Morphological Boundary Based Shape Representation Schemes Biology Essay

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Mathematical morphology is a natural processing approach for the identification of image object since it is a technique based directly on shape [132]. Shape recognition using morphology demands systematical methods of selecting features from a given set of shapes to be classified. A more fundamental criterion for characterizing shape representation techniques involves their classification as boundary-based and region-based schemes. Boundary-based technique represents the type and position of the localized features round the boundary of a region and the objects are represented in terms of their external characteristics. In contrast, region-based scheme represents the interior of the shape. The morphology is well suited for representing shape of the object in both boundaries based and region based approaches.

The present study concentrated on schemes that derive their descriptions by traversing the morphological boundary of a shape. A number of morphological shape representation schemes have been proposed [74], [88,106,118,107,121,155] many of them use the structural approach i.e., a given shape can be expressed in the form of its simpler shape components and the relationships among the components. A set of literature survey on a boundary representation scheme assumed by the present study, improvement is still required. Because the existing schemes of boundary representations [156,157,154,151,150,152] often suffer from one or more of the following drawbacks:

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Sensitive to the starting point of the shape boundary, if the starting point changes the whole boundary sequence will change.

The boundary of the shape is broken down into discrete segments called primitives. Many techniques exists utilizing structural-based shape representation, such as chain code, boundary approximations, scale-space techniques, and syntactic techniques.

Chain code suffers from digitization noise so it is not desirable to use it directly for shape description and matching.

In addition, most of the methods are not rotation invariant.

To address these problems the present study derived a novel scheme of shape representation based on morphological boundary theory. For an efficient and precise classification the present study derived Boundary Moments (BM) on the morphological boundaries [143] from Hu moments (HM). This novel approach resolve well complex object's recognition and classification problem. This approach also resolves the complexity because the BM is evaluated only on boundary objects.

4.1.1 Boundary as a shape

The boundary shows the form of the element and having a different type of shape. Boundary of a shape is the sum of unique finite set of disjoint segments with disjoint boundaries.  Hence, the shape can be treated as maximal and the sum of the boundaries of its maximal segments represents the boundary of a maximal shape. The boundary of a shape can be classified with respect to another shape as to be inside or outside the other shape, or shared in the same way or shared oppositely between the two shapes [46, 11, 31]. The present study defines boundary as the 'outline' or 'form' of a shape. The advantage of boundary generation is one can obtain the original image from the boundary points of the boundary image [47,162]. The boundary points can be used for image compression.

The basic model of morphological boundary based shape representation and classification can be represented as shown in block diagram of Figure 4.1.

Convert gray to binary

Original Images

Extract Boundary for the image

Apply boundary based moments BM (1-7)

Determine the best BM classifier

Analyze Classification Results

Figure 4.1 Algorithm based on the combination of Morphological boundary and boundary based moment scheme.

From this, the present study defines boundary as a feature on shape that combines the information contained in the outline of the shape with information about the boundary confined by the outline of the shape, in such a way that the geometrical and topological properties of its originating object must be preserved. Based on this the present study proposes a novel Morphological Boundary based Shape Representation (MBSR) scheme on BM for classification of textures with similar shape components. The MBSR is based on Morphological Boundary Extraction (MBE), which is a simple and leading morphological shape representation algorithm.

4.2 A NOVEL MBSR SCHEME ON BM FOR CLASSIFICATION OF TEXTURES WITH SIMILAR SHAPE COMPONENTS

4.2.1 A Novel Approach for Shape Representation based on MBE

In MBSR scheme, the boundary of an image (I) is derived in terms of morphological erosions. Boundary extraction of the image (I) is obtained by first eroding I with a structuring element (SE) and then performing the set difference between I and its erosion. The morphological boundary β (I) of MBSR scheme is defined by the Equation 4.1

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(4.1)

-- Denotes morphological erosion operation

SE -- Denotes structuring element consisting of all ones in a 3x3 matrix.

4.2.2 Boundary Moments (BM) on MBSR

A number of moment-based techniques have been developed [112] for characterizing the shape of objects within an image. These techniques are useful in applications for classifying objects or for symbolically describing objects. Non linear functions were introduced by Hu [65] and Flusser and Suk [118], which are invariance in the scale, translation and rotational change of the objects. The image recognition method based on these moments has achieved good results in the majority of 2D and 3D image recognition experiment, which caused researchers widespread attention [11, 31, 47, 162]. Shape parameters are extracted from objects in order to describe their shape, to compare it to the shape of template objects, or to partition objects into classes of different shapes. In this respect the important question arises how shape parameters can be made invariant on certain transformations. For this the present study derived BM on HM.

A variety of classification methods are proposed in the literature for the past three decades. But so far no one has attempted classification of textures based on MBSR schemes especially using BM. However, the digital images in practical application are not continuous and noise-free, because images are quantized by finite-precision pixels in a discrete coordinate. After an in-depth study the present thesis found three fundamental problems related to boundary based shape representation schemes.

Sensitivity to image noise, image quantization and image transformations such as scaling and rotation.

If the images size is increased or decreased, image pixels will be interpolated or deleted.

In addition, the rotation of images also causes the change of image function.

To address this present thesis evaluated BM on MBSR schemes of original, noisy, rotated, and scaled and wavelet based texture images. The present chapter derives 11 BM as given in equations from 4.2 to 4.13.

For a digital image with density distribution function, the two dimensional (p + q) order moment is defined as follows:

(4.2)

Where p, q = 0, 1, 2 … … … … …

The double integrals are to be applied over the whole area of the object of the image including its boundary. When the geometrical moments mpq in equation (2) are referred to the object centroid they become the Central Moments, and are given by equation (4.3)

(4.3)

Where

(4.4)

(4.5)

The normalized central moment of order (p+q) is given by equation (4.6)

ηpq = (4.6)

The set of seven lowest order rotation, translation and scale invariant functions BMi include invariants up to the third order, it is termed as boundary moments and given by

BM1 = η20 +η02 (4.7)

BM2 = (η20 - η02)2 + 4 η112 (4.8)

BM3 = (η30 - η12)2 + (3η21 - η03)2 (4.9)

BM4 = (η30 + η12)2 + (η21 + η03)2 (4.10)

BM5 = (η30 - 3η12)( η30 + η12)[( η30 + η12)2 - 3(η21 + η03)2]+3(η21 - η03)( η21 + η03)[3(η30 + η12)2 - (η21 + η03)2] (4.11)

BM6 = (η20 - η02)[( η30 + η12)2 - (η21 + η03)2] + 4η11(η30 + η12)( η21 + η03) (4.12)

BM7 = (3η21 - η03)( η30 +η12)[( η30 +η12)2 - 3(η21 + η03)2] + (3η12 - η30)( η21 + η03)[3(η30 + η12)2 - (η21 + η03)2] (4.13)

In practical applications images are discrete, but the BM is strictly invariant for the continuous function. Consequently, the BM invariants may change over image geometric transformation. To advocate this problem, an analysis associated with the variation of BM invariants on image geometric transformation is presented, so as to analyze the effect of image scaling and rotation in classification problems. Based on this knowledge, the present thesis did a thorough investigation on the existing literature and found that this issue on shape representation with BM has never been quantitatively studied for image classification problems.

4.3 CLASSIFICATION OF TEXTURES USING MBSR ON BM

For an efficient classification problem the present study considered five different groups of textures namely brick, granite, fabric, mosaic and marble, where each group contains ten textures each. These textures are of similar shape. That is the reason the present study has chosen this texture group, and applied the proposed MBSR scheme on BM. The original images of these textures are displayed in chapter 2 from figure 2.9 for the five different groups of textures each with ten textures of 150x150 resolutions namely brick, granite, fabric, mosaic and marble respectively. The classification algorithm is given in Algorithm 4.1.

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Algorithm 4.1: A Novel MBSR scheme on BM.

The proposed algorithm contains six basic steps as given below.

Step 1: Convert the given color image into grey scale image.

Step 2: Convert the grey scale image into binary image.

Step 3: Apply the proposed MBSR scheme on the generated image of step 2 to obtain boundary that represents the shape of the texture in an efficient way.

Step 4: Evaluate HM on boundary of texture generated by MBSR scheme.

Step 5: Calculate the average BM value for each group of ten textures and place them in database.

Step 6: Plot the classification graph for all seven BM on MBSR scheme and determine the significant BM that classifies accurately and efficiently the given textures.

The proposed algorithm is used on the following five types of images and results are compared.

Original Images

Noise Images

Rotated Images

Scaled Images

Wavelet Transformed Images

4.3.1 Classification of Original Texture Images on MBSR Scheme Using BM

The algorithm 4.1 is applied on the five groups of textures, where each group consists of ten textures each, which results a total of 50 textures. The average BM values for each group of texture is listed in Table 4.1 and based on this a classification graph is plotted in Figure 4.2.

Table 4.1 Average BM on MBSR schemes obtained from original images.

Images

BM1

BM2

BM3

BM4

BM5

BM6

BM7

Brick

0.8857

0.0396

0.0391

0.0035

0.0000

-0.0007

0.0000

Fabric

0.5636

0.0860

0.0003

0.0002

0.0000

0.0000

0.0000

Granite

0.4788

0.0009

0.0009

0.0006

0.0000

0.0000

0.0000

Marble

0.9258

0.0121

0.0587

0.0142

-0.0007

-0.0025

-0.0021

Mosaic

1.2093

0.0202

0.0189

0.0261

0.0010

0.0024

-0.0006

Figure 4.2 Classification graphs on average BM on MBSR schemes obtained from originals images.

4.3.2 Classification of Noisy Texture Images on MBSR Schemes Using BM

The Algorithm 4.1 is applied on the same texture images by adding salt and pepper noise. The average values of the BM are listed in Table 4.2 and the classification graph is plotted in the Figure 4.3.

Table 4.2 Average BM on MBSR scheme obtained from noisy images.

Image

BM1

BM2

BM3

BM4

BM5

BM6

BM7

Brick

1.2818

0.0917

0.1476

0.0142

0.0007

-0.0059

0.0001

Fabric

0.7117

0.0009

0.0003

0.0002

0.0000

0.0000

0.0000

Granite

0.6940

0.0011

0.0017

0.0011

0.0000

0.0000

0.0000

Marble

1.3072

0.0134

0.0920

0.0283

-0.0021

-0.0046

-0.0049

Mosaic

1.8497

0.0499

0.0669

0.0745

0.0060

0.0079

-0.0026

Figure 4.3 Classification graphs on BM on MBSR scheme obtained from noisy images.

Based on the Table 4.2 and plotted graph of Figure 4.3, it is clearly evident that only BM1 on boundary of noisy images obtained from MBSR scheme classifies the given five groups of textures and all other six BM from 2 to 7 shows more or less same value and plotted in the same region of the graph. That is BM from 2 to 7 on boundary of MBSR scheme failed in classification of the textures. The average BM value of Mosaic images is having maximum value.

4.3.3 Classification of Texture Images on MBSR Scheme with Rotations Using BM

To test the transformation property of the BM on the proposed MBSR scheme, the original images are rotated from 00 to 1800 with an incremental rotation of 450. The Algorithm 4.1 is applied on the rotated texture images. The average values of the BM are listed in Table 4.3 and the classification graph is plotted in the Figure 4.4.

Table 4.3 BM on MBSR scheme obtained from rotated images.

Image

BM1

BM2

BM3

BM4

BM5

BM6

BM7

Brick

0.7939

0.0277

0.0196

0.0016

0.0000

-0.0001

0.0000

Fabric

0.4839

0.0008

0.0003

0.0002

0.0000

0.0000

0.0000

Granite

0.4730

0.0008

0.0011

0.0005

0.0000

0.0000

0.0000

Marble

0.9007

0.0107

0.0306

0.0125

0.0000

-0.0014

-0.0012

Mosaic

1.0838

0.0150

0.0141

0.0205

0.0008

0.0022

-0.0005

Based on the Table 4.3 and plotted graph of Figure 4.4, it is clearly evident that Figure 4.4 Classification graphs on BM on MBSR scheme obtained from rotated images.

only BM1 on boundary of rotated images obtained from MBSR scheme classifies the given five groups of textures. All other six BM from 2 to 7 shows more or less same value and plotted in the same region of the graph. That is these BMs on boundary of MBSR scheme failed in classification of the textures. The Mosaic texture images are having maximum value for BM1.

4.3.4 Classification of Texture Images on MBSR Scheme with Scaling Transformation Using BM

To test the transformation property of the BM on MBSR scheme with scaling transformation, the original images are scaled to double and half size and the Algorithm 4.1 is applied on the scaled image textures. The average BM of MBSR scheme is listed in Table 4.4 and classification graph is plotted in Figure 4.5.

Table 4.4 BM on MBSR scheme obtained from scaled images.

Image

BM1

BM2

BM3

BM4

BM5

BM6

BM7

Brick

1.2818

0.0917

0.1476

0.0142

0.0007

-0.0059

0.0001

Fabric

0.7117

0.0009

0.0003

0.0002

0.0000

0.0000

0.0000

Granite

0.6940

0.0011

0.0017

0.0011

0.0000

0.0000

0.0000

Marble

1.3072

0.0134

0.0920

0.0283

-0.0021

-0.0046

-0.0049

Mosaic

1.8497

0.0499

0.0669

0.0745

0.0060

0.0079

-0.0026

Figure 4.5 Classification graphs on BM on MBSR scheme obtained from scaled images.

Based on the Table 4.4 and plotted graph of Figure 4.5, it is clearly evident that only BM1 on boundary of MBSR scheme classifies the given five groups of textures and all other six BM from 2 to 7 shows more or less same value and plotted in the same region of the graph. That is BM from 2 to 7 on boundary of MBSR scheme failed in classification of the textures. The average BM value of Mosaic texture images is having maximum value for BM1.

4.3.5 Classification of Texture Images on Wavelet Based MBSR Scheme Using BM

The present study used the novel MBSR scheme using BM on Haar, Db6, CF6 and Sym8 wavelet transformed texture images, to evaluate the classification rate based on the following Algorithm 4.2.

Algorithm 4.2: A Novel MBSR Scheme on BM on Wavelet images.

The proposed algorithm contains six basic steps as given below.

Step 1: Convert the given color image into grey scale image.

Step2: Convert the grey scale image into various wavelet transformed images.

Step 3: Apply the proposed MBSR scheme on LL subbands of wavelet transformed image to obtain boundary that represents the shape of the texture in an efficient way.

Step 4: Evaluate BM on boundary of LL subband wavelet transformed texture generated by MBSR scheme.

Step 5: Calculate the average BM value for each group of ten textures and place them in database.

Step 6: Plot the classification graph for all seven BM on boundary of texture generated by MBSR scheme and determines the significant BM that classifies accurately and efficiently.

The above algorithm is applied on the one-level LL sub bands of Haar, Db6, CF6 and Sym8 wavelet transformed texture images and average values of BM on the boundary obtained from MBSR scheme is listed in the Tables 4.5 to 4.8 and Tables 4.9 to 4.12 respectively. The associated graphs are shown in the Figures 4.6 to 4.9 and Figures 4.10 to 4.13 respectively. The Haar and Db6 have shown a poor classification rate. A precise classification of textures is resulted for CF6 and Sym8.

Table 4.5 BM values on MBSR scheme for one level Haar wavelet transform on LL- subband.

Images

BM1

BM2

BM3

BM4

BM5

BM6

BM7

Brick

0.682551

0.014363

0.006270

0.001027

-0.000001

-0.000051

0.000001

Fabric

0.407330

0.001117

0.000339

0.000183

0.000000

-0.000005

0.000000

Granite

0.411039

0.000892

0.001083

0.000513

0.000000

0.000007

0.000001

Marble

0.847039

0.012741

0.040737

0.007896

0.000349

0.000399

-0.000093

Mosaic

0.808418

0.011285

0.005490

0.008382

0.000112

0.000830

-0.000075

Table 4.6 BM values on MBSR scheme for one level Db6 wavelet transform on LL- subband.

Images

BM1

BM2

BM3

BM4

BM5

BM6

BM7

Brick

0.658096

0.014257

0.006140

0.001025

-0.000001

-0.000063

0.000000

Fabric

0.401163

0.001020

0.000366

0.000153

0.000000

-0.000006

0.000000

Granite

0.402784

0.000740

0.001056

0.000446

0.000000

0.000007

0.000001

Marble

0.780895

0.008438

0.023055

0.005443

0.000022

-0.000227

-0.000225

Mosaic

0.788425

0.010750

0.004840

0.008685

0.000121

0.000876

-0.000069

Table 4.7 EHM values on MSSR scheme for one level CF6 wavelet transform on LL- subband.

Images

BM1

BM2

BM3

BM4

BM5

BM6

BM7

Brick

0.669638

0.011539

0.004797

0.001163

-0.000002

-0.000046

0.000001

Fabric

0.403748

0.000969

0.000389

0.000152

0.000000

-0.000005

0.000000

Granite

0.407252

0.000760

0.001105

0.000483

0.000000

0.000008

0.000001

Marble

0.800367

0.007861

0.019631

0.005229

0.000089

-0.000214

-0.000155

Mosaic

0.795047

0.010680

0.004726

0.008389

0.000161

0.000784

-0.000061

Table 4.8 EHM values on MSSR scheme for one level Sym8 wavelet transform on LL- subband.

Images

BM1

BM2

BM3

BM4

BM5

BM6

BM7

Brick

0.659251

0.011555

0.004485

0.001070

-0.000003

-0.000066

0.000001

Fabric

0.400782

0.000950

0.000326

0.000134

0.000000

-0.000004

0.000000

Granite

0.401953

0.000624

0.000948

0.000456

0.000000

0.000007

0.000001

Marble

0.761245

0.005550

0.023594

0.004697

0.000143

0.000047

-0.000017

Mosaic

0.782035

0.010133

0.004698

0.007868

0.000143

0.000707

-0.000034

Table 4.9 EHM values on MSSR scheme for four subband averages of Haar wavelet transform.

Images

BM1

BM2

BM3

BM4

BM5

BM6

BM7

Brick

0.429807

0.003614

0.001582

0.000274

0.000000

-0.000013

0.000000

Fabric

0.356683

0.000286

0.000091

0.000048

0.000000

-0.000001

0.000000

Granite

0.354084

0.000227

0.000274

0.000129

0.000000

0.000002

0.000000

Marble

0.470536

0.003202

0.010193

0.001980

0.000087

0.000100

-0.000023

Mosaic

0.479525

0.003295

0.001427

0.002117

0.000028

0.000208

-0.000019

Table 4.10 EHM values on MSSR scheme for four subband averages of Db6 wavelet transform.

Images

BM1

BM2

BM3

BM4

BM5

BM6

BM7

Brick

0.597007

0.003664

0.001907

0.000380

0.000000

-0.000015

0.000000

Fabric

0.5163

0.0004

0.0003

0.0001

0.0000

0.0000

0.0000

Granite

0.4833

0.0002

0.0003

0.0001

0.0000

0.0000

0.0000

Marble

0.6258

0.0024

0.0061

0.0016

0.0000

-0.0001

-0.0001

Mosaic

1.1118

0.0397

0.0182

0.0056

0.0000

0.0011

-0.0001

Table 4.11 EHM values on MSSR scheme for four subband averages of CF6 wavelet transform.

Images

BM1

BM2

BM3

BM4

BM5

BM6

BM7

Brick

0.601591

0.003052

0.001311

0.000488

-0.000001

-0.000012

0.000000

Fabric

0.516344

0.000430

0.000257

0.000113

0.000000

-0.000002

0.000000

Granite

0.474650

0.000226

0.000305

0.000135

0.000000

0.000002

0.000000

Marble

0.627373

0.002354

0.005211

0.001491

0.000022

-0.000056

-0.000039

Mosaic

0.720417

0.008724

0.002176

0.002653

0.000038

0.000148

-0.000019

Table 4.12 EHM values on MSSR scheme for four subband averages of Sym8 wavelet transform.

Images

BM1

BM2

BM3

BM4

BM5

BM6

BM7

Brick

0.596921

0.003028

0.001366

0.000361

-0.000001

-0.000016

0.000000

Fabric

0.511164

0.000407

0.000256

0.000118

0.000000

-0.000002

0.000000

Granite

0.475731

0.000211

0.000274

0.000127

0.000000

0.000002

0.000000

Marble

0.609943

0.001675

0.006133

0.001350

0.000036

0.000010

-0.000004

Mosaic

0.793925

0.023480

0.003707

0.003490

0.000105

0.001275

0.000004

Figure 4.6 Classification graph on BM values on MBSR scheme for one level Haar wavelet transform on LL- subband.

Figure 4.7 Classification graph on BM values on MBSR scheme for one level Db6 wavelet transform on LL- subband.

Figure 4.8 Classification graph on BM values on MBSR scheme for one level CF6 wavelet transform on LL- subband.

Figure 4.9 Classification graph on BM values on MBSR scheme for one level Sym8 wavelet transform on LL- subband.

Figure 4.10 Classification graph on Average of BM values on MBSR scheme for Haar wavelet transform.

Figure 4.11 Classification graph on Average of BM values on MBSR scheme for Db6 wavelet transform.

Figure 4.12 Classification graph on Average of BM values on MBSR scheme for CF6 wavelet transform.

Figure 4.13 Classification graph on Average of BM values on MBSR scheme for Sym8 wavelet transform.