Digital Distance Geometry

Digital Distance Geometry – Applications to Image Analysis

Dr. P. P. [email protected], [email protected]

Interra Systems, Inc. www.interrasystems.com

ICVGIP ’04. Science City. 18-Dec-04

DedicationTo the Memory of Professor Azriel Rosenfeld

Who taught us to “see” through the eyes of the machine

This photo is taken from http://www.cfar.umd.edu/~ar/

Agenda What is Distance? Distance Geometry Digital Distance Geometry

Digital Distance in 2-D Digital Distances in 3-D Digital Distances in n-D

Digital Distance Computation & Approximation

Glimpses of Applications

What is Distance?

What is Distance? Sense of being Near or Far Distances in Space – As-the-crow-flies

Normally follows the basic properties – well-defined, finite, positive, definite, symmetric & triangular.

Exception are not uncommon Not every place is reachable – violates being well-defined /

finite One-way roads do not have symmetry Triangularity on sphere may not hold

Distances in Time

Distances in Personal Space

Proxemics People's use of personal space

Four categories for informal space: Intimate distance

for embracing or whispering (6-18 inches) Personal distance

for conversations among good friends (1.5-4 feet) Social distance

for conversations among acquaintances (4-12 feet) Public distance

used for public speaking (12 feet or more)

Edward T. Hall, 1963

Distances in Other Spheres

Psychological Distance Emotive Distance Separational Distance of Six-

Degrees Everyone on Earth is separated by no

more than six friends of friends of friends

Distance Geometry

Significance of Distance Geometry

Distance is a Fundamental Concept in Geometry Shortest Paths

Straight Lines Geodesic on Earth Parallel Lines

Equidistant Ever Circle

Trajectory of a point equidistant from Center Least Perimeter with Largest Area

Conics are distance defined Geometries can be built on Distances

Taxicab Geometry

Krause 1975

Digital Geometry Necessitated by Automated Analysis Discrete Models are needed for

Machine Image Analysis Navigation Robotics

Digital Geometry – by Rosenfeld et al

Euclidean :: Digital Geometry

Euclidean Geometry

Digital Geometry

Properties that hold 

Euclidean distance is a metric Extendable to higher dimensions

Euclidean distance is a metric Extendable to higher dimensions

Properties that hold after extension

Jordan’s theorem in 2-D

Every shortest path which connects two points has a unique mid-point

Jordan’s theorem in 2-D holds if 4-connectivity is maintained in the object (background) and 8-connectivity in the background (object)

A shortest path has a unique mid-point or a mid-point pair

Euclidean :: Digital Geometry

Euclidean Geometry

Digital Geometry

Properties that do not hold

The shortest path between any pair of points is unique

Only parallel lines do not intersect

Two intersecting lines define an angle between them

The shortest path between pair of points may not be uniqueLines may not intersect but may not be parallelAngle is unlikely. Digital trigonometry has been ruled out (Rosenfeld 1983)

Significance of Digital Distance Geometry

Divergence from Euclidean Geometry. Preservation of “intuitive” Properties. Preservation of Metric Properties. Quality of Approximation

How to work in digital domain with Euclidean accuracy?

“Circularity” of Disks Computational Efficiency

Distance Transformations Medial Axis Transform

Digital Distance Geometry

Basic Notions

Model: Digitization of Space

Digitization Partitioning through Cells Covers the Space No Overlap

Homogeneous – uniformity around every vertex

Regular – identical tiles 2D – 11 Homogeneous, 3 Regular 3D – 1 Regular 4D – 3 Regular n-D – 1 Regular

Tessellations

The Discrete Model Rectangular Tessellations

Easy Algebraic Representation, Zn

Extensible to arbitrary dimensions In 2-D

Hexagonal / Triangular is more “pleasant”

Melter did a mapping to Rectangular

Neighborhood The neighborhood N(x) of a point x is the set of

points defined to be neighbors of x. A digital neighborhood is characterized by a

set of difference vectors {e1,e2, …, ek} Zn such that y N(x) or y Zn is a neighbor of x Zn

iff (x-y) {e1,e2, … , ek}.

With every neighbor ei we also associate a cost (ei). Most often this cost is taken to be unity.

Neighborhood: Examples Cityblock or 4-neighbors:

{(±1,0), (0, ±1)} k = 4

Chessboard or 8-neighbors: {(±1, 0), (0, ±1), (±1, ±1)}, k = 8

Knight’s neighbors:{(±1, ±2), (±2, ±1)}, k = 8

Neighborhoods in 3-D are 6, 18 and 26.

Neighborhoods: 2D

Neighborhoods: 3D

Digital Neighborhood: 5 Factors

Proximity: Any two neighbors are proximal sharing a common hyperplane.

That is, |eij| ≤ 1, 1 ≤ i ≤k, 1 ≤ j ≤n

Separating dimension: The dimension d of the separating hyperplane is bounded by a constant r such that 0≤r≤d≤n-1.

4-neighbors have r = 1 – line separation8-neighbors have r = 0 – both point- and line-separations.

That is, n - d = 1n|ei

j| ≤n – r, 1 ≤ i ≤ k,

Separating cost: The cost between neighbors - usually unity.

That is, (ei) = 1, 1≤i≤k.

Digital Neighborhood: 5 Factors

Isotropy and symmetry: The neighborhood is isotropic in all (discrete) directions.

That is, ej is a permutation and / or reflection of ei, 1 ≤ i, j ≤k.

Uniformity: The neighborhood relation is identical at all points along a path and at all points of the space

Exceptions are not uncommon.

Path – Graph-Theoretic Notion

Given a neighborhood N(·), a digital path π(u,v) connecting two points u and v is defined to be a sequence of points where all pairs of consecutive points are neighbors.

π(u, v) is {u = x0, x1, x2, …, xM = v} where xi N(xi+1), 0≤i<M.

The length of the path |π(u, v)| = (xi+1 – xi). For unit cost this is the number of points on the

path (excluding either u or v). Of all paths that connect u to v the one having the

smallest length is called the minimal path π*(u, v).

2D Paths

3D Paths

Distance Function The distance d(u, v) between u, v (w.r.t. to a

neighborhood N(·)) is the length of the shortest path connecting them.

That is d(u, v) = |π*(u, v)|. Distance Function:

d: Rn x Rn R+

Digital Distance Function:d: Zn x Zn P

Also, d(u, v) = d(0, u – v) = d(0, |u – v|), where 0 is the origin. That is, d(u, v) = d(x), where x = |u – v|.

MetricA distance function d is said to be a metric if it is:

Total: d(u, v) is defined and finite;Note that for all u, v Zn, π(u, v) may not exist, and hence d(u, v) may

not be defined. Ref. super-knight’s distance. Positive: d(u, v) ≥ 0; Definite: d(u, v) = 0, iff u = v; Symmetric: d(u, v) = d(v, u), and; Triangular: d(u, v) + d(v, z) ≥ d(u, z); for all u, v, z Zn.

Euclidean distance: En(u, v) = (i(ui – vi)2)1/2

Also En(x) = En(u, v) for x = |u – v|.

Digital Distances in 2-D

Basic Digital Distances

For u, vZ2 and x = |u – v|,Cityblock: d4(u, v) = x1 + x2,

Chessboard: d8(u, v) = max(x1, x2),

Octagonal (uses non-uniform alternating neighborhoods):doct(u, v) = max(x1, x2, 2(x1 + x2)/3) . 

Rosenfeld and Pfaltz ’68

Octagonal Distance

2 2 2

2 2 1 2 2

2 1 * 1 2

2 2 1 2 2

2 2 2

Neighborhood sequence {1,2}

d(a,b) = 10

a

b

Generalized Octagonal Distances

For u, v Z2, B being a neighborhood sequence.d(u, v; B) = d(x; B) = max(d1(x; B), d2(x; B)), where x = |u – v|,

ƒ(i) = 1≤j≤i b(j),

g(i) = ƒ(p) - ƒ(i - 1) – 1, p = |B|, d1(x; B) = max(x1, x2), and

d2(x; B) = 1≤j≤p ((x1 + x2) + g(j))/ƒ(p) .

 Example: Let B = {1, 2}, So, p = 2, ƒ(0) = 0, ƒ(1) = 1, ƒ(2) = 3 and g(1) =

2, g(2) = 1.Thus d1(x) = max(x1, x2) and

d2(x) = (x1 + x2 + 2)/3 + (x1 + x2 + 1)/3 = 2(x1 + x2)/3. 

Das et al 1987; Das & Chatterji 1990

Properties of Octagonal Distances

Metric:d(B) is a metric iff B is well-behaved, that is,

ƒ(i) + ƒ(j) ≤ ƒ(i + j), if i + j ≤ p; ≤ ƒ(p) + ƒ(i + j - p), if i + j ≥ p.

Lattice:Octagonal Distances form a Lattice where the

Partial Order is defined between B1 & B2 iff d(x; B1) ≤ d(x; B2) for all x Z2.

Das 1990

Simple Octagonal Distances

An Octagonal Distance with a single Integer Function is Simple.

d(x; B) is Simple iff b(j) = j.f(p)/p - (j-1).f(p)/p, gcd(p, f(p)) = 1.

d(x; B) = max(|x1|, |x2|, (|x1|+ |x2|)/m), where m = f(p)/p, called the effective neighborhood.

 Example: Let p = 5, f(p) = 7. Then B = {1, 1, 2, 1, 2},

d(x; B) = max(|x1|, |x2|, 5(|x1|+ |x2|)/7)

Simple Octagonal Distances are always Metric. 

Das 1992

Simple Octagonal Distances

Properties of Simple Octagonal Distances

Chamfer Computable repeated iterations of forward and reverse scans

Simple in Functional Form Simple in Neighborhood Unity in Cost Has good Circularity for Disks Works as Good Approximations for Euclidean Distance 

Best Simple Octagonal Distances

A Simple Octagonal Distance that “Best” approximates the Euclidean Distance.

Best Distance is: B = {1, 1, 2, 1, 2}

d(x; B) = max(|x1|, |x2|, 5(|x1|+ |x2|)/7)

Other Good Candidates:B = {1, 1, 2}B = {1, 2}B = {2}

Das 1992

Fun Distances

Knight’s Distance in 2D 

dknight(x) = max(x1/2 , (x1 + x2)/3) +

((x1 + x2) –

max(x1/2, (x1 + x2)/3)) mod 2,

if x (1, 0), (2, 2), = 3, if x = (1, 0), = 4, if x = (2, 2).

Das & Chatterji 1988

Knight’s Circle & Disk

Properties of Knight’s Distance

A Metric A non-Proximal Distance Results in Porous Disks Generalizations – Super-Knight's

Distances may not be well-defined

Digital Distances in 3-D

Digital Distances:For u, v Z3 and x = |u – v|,d6(u, v) = x1 + x2 + x3, (grid distance)

d18(u, v) = max(x1, x2, x3, (x1 + x2 + x3)/2)

d26(u, v) = max(x1, x2, x3) (lattice distance).

Yamashita and Ibaraki ‘86

Digital Distances in n-D

n-D Extension of Neighborhood

m-neighborhood distancesType – m neighbor:

im

i

n

i

i

nmO

miviu

2)(

)()(

0

1

Das, Chakrabarti and Chatterji ‘87

m-Neighbor Distance

m

vuvuvu

iviuvu

iviuvu

ddd

d

d

n

nn

m

ni

n

n

i

n

n

n

),(),,(max),(

)()(max),(

)()(),(

1

1

1

1

•All are metric

•n metrics in n-D

•Generalizes simple distances for 2-D & 3-D

t-cost Distance Cost between neighbors:

(u – v) = mini(t, |u(i) – v(i)|)

Cost bound: t, 1 ≤ t ≤ n i th maximum function

ntvufvu

uiu

P

t

ii

n

t

i

n

i

D

fZf

1 ,,

ofcomponent maximum th

,:

1

Das, Mukherjee and Chatterji ‘92

1 1 1

1 o 1

1 1 1

2 1 2

1 o 1

2 1 2

1 1 1

1 1 1

1 1 1

2 2 2

2 1 2

2 2 2

3 2 3

2 2 2

3 2 3

1 1 1

1 o 1

1 1 1

2 1 2

1 o 1

2 1 2

2 1 2

1 o 1

2 1 2

1 1 1

1 1 1

1 1 1

2 2 2

2 1 2

2 2 2

3 2 3

2 1 2

3 2 3

2D

3D

D2

1 D2

2

D3

3D3

2D3

1

t-cost neighbors

Path by t-Cost Distance

Properties of t-Cost Distance t-Cost Distances are Metrics Dn

1 = dnn

Dnn = dn

1

New Distance in 3-D:D3

2 = max(x1, x2, x3)+

max(min(x1, x2), min(x1, x3), min(x2, x3))

Generalized Octagonal Distance in n-D

Cyclic neighborhood sequence:

All sorted (in non decreasing order) sequences yield to metric.

There exists a necessary as well as sufficient condition on B for metricity.

Relative proportions of different neighborhood types in the sequence influence the shape of the discs.

},....,2,1{)(

)}(),.....,2(),1({

nib

nbbb

Das and Chatterji ‘ 90

Generalized Octagonal Distance in n-D: Special Cases

d({m}) = dmn, |B| = 1.

d(u, v; {m, m + 1}), |B| = 2 = d(x) = max(maxi xi, 2i xi/(2m + 1) )

(recollect doct where n = 2 and m = 1).

Das and Chatterji ‘ 90

Generalized Octagonal Distances in 3-D

Digital Distance Computation & Approximation

Chamfering for computing Distance Transform

o

a b a

Forward Scanning From Left to Right and Top to

Bottom

o

a b a

b

Backward Scanning From Right to Left and Bottom to

Top

b

Distance at o = min (Distance value at Neighboring pixel + local distance between them)

1. Initialize all distance values to a Maximum Value.

2. At every point o compute the distance value from its visited neighbors as follows:

Extend this concept with larger neighborhood and dimension.

Chamfer/Weighted distances

2c;db2b;cad;c2a;b : 4D

2b;cac;b2a;b : 3D

;2; :2D

:conditions Metric

0

.:,,, : 4D

.:,,:3D

.:, :2D

abba

xyzw

xayabzbcwcddcba

xayabzbccba

xayabba

Borgefors and her colleagues 1984-2004

b a b

a o a

b a b

c b c

b a b

c b c

b a b

a o a

b a b

c b c

b a b

c b c

2D

3D

Weighted Distance Template

s

Benchmarking with Euclidean Metric: Analytical Approach

Maximum Absolute Error(MAE)

Mean Square Error(MSE)

),(),(max vudvuEE na

vudvuEE navgm ,,

2

Borgefors ’84-04, Das and Chatterji ’92

Benchmarking Euclidean Metric: Geometric Approach

Comparing Geometric Properties of hyper-spheres.

Perimeter, area, shape-descriptors in 2-D

Surface Area, Volume, Shape-descriptors in 3-D.

Danielsson’93, Kumar et al’95, Butt and Maragos’98, Mukherjee et al’2000

Optimal m-neighbor Distance in Bounded Images

Solution of the following equation:

m: neighborhood typeM: maximum size along a dimensionn: dimension

nn

m

m

nMMm

111

11

Relative Error: t-cost distance

n

tt

vu

vuvu

EDE

n

n

tn1,1max

,

,,

3 1 ,1 ,

:boundupper theminimisingin t – Optimal

topt nn

n 1 2 3 4 … 55 56 …

topt 1 1 2 2 … 2 3 …

Finding Best Octagonal Distance

Compare the area, perimeter, volume, surface etc. with the Euclidian Discs.

Best in 2D: {1,1,2} Best in 3D: {1,1,3} They also minimize the MAE & MSE

in a finite space

Mukherjee et al’2000

Benchmarking 2D Distances

Distance Function

MAE MSE

{1,1,2} 0.087 0.026

{1,1,1,2} 0.187 0.056

{1,2} 0.118 0.043

<2,3> 0.134 0.068

<3,4> 0.081 0.025

<5,7> 0.083 0.035

<8,11> 0.073 0.030

Benchmarking 3D Distances

Distance Function

MAE MSE

{1,1,3} 0.105 0.027

{1,1,1,2,3} 0.146 0.034

{1,2} 0.269 0.070

<3,4,5> 0.118 0.043

<8,11,13> 0.107 0.043

<13,17,22>

0.107 0.043

<13,17,23>

0.107 0.034

Good Weighted distancesDimension Weights MSE MAE

2

<2,3><3,4><5,7>

<8,11>

0.0680.0250.0350.03

0.1340.0810.0830.073

3

<3,4,5><8,11,13>

<13,17,22><13,17,23><16,21,27><16,21,28>

0.0430.0430.0430.0340.0360.043

0.1180.1070.1070.1070.1030.103

4

<3,4,5,6><6,8,10,11><6,9,10,11><7,10,12,13

><8,11,13,15

>

-----

0.1840.1670.1670.1550.143

Neighborhood :: Digital Distance

m-Neighbor

t-Cost Sequence-based

(Octagonal)

Others (Knight’s /

Super-Knight’s)

Chamfer / Weighted

Proximity Proximal Proximal Proximal Non-Proximal

Proximal

Separating Dimension

Graded Maximal Graded Undefined Maximal

Separating Cost

Unity Non-Unity Unity Unity Fn of Separating Dimension

Isotropy & Symmetry

Isotropic Isotropic Isotropic Isotropic Isotropic

Uniformity

Position Independent

Position Independent

Position Dependent

Position Independent

Position Independent

Glimpses of Applications

Distance Transform

Minimum distance of a feature point from the back ground.

SuvuduDTSv

)},,({ min)(

Medial Axis Transform

A set of maximal blocks contained in the pattern.

Computation of Medial Axis Transform

• Compute the distance transform.

• Compute local maxima in the distance transformed image.

Computation of minimal set of maximal disks

1. Compute Local Maximum Blocks from the distance transformed image.

Form a relational table expressing the relationships between boundary pixels and individual disks.The problem is mapped to the covering of the list of boundary pixels with the optimal set of maximal blocks.

2.

3.

Nilson-Danielsson’96

Digital discs

RR2D

3D

d6 d26d18

RxOdxROB ,;

d4d8

Vertices of octagonal discs:

R) R, R,of(n permutatio : :3D

R)R,of(n permutatio : :2D

321

21

cba

c

cba

cbba

a

cba

ba

Application of MAT in Image Analysis

• Geometric Transformation (Kumar et al ’96)• Computation of Normals (Mukherjee et al ’02)• Thinning of binary pattern (Costa ’00, Pudney

’98)• Computation of cross-sections of 3D objects

(Mukherjee et al ’00)• Visualization of 3D objects (Mukherjee et al ’99,

Prevost and Lucas ‘00)• Image compression (Kumar et al ’95).• Shape Description (Baja and Svensson ’02)

Thinning from Distance Transform

Compute the set of Maximal Blocks.

Use them as anchor points while iteratively deleting boundary points preserving the topology.

Vincent ’91, Ragnemalm ’93, Svensson-Borgefors-Nystrom ’99

Normal Computation Normal at a point p computed by

computing the resultant vector from that point to the neighboring medial circles.

Normal Computations: Examples

Discrete Shading

Render individual medial sphere independently using Z-buffering.

Discrete Shading: Examples

Discrete Shading: Examples

Discrete Shading: Examples

Decomposition of 3D Objects

Identification of seed of a component from inner layers of Distance Transformed Image.

Seed-fusion by expansion and shrinking

Region growing by reversed DT.

Surface smoothing and merging.

Svensson-Saniti di Baja’02

Cross-sectioning

Cross-sectioning with different distance functions.

A set of objects for experimentation

Cross-sectioning: Voxel data, MAT & Sphere Approx.

Voxel Data MAT Euclidean Sphere

Approximation

Acknowledgement Azriel Rosenfeld B N Chatterji Gunilla Borgefors Hanan Samet Jayanta Mukhopadhayay P P Chakrabarti R A Melter Y V Venkatesh Innumerable others whose results have been used in this

presentation and all other works in the area. And the ICVGIP 2004 Committee for the opportunity to

present.

Thank you !