CHAPTER 4 4. METHODS FOR MEASURING DISTANCE IN …shodhganga.inflibnet.ac.in/bitstream/10603/33597/12/12... · 2018-07-02 · 4. METHODS FOR MEASURING DISTANCE IN IMAGES 4.1. INTRODUCTION

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CHAPTER 4

4. METHODS FOR MEASURING DISTANCE IN IMAGES

4.1. INTRODUCTION

In image analysis, the distance transform measures the distance of each object point

from the nearest boundary and is an important tool in computer vision, image processing and

pattern recognition. In the distance transform, binary image specifies the distance from each

pixel to the nearest non-zero pixel. The euclidean distance is the straight-line distance

between two pixels and is evaluated using the euclidean norm. The city block distance metric

measures the path between the pixels based on a four connected neighbourhood and pixels

whose edges touch are one unit apart and pixels diagonally touching are two units apart.

The chessboard distance metric measures the path between the pixels based on an

eight connected neighbourhood. The quasi-euclidean metric measures are the total euclidean

distance along a set of horizontal, vertical, and diagonal line segments. A central problem in

image recognition and computer vision is determining the distance between images and

efforts which have been made to define image distances to provide intuitively reasonable

results. Estimating distances in digital image is useful in different shape representation and

shape recognition tasks [Borgefors 1994].

4.2. DISTANCE METRICS

The distance transform provides a metric or measure of the separation of points in the

image. The bwdist function calculates the distance between each pixel and the nearest

nonzero pixel for binary images. The bwdist function supports several distance metrics listed

in table 4.1.

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Table 4.1 various distance metrics

Distance Metric

Description Illustration

Euclidean The Euclidean distance is the

straight-line distance between two

pixels.

0 0 0

0 1 0

0 0 0 Image

1.41 1.0 1.41

1.0 0.0 1.0

1.41 1.0 1.41 Distance Transform

City Block

The city block distance metric

measures the path between the

pixels based on a 4-connected

neighborhood. Pixels whose edges

touch are 1 unit apart and pixels

diagonally touching are 2 units

apart.

1 0 0

0 1 0

0 0 0 Image

2 1 2

1 0 1

2 1 2 Distance

Transform

Chessboard

The chessboard distance metric

measures the path between the

pixels based on an 8-connected

neighborhood. Pixels whose edges

or corners touch are 1 unit apart.

1 0 0

0 1 0

0 0 0 Image

1 1 1

1 1 1

1 1 1 Distance

Transform

Quasi-

Euclidean

The quasi-Euclidean metric

measures the total Euclidean

distance along a set of horizontal,

vertical and diagonal line segments.

0 0 0 0 0

0 0 0 0 0

0 0 1 0 0

0 0 0 0 0

0 0 0 0 0 Image

2.8 2.2 2.0 2.2 2.8

2.2 1.4 1.0 1.4 2.2

2.0 1.0 0 1.0 2.0

2.2 1.4 1.0 1.4 2.2

2.8 2.2 2.0 2.2 2.8 Distance Transform

Chamfer

distance

The chamfer distance dM between 2

points A and B is the minimum of

the associated costs to all the paths

starting from A to B

c2 c1 c2 c1 0 c1 c2 c1 c2

c2 c1 c2 c1 0

0 c1 c2 c1 c2

Parallel, forward & backward chamfer


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4.3. EUCLIDEAN DISTANCE

4.3.1. Euclidean distance

The euclidean distance is the distance between two points in euclidean space. The two

points P and Q in two dimensional euclidean spaces and P with the coordinates (p1, p2), Q

with the coordinates (q1, q2). The line segment with the endpoints of P and Q will form the

hypotenuse of a right angled triangle. The distance between two points p and q is defined as

the square root of the sum of the squares of the differences between the corresponding

coordinates of the points. The two-dimensional euclidean geometry, the euclidean distance

between two points a = (ax, ay) and b = (bx, by) is defined as :

, by

4.3.2. Euclidean distance algorithm

Euclidean distance algorithm computes the minimum distance between a column

vector x and a collection of column vectors in the code book matrix cb. The algorithm

computes the minimum distance to x and finds the column vector in cb that is closest to x.

Figure 4.1 shows Euclidean distance algorithm.

, | |

1 1 2 2 ⋯ n n

i i

In one dimension, the distance between two points, x1 and x2, on a line is simply the

absolute value of the difference between the two points as:


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2 1 |2 1| In two dimensions, the distance between P = (p1, p2) and q = (q1, q2) as:

1 1 2 2

Step1: load the column vector x;

Step2: load the code book;

Step3: minimum distance is initially set to the first element of cb.

Step4: i.e. set idx=1;

Step5: compute distance by normalized values of (x-cb) for all cb;

Step6: if d is less than distance set distance is equal to d;

Step7: set idx=index;

Step8: end

Figure 4.1 Euclidean distance algorithm

4.3.3. Euclidean function

The input source data is a feature class; it will first be converted internally to a raster

before the euclidean analysis is performed. The resolution will be smaller of the height or

width of the extent of the feature class, divided by 250 and the resolution can be set with the

output cell size parameter. Figure 4.2 shows how euclidean function works.

Figure 4.2 Euclidean function


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The euclidean distance is calculated from the center of the source cells to the center of

each of the surrounding cells and true distance is calculated to each cell in the distance

functions. The euclidean algorithm works as follows: for each cell, the distance is calculated

to each source cell by calculating the hypotenuse, with the x-max and y-max. Figure 4.3

shows sample sonographic image of appendicitis.

Figure 4.3 Sonographic image of Appendicitis

Figure 4.4 Euclidean Distance Measure


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The shortest distance to a source is determined if it is less than the specified

maximum distance and the value is assigned to the cell location on the output raster. The

output values for the euclidean distance raster are floating-point distance values and the cell

is at an equal distance to two or more sources, which assigned to the source as encountered

first in the scanning process. The euclidean distance raster tells how close each cell is to the

nearest source and raster defines which source zone and cell value is the closest. The

euclidean direction identifies the direction to the closest source cell [Esri, 2007]. Figure 4.4

shows euclidean distance measure.

4.3.4. Image and Euclidean Distance

M X N images can be easily discussed in an MN dimensional euclidean space, called

image space. It is natural to adopt the base e1, e2 ,….., eMN to form a coordinate system of

the image space, where ekN+ l corresponds to an ideal point source with unit intensity at

location (k, l). Thus an image x = (x1, x2 ,………., xMN ) , where xkN+l is the gray level at the

(k, l)th pixel, is represented as a point in the image space, and xkN+l with respect to ekN+ l . The

origin of the image space is an image whose gray levels are zero everywhere

[Liwei Wang et al., 2005].

Although the algebra of the image space can be easily formulated as above, the

euclidean distance of images could not be determined until the metric coefficients of the basis

are given. The metric coefficients gij i, j =1, 2,…..,MN , are defined as

where <, > is the scalar product, and θij is the angle between ei and ej . Note that, if

< ei ei > = < ej ej> =…, i.e. all the base vectors have the same length, then gij depends


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completely on the angle θij. Given the metric coefficients, the euclidean distance of two

images x, y is written by

where the symmetric matrix G = (gij)MNxMN will be referred to as metric matrix.

For images of fixed size M by N , every MNth order and positive definite matrix G

induces a euclidean distance. But most of them are not appropriate for measuring image

distances. For example, suppose any two base vectors ei ,ej (i ≠ j) , no matter which pixels

they correspond to are mutually perpendicular, the basis then forms a Cartesian coordinate

system. Accordingly, G is the identity matrix, and it induces the traditional euclidean distance

given by

Geometrically, this defect is due to the orthogonality of the base vectors e1, e2

,…..,eMN which correspond to pixels. Clearly, the information about the spatial relationship,

i.e. the distances between the pixels, cannot be reflected by all mutually perpendicular base

vectors. Such information, however, often appears in intuitive image distance as in the

following statement: A slightly deformed image is very similar to the original one. Here,

slightly deformed means that pixels in the deformed image are close to the corresponding

pixels in the original image. This implies that a good euclidean distance for images should

contain the information of pixel distances. Accordingly, the metric coefficients, which define

the euclidean distance, have to be related to the pixel distances.


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If the metric coefficients depend properly on the pixel distances, the obtained

euclidean distance is insensitive to small deformation. The two distances that are considered

are: one is the image distance measured in the high dimensional image space, the other is the

pixel distance. Let PiPj , i, j = 1, 2,….., MN , be pixels. The pixel distance, written as

| Pi − Pj |, is the distance between Pi and Pj on the image lattice. For example, if Pi is at

location (k, l), and Pj is at (k′, l′) , | Pi − Pj | may be . Let euclidean

distance, set θ ij = π/2 . Then gij = cos θij = 0 hence

thus obtained euclidean distance is

4.4. CHAMFER DISTANCE

The chamfer distance relatively well approximates the euclidean distance and is

widely used because of its relatively small computational requirements as it imposes only

2 scans of the n-dimensional image independently of the dimension of the image. The

chamfer distances are widely used in image analysis of the euclidean distance with integers

[Eric Remy et al., 2000]. The chamfer distance dM between 2 points A and B is the

minimum of the associated costs to all the paths PAB from A to B :

Chamfer distance was first proposed an evaluation of two dimensional asymmetric

distance between two set of edge points. Given a template T positioned at location x in an


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image I and a binary edge map E of the image I, the basic form of chamfer distance is

calculated as

where || . ||2 is l2 norm and |T| denotes number of points in template T. Chamfer

distance can be efficiently computed as:

where DTE is a distance transform defined for every image point x ∈ I as

Meanwhile, in practice, distance transform is truncated to a constant τ

Chamfer distances are used for computing geodesic diameters to propagate chamfer

distances from a binary image, constrained to another binary image.

Figure 4.5 Chamfer Distance [Muhammad Akmal et al., 1998]


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The chamfer distance transformations rely on the assumption that it is possible to

deduce the value of the distance at a pixel from the value of the distance at its neighbours.

They offer a good approximation to the desired euclidean distance transform which is

computationally very intensive [Shengwen Guo et al., 2009].

Chamfer distance algorithms are a class of discrete algorithms that offer a good

approximation and euclidean distance transform at a lower computational cost. The distance

approximates the global distance computation with repeated propagation of local distances

within a small neighborhood mask. The approximation error depends upon the size of the

neighborhood and the selection of the local distances [Muhammad Akmal et al., 1998].

Figure 4.5 shows the sample chamfer distance.

Figure 4.6 Euclidean and chamfer distances on appendicitis image

Chamfer distances are local distances which permit to deduce a distance from the

distances of close neighbours unlike dE. The computation of the medial axis is done by a local

test and all computations are done by using integer numbers and linear operations such as +, -

and <. Figure 4.6 shows euclidean and chamfer distances on appendicitis image.


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4.5. GEODESIC DISTANCE

Geodesics are locally shortest paths in the sense that any perturbation of a geodesic

curve will increase the length. The minimal length path between two points on the surface is

the minimal geodesics connecting those points. Geodesics minimize the euclidean distance

on a surface and a geodesic distance between two vertices of a triangle mesh surface, for

instance is computed using a shortest path algorithm on the mesh graph, where the weight

associated with an edge is its length.

The geodesic distances on a three dimensional surface, is important in many fields of

computer graphics and geometric modeling. Normally, geodesic paths on a surface are

critical in surface segmentation and editing methods since cutting the surface along the

geodesic paths produces better results and minimum distortion parameterization or remeshing

of 3D models are based on the knowledge of geodesic distances . The applications include

isometry-invariant shape classification, skinning, medical imaging and geophysics

[Katz et al., 2003]. Figure 4.7 shows difference between euclidean distance and geodesic

distance on a circle.

Figure 4.7 Euclidean and geodesic distances on a circle [Balu et al., 2012]


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Geodesic has shortest paths, and perturbation of a geodesic curve will increase its

length. A geodesic distance between two vertices of a triangle mesh surface can be computed

using a shortest path algorithm on the mesh graph with an edge is its length. The Dijkstra’s

algorithm can lengths of paths very quickly, but produce paths quite different from true

geodesics and the paths created pass only through the mesh vertices. The research on

geodesic computation on a surface focuses on solving the eikonal equation. Kimmel and

Sethian proposed an optimal time algorithm for computing the geodesic distances and

extracting shortest paths on triangulated manifolds using the fast marching method

[Sifri et al., 2003].

Figure 4.8 Euclidean and geodesic distances on an appendicitis image

In figure 4.8, white lines are represent euclidean distance and blue line represents

geodesic distances on an appendicitis image. Mitchell et al (1987) presented an algorithm for

determining the shortest path between a source and a destination on an arbitrary polyhedral

surface, and seeking to approximate distance maps on a parametric surface and the eikonal

equation on a discrete grid obtained by sampling the parametric domain. Elad et al., (2003)

presented an efficient O(n) numerical algorithm for first-order approximation of geodesic

distances on parametric surfaces, where n is the number of points on the surface.


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4.5.1. Geodesic distance from a binary region on an image

Let I(x) : ψ Rd be an image (d = 3 for a color image), whose support ψ ⊂ R2 is

assumed to be continuous. Given a binary mask M (with M(x) ∈ 0, 1, ∀ x ∈ ψ) associated

to a “seed” region (or “object” region) Ω = x ∈ ψ : M(x) = 0, the unsigned geodesic

distance transform D0 (. ; M, ∇I) assigns to each pixel x its geodesic distance from Ω defined

as :

Where Pa,b is the set of all possible differentiable paths in ψ between the points a and

b and Γ(s) : R R2 indicates a path, parametrized by its arc length s ∈ [0, l(Γ)]. The spatial

derivative Γ ` (s) = ∂ Γ(s) / ∂s is the unit vector tangent to the direction of the path. The dot-

product is ensuring the maximum influence for the gradient ∇I when it is parallel to the

direction of the path Γ. The geodesic factor γ weighs the contribution of the image gradient

versus the spatial distances. Figure 4.9 shows the difference between euclidean distance and

geodesic distance.

Figure 4.9 Euclidean Distance vs Geodesic Distance [Hilaga et al., 2001]


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4.6. MANHATTAN DISTANCE

The distance between two points in a grid is based on a strictly horizontal and/or

vertical path as opposed to the diagonal. The manhattan distance is the simple sum of the

horizontal and vertical components, whereas the diagonal distance might be computed by

applying the Pythagorean Theorem [Wikipedia, 2010]. It is also called the L1 distance and of

u = (x1 , y1) and v = (x2 , y2) are two points, then the manhattan distance between u and v is

given by

------------------------------------------------ (1)

Instead of two dimensions, if the points have n-dimensions, such as a = (x1 , x2 ,..., xn)

and b = (y1 , y2 ,..., yn) then, equation (1) can be generalized by defining the manhattan

distance between a and b as

Figure 4.10 Euclidean and Manhattan distances on a appendicitis image


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Figure 4.10 shows euclidean and manhattan distances on an appendicitis image. The

yellow lines are representing euclidean and green lines are representing manhattan distance.

The Manhattan distance function computes the distance that would be travelled from one data

point to the other if a grid-like path is followed and manhattan distance between two items is

the sum of the differences of their corresponding components. Figure 4.11 shows the

difference between euclidean distance and manhattan distance.

| |

Figure 4.11 Euclidean distance vs manhattan distance [Balu et al., 2012]

4.7. CITY BLOCK DISTANCE

The city block distance is always greater than or equal to zero. The measurement

would be zero for identical points and high for points that show little similarity. The city-

block distance measuring horizontal and vertical directions and the chessboard distance takes

diagonal directions. The chamfer distance is faster than the city-block distance and city-

block distance largely over estimates distances towards 45 directions. This makes the needed

rectangular area around the moving object for the city-block distance than the chamfer

distance [Theo E. Schouten et al., 2005].


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4.7.1. The City Block Distance Transform

Consider a black and white n x n binary image: i.e. a two dimensional array where aij =

0 or 1, for i, j = 0, 1, 2, n -1. The index i stands for the row and the index j for column and

pixel (0, 0) is the upper left point in the image. The city block distance transform is to find for

each point (i , j ) its city block distance from the set of all black pixels B = ( i, j) : aij = 1.

In other words, authors compute the array

The city block distance transform is a basic operation in computer vision, pattern

recognition and robotics. For instance, if the black pixels represent obstacles, then dij tells us

how far the point (i, j) is from these obstacles. This information is useful when one tries to

move a robot in the free space hit pixels of the image and to keep it away from the obstacles.

Many algorithms have been proposed for computing the distance transform using different

distance metrics. Figure 4.12 shows difference between euclidean distance and city block

distance.

Figure 4.12 Euclidean distance vs city block distance [Sarah et al., 1999]


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4.8. CHESS BOARD DISTANCE

The chessboard distance is a metric defined on a vector space where the distance

between two vectors is the greatest of their differences along any coordinate dimension. In

two dimensions, i.e. plane geometry, if the points P and Q have Cartesian coordinates (x1, y1)

and (x2, y2), their chessboard distance is:

DChess = max ( | x2 – x1| , | y2 – y1| )

Figure 4.13 shows difference between euclidean distance and chess board distance.

Danielsson (1980) asserted that both manhattan and chessboard distance are rarely used. The

recent publication of a few transforms use manhattan distance in particular, due to the speed

of calculation but result in a coarse solution. Figure 4.14 shows difference between euclidean

distance and chess board distance matrix.

Figure 4.13 Euclidean Distance vs Chess Board Distance [Sarah et al., 1999]

Figure 4.14 Euclidean Distance vs Chess Board Distance Matrix [Antoni Moore, 2002]


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4.9. SUMMARY

The relationship and understanding among different distance measures is helpful in

choosing a proper measure for a particular application. Various distance measures, euclidean

distance, chamfer, geodesic, manhattan, city block and chess board distance are described in

detail. This chapter highlighted euclidean distance, how it is useful and described chamfer

distance as available from literature. In addition, this chapter illustrated the difference

between geodesic and euclidean distance and also demonstrated differences between

manhattan and euclidean distance, euclidean and city block distance.


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CHAPTER 4 4. METHODS FOR MEASURING DISTANCE IN …shodhganga.inflibnet.ac.in/bitstream/10603/33597/12/12... · 2018-07-02 · 4. METHODS FOR MEASURING DISTANCE IN IMAGES 4.1. INTRODUCTION