two dimensional shape and texture quantification - medical image processing

Two-Dimensional Shape and Texture Quantification

Medical Image Processing-BM4300

120405P

Shashika Chamod Munasingha

Giving a numerical representation for a special attribute that maps the human sense

Ex: Let’s think of the shape circle. Radius/diameter is a quantitative value to represent how big the circle is.

Quantification

r1

r2

Methods

1.1 Compactness

1.2 Spatial Moments

1.3 Radial Distance Measures

1.4 Chain Codes

1.5 Fourier Descriptors

1.6 Thinning

1. Shape Quantification

Computed using the perimeter (P) and the area (A) of the segmented region

𝑪 = 𝑷𝟐/𝑨

Quantifies how close a shape is to the smoothest shape; circle

When the C is getting larger for a particular shape, it moves away from the shape circle.

Perfect circle has the smallest value for C

More details: ‘State of the Art of Compactness and Circularity Measures’ by Raul S. Montero

and Ernesto Bribiesca, 2009

1.1 Compactness (C)

Calculating C

1.1 Compactness

𝐶 =(2𝜋𝑟)2

𝜋𝑟2

𝐶 = 4𝜋 𝐶 = 12.57

r

a

a

𝐶 =?

ITK Implementation

1.1 Compactness

Region growing segmentation Shape attributes class Get the area and perimeter Calculate the compactness

Advantages Invariant descriptor for

scale, rotation and translation.

Computational simplicity

Often a good estimation for shape roughness

1.1 Compactness

Drawbacks

Not always a robust estimator of shape complexity

C=15.4 C=27.6

Quantitative measurement of the distribution and the shape of set of points

A 2D digital image can be represented by 3 parameters (i ,j ,f ) ;

( i , j ) spatial coordinates and f( i, j ) be the intensity of pixels

More details : ‘Application of Shape Analysis to Mammographic Calcifications’ by Liang

Shen, Rangaraj M. Rangayyan, and J. E. Leo Desautels

1.2 Spatial Moments

The first moments of a 2D image

Infinite set of moment coefficients

Can be retrieved the original image

But not practical

Finite set of moments can be used to retain useful information about the image

For binary images it quantifies the shape

For gray scale images it quantifies the shape as well as the intensity distribution

1.2 Spatial Moments

Central moments

Translation invariant

Think of an image scaled by a factor of s (>1), the central moment will be

How could we have a scale invariant descriptor?

Scale down the image axes by a factor of µ00

( µ00

=area of the

object),then calculate the central moments or

Direct calculation

1.2 Spatial Moments

What's more?

Scale , translation and rotation invariant descriptor

Orientation of a shape

Eccentricity of a shape

1.2 Spatial Moments

Results

1.2 Spatial Moments

X 2

Rotation

New : ‘Carotid artery image segmentation using modified spatial fuzzy c-means and ensemble clustering’, by Mehdi Hassan ,Asmatullah Chaudhry, Asifullah Khan & Jin Young Kim (2012)

Shape analyzing using its boundary ; variations and its curvature

Calculating the radial distance

Let (x(n),y(n)) be the boundary pixel values

(𝑥𝑐 , 𝑦𝑐) = Centroid of the

segmented area

The normalized radial distance(r(n)) can be obtained by dividing the d(n) by the maximum d(n) value.


The r(n) sequence can be used to obtain the statistical moments

First moments

Central moments


Normalized moments

Invariant to translation, rotation and scaling

Important: The moment orders greater than 4 are not used as they are highly sensitive to noise


Results

1.3 Radial Distance Moments

X 2

Rotation

More..

The features,

Invariant as well as 𝑓21 = 𝑓2 − 𝑓1 monotonic increase with the shape complexity.


The number of times the r(n) signal crosses the mean is a measurement of the shape smoothness.

The r(n) can also be analyzed in Fourier domain (spectral) .

Coefficients with the highest value contain the essential shape information.


C

r(n) 1

N-1 n

r(n) 1

n

Also a shape boundary measurement

A shape can be quantified by the relative position of consecutive points in its boundary

Method

Each point in the boundary is assigned a label according to the relative position from the previous point

Clockwise or counter clockwise walking can be chosen

4- connectivity or 8-connectivity base can be used

1.4 Chain Codes

3 2 1

4 P 0

5 6 7

1

2 P 0

3

High resolution grid is not suitable in obtaining the chain codes

Why?

Because high resolution means high sensitivity to noise as well

Hence low resolution grid is imposed on the extracted boundary of the shape

1.4 Chain Codes

Tip like structure is due to the noise

Example : 8-connected chain code in counter clockwise direction

What about the starting point (s) of the chain code? When S changes the chain code will also be changed Hence the chain code which gives the minimum value for the sum is considered to be chain code of that shape .

1.4 Chain Codes

4 3

4 6 2

6 0 2 0

s 0

0 -> 2 ->0 ->2 ->3 ->4 ->6 ->4 ->6 ->0

A rotation invariant code can be obtained using the first difference of the chain code. Counter clockwise direction – Positive

Clockwise direction – Negative

Different for different boundaries, hence it can be used to distinguish different shapes.

But cannot be used to compare the scale

1.4 Chain Codes

2, -2, 2, 1, 1, 2, -2, 2, -6

Is that all?

If the differential chain code shows very small values (0,1) over a local section of a boundary that section tends to be smoother.

If the shape is symmetric or nearly symmetric, the differential chain code will also have symmetric parts too.

Concave and Convex regions can also be identified using differential chain code Continuous positive differences – Convex region

Continuous negative differences – Concave region

New:

‘A measure of tortuosity based on chain coding’ by E Bribiesca - Pattern Recognition, 2013 – Elsevier

‘An automated lung segmentation approach using bidirectional chain codes to improve nodule detection accuracy’ by S Shen, AAT Bui, J Cong, W Hsu - Computers in biology and medicine, 2015 - Elsevier

1.4 Chain Codes

Each pixel on a selected contour can be represented by a complex number.

The DFT

Inverse DFT

Except the first coefficient d(0) in DFT (centroid), all other coefficients are translation invariant

Important information is on lower order coefficients of d(u)


d(u) is dependent upon the starting point of pixel array

𝑑𝑠=coefficients after moving the starting point by 𝑛𝑠 pixels

FT properties can be effectively used in obtaining invariant descriptors

The rotation of the object by angle 𝜃, shift of the starting point and scale by a factor of a yields the DFT series


The starting point, translation, scale and rotation invariant descriptors can be obtained with

The magnitudes of the normalized descriptors are scale invariant

The descriptor so call the ‘shape factor (FF)’ is invariant to all above , low sensitive to noise and a good quantifier of shape complexity/roughness.


We have talked a lot about Fourier Descriptors. But,

All the results are based on the assumption; equidistance between pixel values of the contour

Can be achieved by 4-connected pixels

Then the diagonal orientations are over estimated.

If the region is large enough equidistance points can be selected along the contour, not necessarily be the adjacent pixels

New: ‘Thermography based breast cancer analysis using statistical features and fuzzy classification’ by Gerald Schaefer


The essential shape information can be represented by reducing the shape to a graph(skeleton)

Skeleton is made of the medial lines along the main shape

Medial lines are obtained via the medial axis transformation (MAT).

The computational complexity is high for the MAT algorithm.

Some iterative algorithms are introduced to reduce the computational complexity

Algorithm by Zhang and Suen is the widely used one among them

1.6 Thinning/Skeleton

Zhang and Suen Algorithm

Iterative method to remove the boundary pixels of a shape and obtain the skeleton

The segmented area/shape should be converted to binary image where inside of the contour is black (value 1) and outside is white (value 0) or the inverse.

1.6 Thinning

0 1

Z & S algorithm conti..

Talking about two sums

Non –zero neighbor sum n(𝑝1)

0 to 1 transitions sum s(𝑝1)

Ex:

1.6 Thinning

n(𝑝1) = 4 s(𝑝1) = 2

Pixel labels

Z & S algorithm conti..

Step 1 : Determining the boundary pixels 1 valued pixel with at least a 0 valued neighbor pixel is a boundary pixel

Step 2: Delete the pixel, if following 4 conditions are satisfied simultaneously Condition 1: 𝟐 ≤ 𝒏(𝒑𝟏) ≤ 𝟔

if 𝑛(𝑝1) = 1 then the pixel is an end pixel

if 𝑛(𝑝1) = 7 then the deleting may split the region

1.6 Thinning

0 1 1

1 1 1

0 1 1

0 0 0

0 1 0

0 1 0

Z & S algorithm conti.. Condition 2: s(𝒑𝟏) = 1

If more than 0 to 1 transitions ,deleting 𝒑𝟏 may lead to split the region

Condition 3: 𝒑𝟐 ∗ 𝒑𝟒 ∗ 𝒑𝟔 = 𝟎

Condition 4: 𝒑𝟒 ∗ 𝒑𝟔 ∗ 𝒑𝟖 = 𝟎

Mark 𝒑𝟏for deletion (Do not delete!) -> Delete after evaluation of all the pixels

1.6 Thinning

𝒑𝟔 = 0 or 𝒑𝟒 = 𝟎 or 𝒑𝟖 = 𝒑𝟐 = 𝟎

𝒑𝟒 = 𝟎 East border

𝒑𝟔 = 0 South border

𝒑𝟒 = 𝒑𝟐 = 𝟎 Northwest corner

Step 3 :Apply the step2, first two conditions along with the new two conditions to the result of the step2

Condition 3: 𝒑𝟐 ∗ 𝒑𝟒 ∗ 𝒑𝟖 = 𝟎

Condition 4: 𝒑𝟐 ∗ 𝒑𝟔 ∗ 𝒑𝟖 = 𝟎

These two conditions yield the north, west borders and the southeast corner in the similar manner

Mark for deletion

Delete the marked after running the step 3 to all the boundary pixels

Iteratively apply the step 2 to the results of step 3

Iteration stops when no pixel is marked in both step 2 and step 3

1.6 Thinning

𝒑𝟐 = 0 or 𝒑𝟖 = 𝟎 or 𝒑𝟒 = 𝒑𝟔 = 𝟎

Results

1.6 Thinning

Small section of an image After applying the thinning algorithm

New: ‘Human body and posture recognition system based on an improved thinning algorithm’ by F Xie, G Xu, Y Cheng, Y Tian – IET image processing, 2011 - ieeexplore.ieee.org

Texture – A regular repetition of an element or pattern on a surface. (‘Statistical Texture Analysis’

by G. N. Srinivasan, and Shobha G.)

The nature of a surface as defined by the irregularities on the surface.

Methods

2.1 Statistical Moments

2.2 Co-Occurrence Matrix Measures

2.3 Spectral Measures

2.4 Fractal Dimensions

2.5 Run-length Statistics

2. Texture Quantification

Based on,

The smooth region contains pixel value close to each other

Where as a rough region has wide variation in pixel values

Intensity histograms

Consider an image with K gray levels (0,K-1).

Region with histogram h(k)

mean value of 𝜇 = ℎ 𝑘 .𝑘𝐾−1𝑘=0

ℎ 𝑘𝐾−1𝑘=0


The nth moment about the mean is defined as

The second moment 𝒎𝟐 (variance) is an important texture measurement.

It correlates with the visual roughness perception.

But,

The third and fourth moments, skewness (symmetric nature) and kurtosis (Peakedness) respectively cannot be used as texture measurements.

Because they do not always correlate with texture

In special cases where their potential value is pre quantified we can use them as well.



Ultrasonic image of normal liver(left) , fatty liver(middle) and cirrhosis(right)

Drawbacks

No spatial information

Hence may lost important information like grainy structure

What about the following images? (50% white and 50% black pixels)


New: ‘Breast tissue classification using statistical feature extraction of mammograms’ by HS Sheshadri, A Kandaswamy

‘Gray Level Co-Occurrence Matrix’ (GLCM)

Co-Occurrence matrix

Pair of pixels with specific values

In specific orientation and distance

How often they occur

Quantification the texture by extracting the statistical measures from the GLCM

Typical orientation values 𝟎𝒐, 𝟗𝟎𝒐, 𝟒𝟓𝒐 𝒂𝒏𝒅 𝟏𝟑𝟓𝒐

Typical distance values 𝒅 = 𝟏 (0𝑜, 90𝑜) and 𝒅 = 𝟐(45𝑜, 135𝑜)

2.2 C0-Occurrence Matrix Measures

Ex: Consider an image of 6x6=36 pixels (P) and 4 gray levels(K) (0,1,2 and 3)

The matrix of 4x4 (based on gray levels) which contains count for specific d and 𝜽 H(d, 𝜽)

H( 𝟐, 𝟒𝟓𝒐)

2.2 C0-Occurrence Matrix Measures

This is not the GLCM 𝒉𝒊𝒋


Number of pixel pairs in H(d, 𝜽) < Number of pixels in the picture P’ < P (3+3+2+2+1+9+1+4 = 25) < 36 WHY? The edge pixels do not contribute in creating pairs. When d is getting larger and larger the lesser the # pairs (large number of rows and columns will be omitted) For M x N image,

Co-occurrence matrix (C(d, 𝜽) )

𝒄𝒊𝒋 = 𝒉𝒊𝒋/𝑷′ Showing the probability

Size of Co-occurrence matrix depends on number of gray levels

Very large number of gray levels Computational time increases ? ? ?

Solution

Mapping into new smaller gray scale, but

Larger steps at insignificant gray levels and smaller steps at critical gray levels


What is derived by Co-Occurrence Matrix ? ?

Angular second moment (Energy) (Homogeneity)

Inertia (Contrast)

Entropy (Randomness)


Inhomogeneous (many intensity transformations) lesser value for 𝑡1

Homogeneous lesser value for 𝑡2

Structured (less random) less value for 𝑡4

Co-Occurrence Matrix Marginal distributions Other texture matrices

Marginal distributions

Sum of probabilities of intensity sums and differences


Means and variances

Results


Ultrasonic image of normal liver(left) , fatty liver(middle) and cirrhosis(right)

New: ‘Gray-Level Co-occurrence Matrix Bone Fracture Detection’ by HUM YAN CHAI, LAI KHIN WEE , TAN TIAN SWEE , SHEIKH HUSSAIN

Some textures have periodic or almost periodic structure

They can be quantified using the Fourier transform

The DFT of M x N image f(x,y) given by

Texture orientation, grain size and texture contrast information can be retrieved from DFT

Repetitive global patterns are hard to describe in spatial domain, but shown peaks in spectrum


Magnitude of the DFT (Power spectrum)

Power spectrum Polar coordinates

𝑆 𝑢, 𝑣 ↔ 𝑄(𝑟, 𝜃)

𝑟 = 𝑢2 + 𝑣2 = frequncy of the pixel

𝜃 = tan−1(𝑣/𝑢) = 𝑜𝑟𝑖𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑝𝑎𝑡𝑖𝑎𝑙 𝑑𝑜𝑚𝑎𝑖𝑛

For a texture with given periodicity and direction, spectrum exhibits peak at that (r, 𝜃)


The presence of a texture with given periodicity(𝑟𝑡) in any direction

Texture of any size in a desired orientation (𝜃𝑡)

New: ‘Automated Diagnosis of Glaucoma Using Texture and Higher Order Spectra Features’ by U. Rajendra Acharya , Ngee Ann, Sumeet Dua, 2011

2.3 Spectral Moments

The known Euclidean spaces - 1D, 2D and 3D


𝑁 = 𝑙𝐷

log𝑁 = log 𝑙𝐷

𝐷 = log 𝑁

log 𝑙= Hausdroff dimension

N = Copies l = Scale D= dimension

a

𝜀


Sierpinski triangle

𝑁 = 𝑙𝐷

log𝑁 = log 𝑙𝐷

𝐷 = log𝑁

log 𝑙

𝐷 = log 3

log 2= 1.585 ? ? ?

(Between a 1D line and 2D shape)

D = Fractal Dimensions

When curve roughness increases the fractal dimension’s fractional part increases.

* The same observation is applied to the 3D space as well.

* An image can be represented in 3D space (2 dimensions and intensity associated)

* The area of this intensity surface

* The FD (=D) can be estimated by linear regression on the above graph (𝐴(𝜀)) can be calculated by box-counting concept)


Another method to calculate FD

Fractional Brownian Motion model

Distance between two pixels

Absolute difference of intensity values

Above two values are related with

FD

Imp: 𝐻 𝑖𝑠 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝑏𝑦 𝑎𝑝𝑝𝑙𝑦𝑖𝑛𝑔 𝑙𝑖𝑛𝑒𝑎𝑟 𝑟𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑡𝑜

Rough textures large values for D


E= Expectation Operator b= proportionality constant H= Hurst coefficient


Fractal Dimension analysis of images of human eye retina (IJRET: International Journal of Research in Engineering and Technology)

New: ‘Detection of glaucomatous eye via color fundus images using fractal dimensions’ R Kolar, J Jan - radioengineering, 2008 - dspace.vutbr.cz

A run – String of consecutive pixels with same gray values in a given orientation

Orientation is same as the Co-Occurrence matrix (𝟎𝒐, 𝟗𝟎𝒐, 𝟒𝟓𝒐 𝒂𝒏𝒅 𝟏𝟑𝟓𝒐)

Run length Matrix

Defined by specifying direction and then count the occurrence of runs for each gray levels and length in this direction

i = gray level j = run


ℎ𝜃 𝑗, 𝑖 = 𝐻𝑖𝑠𝑡𝑜𝑔𝑟𝑎𝑚 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛

Longest run occurs in the direction of striation

Shortest run occurs is orthogonal to that

Texture quantification by analyzing run-length histograms of different orientations

The run percentage

𝑝𝜃 = 1

𝑁 ℎ𝜃 𝑗, 𝑖

𝑀𝑗=1

𝐾𝑖=1

Four 𝑝𝜃 values form a feature vector

Mean and S.D. can be used for texture quantification


Thank you!