton edges survive longer during thinning
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CSE554 Cell Complexes Slide 30
Medial Cells (3D)
“Meaningful” skeleton edges and faces survive longer during thinning
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CSE554 Cell Complexes Slide 31
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
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CSE554 Cell Complexes Slide 32
Medial Cells (2D)
• Isolation iteration (I(x)): # thinning iterations before cell is isolated
– Measures “thickness” of shape
• Removal iteration (R(x)): # thinning iterations before cell is removed
– Measures “length” of shape
I R
0 35
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CSE554 Cell Complexes Slide 33
Medial Cells (2D)
• Medial-ness: difference between R(x) and I(x)
– A greater difference means the shape around x is more tubular
I R
0 35
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CSE554 Cell Complexes Slide 34
Medial Cells (2D)
• Medial-ness: difference between R(x) and I(x)
– A greater difference means the shape around x is more tubular
Absolute: R-I Relative: (R-I)/R
0 35 0 1
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CSE554 Cell Complexes Slide 35
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:
1-I/R: 0 1
0 15
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CSE554 Cell Complexes Slide 36
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:
1-I/R:
I
R
0 1
0 30
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CSE554 Cell Complexes Slide 37
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
– t1abs, t1rel
• 3D: thresholds for both medial 1-cells and 2-cells
– t1abs, t1rel
– t2abs, t2rel
• 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)
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CSE554 Cell Complexes Slide 38
Thinning Algorithm (2D)
// Thinning on a 2D cell complex C
// Thresholds t1abs and t1rel 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 disjoint simple pairs in C
2. Repeat for each pair {x,y} in S:
1. If x is 1-cell and ( k-I(x)>t1abs and 1-I(x)/k>t1rel ),
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
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CSE554 Cell Complexes Slide 39
Thinning Algorithm (3D)
// Thinning on a 3D cell complex C
// Thresholds t1abs and t1rel for medial 1-cells
// Thresholds t2abs and t2rel 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 disjoint simple pairs in C
2. Repeat for each pair {x,y} in S:
1. If x is 1-cell and ( k-I(x)>t1abs and 1-I(x)/k>t1rel ),
exclude {x,y} from S.
2. If x is 2-cell and ( k-I(x)>t2abs and 1-I(x)/k>t2rel ),
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
![Page 40: CSE 554 Lecture 3: Shape Analysis (Part II) - cs.wustl.edutaoju/cse554/lectures/lect03_CellComplex.pdf · CSE554 Cell Complexes Slide 3 Review •Thinning on binary pictures –Removable](https://reader031.vdocuments.site/reader031/viewer/2022022509/5ad5a7b87f8b9a1a028d5c48/html5/thumbnails/40.jpg)
CSE554 Cell Complexes Slide 40
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)
t1abs= ∞
Skeletons computed at threshold
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CSE554 Cell Complexes Slide 41
t1rel= 0.5 t1rel= 0.6 t1rel= 0.7 t1rel= 0.8
T1abs = 1
T1abs = 3
T1abs = 5
T1abs = 7
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CSE554 Cell Complexes Slide 42
More Examples
• 2Dt1abs= ∞
t1rel= 1
t1abs= 4
t1rel= .4
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CSE554 Cell Complexes Slide 43
More Examples
• 3Dt1abs=∞
t1rel=1
t2abs=∞
t2rel=1
t1abs=4
t1rel=.4
t2abs=∞
t2rel=1
t1abs=∞
t1rel=1
t2abs=4
t2rel=.4
t1abs=4
t1rel=.4
t2abs=4
t2rel=.4
![Page 44: CSE 554 Lecture 3: Shape Analysis (Part II) - cs.wustl.edutaoju/cse554/lectures/lect03_CellComplex.pdf · CSE554 Cell Complexes Slide 3 Review •Thinning on binary pictures –Removable](https://reader031.vdocuments.site/reader031/viewer/2022022509/5ad5a7b87f8b9a1a028d5c48/html5/thumbnails/44.jpg)
CSE554 Cell Complexes Slide 44
More Examples
• 3D t1abs=∞
t1rel=1
t2abs=∞
t2rel=1
t1abs=∞
t1rel=1
t2abs=5
t2rel=.4
t1abs=5
t1rel=.4
t2abs=∞
t2rel=1
t1abs=5
t1rel=.4
t2abs=5
t2rel=.4