Biomedical Imaging by Youxin Mao - HTML preview

/ Home / Science (Academic) / Biomedical Imaging

PLEASE NOTE: This is an HTML preview only and some elements such as links or page numbers may be incorrect.
Download the book in PDF, ePub, Kindle for a complete version.

used to study the properties of textured scenes, for example the power spectrum reveals

calculation of a horizontal GLRLM is shown in Fig. 7.

information on the coarseness/fineness (periodicity) and directionality of a texture. Texture

directionality is preserved in the power spectrum because it allows directional and non-

Run-Length

directional components of the texture to be distinguished (Bajscy, 1973). These observations

have given rise to two powerful approaches for extracting texture primitives from the Fourier

0 1 2 2

power spectrum, namely, ring and wedge filters. Working from the origin of the power

spectrum the coarseness/fineness is measured between rings of inner radius r and r . The

0 3 2 2

1 2 2 2

size of the rings can be varied according to the application. The directionality of the texture is

3 0 3 0

rey-Level

0 1 2 3 4

0 4 0 0 0

1 2 0 0 0

2 0 2 1 0

3 3 0 0 0

found by measuring the average power over wedge-shaped regions centred at the origin of the

Fig. 7. Simple example demonstrating the formation of a GLRLM. Left, 4 4 image with four

power spectrum. The size of the wedge    depends upon the application. Fig. 8

unique grey-levels. Right, the resulting GLRLM in the direction 0

0 .

illustrates the extraction of ring and wedge filters from the Fourier power spectrum of a

32  32 test image consisting of black pixels everywhere except for a 3  3 region of white

A set of seven numerical texture measures are computed from the GTRLMs. Three of these

pixels centred at the origin.

measures are presented here to illustrate the computation of feature information using this

framework.

Short Run Emphasis,

1 N 

g 1 Nr r (

 i, j| )

f 

 

(5)

i0 j1

Long Run Emphasis,

1 Ng1 Nr 2

f 

 



j r ( i, j| ).

(6)

i0 j1

Fig. 8. Fourier power spectrum showing the extraction of ring and wedge filters. The

Grey-Level Distribution,

spectrum was generated on a 32  32 test image consisting of black pixels everywhere

except for a 3 3 region of white pixels centred at the origin.

Texture Analysis Methods for Medical Image Characterisation

where, N

  



N 1

q is the number of distinct grey-levels in the input and

and

are the

1 g  r



x ,

y ,

f 

r i j 

  ( , | ) ,

(7)

means and standard deviations of p ( i, j) . Throughout, p ( i, j)  P( i, j) R , where P( i, j) is T i0

 j1



the average of ( P  P  P  P

and R is the maximum number of resolution cells in a

RD )

where N is the maximum number of grey-levels, N is the number of different run lengths

GTSDM.

in the matrix and,

3.3 Higher-Order Statistical Texture Analysis

N 

g 1 N

T 

r r i j 

  ( , | ).

(8)

The grey-level run length method (GLRLM) is based on the analysis of higher-order

i0 j1

statistical information (Galloway, 1975). In this approach GLRLMs contain information on

the run of a particular grey-level, or grey-level range, in a particular direction. The number

of pixels contained within the run is the run-length. A coarse texture will therefore be

serves as a normalising factor in each of the run length equations.

dominated by relatively long runs whereas a fine texture will be populated by much shorter

runs. The number of runs r  with gray-level i , or lying within a grey-level range i , of run-

3.4 Fourier Power Spectrum

length j in a direction is denoted by (

R )  [ r ( i, j|)]. This is analogous to the GTSDM

Two-dimensional transforms have been used extensively in image processing to tackle

technique (Haralick et al., 1973) as four GTRLMs are commonly used to describe texture

problems such as image description and enhancement (Pratt, 1978). Of these, the Fourier

transform is one of the most widely used (Gonzalez and Woods, 2001). Fourier analysis can be

runs in the directions (00 ,450 ,900

135

and

0 ) on linearly adjacent pixels. An example of the

used to study the properties of textured scenes, for example the power spectrum reveals

calculation of a horizontal GLRLM is shown in Fig. 7.

information on the coarseness/fineness (periodicity) and directionality of a texture. Texture

directionality is preserved in the power spectrum because it allows directional and non-

Run-Length

directional components of the texture to be distinguished (Bajscy, 1973). These observations

have given rise to two powerful approaches for extracting texture primitives from the Fourier

0 1 2 2

power spectrum, namely, ring and wedge filters. Working from the origin of the power

spectrum the coarseness/fineness is measured between rings of inner radius r and r . The

0 3 2 2

1 2 2 2

size of the rings can be varied according to the application. The directionality of the texture is

3 0 3 0

rey-Level

0 1 2 3 4

0 4 0 0 0

1 2 0 0 0

2 0 2 1 0

3 3 0 0 0

found by measuring the average power over wedge-shaped regions centred at the origin of the

Fig. 7. Simple example demonstrating the formation of a GLRLM. Left, 4 4 image with four

power spectrum. The size of the wedge    depends upon the application. Fig. 8

unique grey-levels. Right, the resulting GLRLM in the direction 0

0 .

illustrates the extraction of ring and wedge filters from the Fourier power spectrum of a

32  32 test image consisting of black pixels everywhere except for a 3  3 region of white

A set of seven numerical texture measures are computed from the GTRLMs. Three of these

pixels centred at the origin.

measures are presented here to illustrate the computation of feature information using this

framework.

Short Run Emphasis,

1 N 

g 1 Nr r (

 i, j| )

f 

 

(5)

i0 j1

Long Run Emphasis,

1 Ng1 Nr 2

f 

 



j r ( i, j| ).

(6)

i0 j1

Fig. 8. Fourier power spectrum showing the extraction of ring and wedge filters. The

Grey-Level Distribution,

spectrum was generated on a 32  32 test image consisting of black pixels everywhere

except for a 3 3 region of white pixels centred at the origin.

Biomedical Imaging

In image analysis the Fourier transform F( u, v) is considered in its discrete form and the

object numerically, which in a sense is similar to the description of objects using standard

power spectrum (

P u, v) is calculated from,

Euclidean geometry. That is, the higher the dimension the more complicated the object.

However, fractal descriptors allow the description of objects by non-integer dimensions.

(

P u, v)  F( u, 2

v .

(9)

A variety of techniques are used to estimate the fractal dimension of objects which, despite

providing the same measure, can produce different fractal dimension values for analysis of

The average power contained in a ring centred at the origin with inner and outer radii r and

the same object. This is due to the unique mechanism used by each technique to find the

r respectively, is given by the summation of the contributions along the direction ,

fractal dimension (Peitgen et al., 1992; Turcotte, 1997). Two approaches commonly used to

calculate the fractal dimension of an image are discussed. The first is the box-counting

approach (Peitgen et al., 1992). The second, which treats the input as a textured surface by

(

P r)



 2  (

P r, .)

(10)

plotting the intensity at each x and y position in the z plane, calculates the fractal dimension

 0

using the Korcak method (Russ, 1994).

The contribution from a wedge of size 

w is found from summation of the radial

components within the wedge boundaries. That is,

4.1 Fractal Dimension from Box-Counting

The box-counting dimension is closely related to the concept of self-similarity where a

structure is sub-divided into smaller elements, each a smaller replica of the original

n 2

(

P 

P r 

w )   ( , ),

(11)

structure. This sub-division characterises the structure by a self-similarity, or fractal,

r0

dimension and is a useful tool for characterising apparently random structures. This

where n is the window size.

approach has been adopted in a variety of applications, for example in the characterisation

of high resolution satellite images (Yu et al., 2007) and in the detection of cracks in CT

images of wood (Li & Qi, 2007). The box-counting dimension Db of any bounded subset of A

4. Fractal Texture Analysis

in n

R , which is a set in Euclidean space, may be formally defined as follows (Stoyan &

Until the introduction of fractals it was difficult to accurately describe, mathematically,

Stoyan, 1994; Peitgen & Saupe, 1988). Let Nr( )

A be the smallest number of sets of r that

complex real-world shapes such as mountains, coastlines, trees and clouds (Mandelbrot,

cover A . Then,

1977). Fractals provide a succinct and accurate method for describing natural objects that

would previously have been described by spheres, cylinders and cubes. However, these

log N r ( A)



b ( A)

lim

(12)

descriptors are smooth, which makes modelling irregular natural scenes, or surfaces, very

r0

log(1 r)

difficult. The popularity of fractals has grown considerably over the past three decades since

the term was first coined by Mandelbrot to describe structures too complex for Euclidean

provided that the limit exists. Subdividing n

R into a lattice of grid size r  r where r is

geometry to describe by a single measure (Mandelbrot, 1977). The fractal dimension

continually reduced, it follows that N r( )

A is the number of grid elements that intersect A

describes the degree of irregularity or texture of a surface. With this approach rougher, or

more irregular, structures have a greater fractal dimension (Feder, 1988; Peitgen & Saupe,

and Db( )

A is,

1988; Peitgen et al., 1992).

log N  r( A)



b ( A)

lim

(13)

The property of self-similarity is one of the central concepts of fractal geometry (Turcotte,

r0

log(1 r)

1997). An object is self-similar if it can be decomposed into smaller copies of itself. This

fundamental property leads to the classification of fractals into two distinct groups, random

provided that the limit exists. This implies that the box counting dimension Db( )

A and

and deterministic. A good example of self-similarity is exhibited by an aerial image of an

Nr( )

A are related by the following power law relation,

irregular coastline structure that has the same appearance within a range of magnification

factors. At each magnification the coastline will not look exactly the same but only similar.

This particular feature is common to many classes of real-life random fractals, which are not

N A 

r (

)

(14)

exactly self-similar. These are referred to as being statistically self-similar. In contrast,

D ( A)

objects that do not change their appearance when viewed under arbitrary magnification are

termed strictly self-similar. These are termed deterministic fractals due to their consistency

Proof of this relation can be obtained by taking logs of both sides of equation (14) and

over a range of magnification scales. The fractal dimension describes the disorder of an

rearranging to form equation (15),