Structural Synthesis based on PCA: Methodology and Evaluation

Sriniwas Chowdhary Maddukuri

2

, Wolfgang Heidl

1

, Christian Eitzinger

1

and Andreas Pichler

2

1

Department of Machine Vision, Profactor GmbH, Im Stadtgut A2, Steyr-Gleink, Austria

2

Department of Robotics and Assistive Systems, Profactor GmbH, Steyr-Gleink, Austria

Keywords: Structural Synthesis, Surface Inspection, Decision Boundary, Principal Components Analysis, Elastic Net

Regularization.

Abstract: In recent surface inspections systems, interactive training of fault classification is becoming state of the art.

While being most informative for both training and explanation, fault samples at the decision boundary are

rare in production datasets. Therefore, augmenting the dataset with synthesized samples at the decision

boundary could greatly accelerate the training procedure. Traditionally, synthesis methods had proven to be

useful for computer graphics applications and have only been applied for generating samples with stochastic

and regular texture patterns. Presently, the state of the art synthesis methods assume that the test sample is

available and are feature independent. In the context of surface inspection systems, incoming samples are

often classified to several defect classes after the feature extraction stage. Therefore, the goal in this

research work is to perform the synthesis for a new feature vector such that the resulting synthesized image

visualizes the decision boundary. This paper presents a methodology for structural synthesis based on

principal components analysis. The methodology expects the samples of the training set as an input. It

renders the synthesized form of the input samples of training set through eigenimages and its computed

coefficients by solving a linear regression problem. The methodology has been evaluated on an industrial

dataset to validate its performance.

1 INTRODUCTION

Synthesis is defined as the process of generating a

sample which is close to the visual appearance of the

input sample. In the context of surface inspection

systems, synthesis plays a vital role in rendering

fault instances that lie on the decision boundary.

Assuming the surface inspection system is

embedded with a dynamic classifier. The dynamic

classifier uses the extracted features of the incoming

samples and performs the classification to existing

or new defect classes. The dynamic classifier

updates the decision boundary through which it

communicates its decision to the end-users (i.e.

machine operators).

Communicating classifier decision to end-users

often requires the synthesis of fault instances to

visualize the decision boundary between classes.

This led to the motivation of generating synthesized

images which are located either close to or on the

decision boundary within the n-dimensional feature

space. Therefore, a methodology for structural

synthesis based on principal components analysis is

presented in this research work.

1.1 Related Work

In general, synthesis methods assumes that the test

sample is available where small patch region is

obtained from it and then synthesized to render a

sample of arbitrary size with texture similar to that

of the test sample or fill the missing regions of the

test sample. Parametric texture synthesis methods

explicitly measure some set of statistics in the input

sample in an analysis step. In the synthesis step, an

arbitrary image (e.g. initialized with random noise)

is altered according to constraints derived from the

previously extracted statistics. (Heeger and Bergen,

1995) uses a random noise image which is modified

to match filter response histograms, obtained at

different scales in the analysis step. Good results for

stochastic textures were reported, but the method

failed to capture structured or regular texture

patterns. More recently (Portilla and Simoncelli,

2000) proposed a texture synthesis method based on

350

Maddukuri, S., Heidl, W., Eitzinger, C. and Pichler, A.

Structural Synthesis based on PCA: Methodology and Evaluation.

DOI: 10.5220/0005721403480355

In Proceedings of the 11th Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2016) - Volume 3: VISAPP, pages 350-357

ISBN: 978-989-758-175-5

Copyright

c

2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

the first and second order statistics of joint wavelet

coefficients. Similar to Heeger and Bergen, these

coefficients are obtained for different scales of the

input texture. While the synthesis quality for regular

textures could be improved, this approach still lacks

capabilities to reproduce structured textures. The

major advantage of parametric texture synthesis

methods is that the statistics of the synthesized

image can be explicitly controlled. For this reason

parametric texture synthesis is highly relevant for

experimental design in the field of human perception

(Balas, 2006) (Balas et al., 2009). All parametric

texture synthesis approaches share the disadvantage,

that several iterations are required until the multi-

scale representation of the output texture is altered to

match the statistics of the input sample. This

optimization process is time consuming, in particular

when compared to pixel based or patch based

synthesis methods (Moore and Lopes, 1999).

Current pixel-based synthesis methods are based

on (Efros and Leung, 1999) which follow a very

simple idea. Starting from an input exemplar, the

output is first initialized by copying a small seed

region from the input. The synthesized region is then

gradually grown from the initial seed by assigning

the output pixels one by one in an inside-out, onion

layer fashion. The colour of each output pixel is

determined by a neighbourhood searching for

similar neighbourhoods in the input image. From

this candidate set of matching neighbourhoods the

output pixel is chosen from the centre of a randomly

selected neighbourhood from the candidate set. This

process is repeated until all the output pixels are

assigned. In order to speed up this process, the

search for matching neighbourhoods (Wei and

Levoy, 2000) may be made more efficient using tree

search (Kwatra et al., 2005), jump-maps (Zelinka

and Garland, 2004) or k-coherence (Tong et al.,

2002); (Thumfart, 2012).

Restoring of structure in image with

missing/damaged regions is investigated e.g. in (Guo

et al., 2008) using morphological erosion and

structure feature replication or (Chen and Xu, 2010)

which builds upon a primal sketch representation

model for the reconstruction of missing structure. In

contrast to (Guo et al., 2008), the methodology

presented here describes the use of eigenimages to

synthesize the test sample. In the absence of test

sample, the proposed methodology tries to restore

the structure of the test sample using the

neighbouring samples which are either close to or on

the decision boundary. Therefore, the principal

components analysis approach of utilizing the

eigenimages was more suitable for our application.

1.2 Paper Organization

Section 2 describes the methodology for structural

synthesis. Section 3 will explain the evaluation

procedure, simulation setup and its results will be

discussed. A few details about the dataset on which

the evaluation procedure was performed will also be

given in Section 3. Section 4 concludes the research

work and provides an outlook regarding the future

work.

2 METHODOLOGY

The methodology proposed in this section focuses

on the structural synthesis of classified images. The

methodology assumes the following pre-requisites to

be fulfilled for performing structural synthesis:

Acquire a training set of classified images

Resize the training set images with constant

width and height

Extracted features of the training set images must

be available

2.1 Principal Components Analysis

Principal components analysis (PCA) is one of the

methods applied for dimensionality reduction of a

dataset while retaining most of the information in

the dataset. The principal components analysis

method is based on the following assumptions:

Dimensionality of the dataset can be efficiently

reduced by linear transformation

Information of the dataset is concentrated in

those directions where input data variance is

maximum

Assume ‘N’ number of resized images are contained

in the training set. Every resized image I(x, y) is a

matrix of size n x n of 8-bit gray scale intensity

values. Initially, the training set images are formed

into a single rectangular matrix

of size n

2

x N.

⋯

(1)

Using Eq. (1) as an input, eigenimages are then

computed by applying the PCA method (Belhumeur,

Hespanha, and Kriegman, 1997). The result of

principal components method is given by Eq. (2).

(2)

where , and represent eigenimages matrix,

score matrix and eigenvalues vector respectively. In

the next step, images are projected onto the subspace

Structural Synthesis based on PCA: Methodology and Evaluation

351

of eigenimages to compute the coefficient matrix

(Turk and Pentland, 1991). The subspace of

eigenimages is selected from the number of non-zero

elements contained in the eigenvalues vector. The

coefficient matrix is represented by Eq. (3) and Eq.

(4).

μ

(3)

…

,

(4)

where ,

and μ represent coefficient matrix,

subspace of eigenimages and column wise mean of

respectively. m corresponds to the number of

non-zero elements of . The subspace of the

coefficient matrix is dependent upon the selection of

desired number of eigenimages in

. The selection

of desired number of eigenimages in

is given by

Eq. (5).

, 01

(5)

where

and represent the scalar parameters

responsible for deriving the subspace of coefficient

matrix. The preferable choice for parameter should

be close to its higher limit (Noortiawati et al., 2006).

The subspace of coefficient matrix and the desired

number of eigenimages in

are represented by Eq.

(6) and Eq. (7).

⊆

…

,

(6)

⊆

…

(7)

where

and

represent the subspace of

coefficient matrix and desired number of

eigenimages respectively.

and

are then

applied in the later stages of the methodology for

synthesizing an image for a given new feature

vector. In the subsequent section, solving of linear

regression problem of C

s

will be described.

2.2 Linear Regression

Linear regression is a statistical approach for

modelling the behaviour of dependent variables

denoted by Y and independent variables denoted by

X. Here, the subspace of coefficient matrix C

s

resulting from the Section 2.1 is considered as the

subspace of the dependent variable Y and the

Feature matrix is considered as the independent

variable X. A linear regression model is assumed to

fit the dependent variable Y. The linear regression

model is represented by Eq. (8).

(8)

where β is the parameter matrix. , and are

represented by Eqs. (9) – (11).

1

,

⋯

,

1

,

⋯

,

⋮

1

⋮

,

⋱

⋯

⋮

,

(9)

,

,

,

⋮

,

,

⋮

,

⋯

,

⋯

⋱

⋯

,

⋮

,

⋮

⋮

(10)

,

,

⋯

,

,

,

⋯

,

⋮

,

⋮

,

⋱

⋯

⋮

,

(11)

where a and b represent the original width and

height of the resized images of the training set. In

the first instance, least squares method was applied

to estimate the parameter matrix β. The least squares

method seeks to minimize the sum of squared

residual errors

.

min

∈

(12)

where

and

represent the column vectors of

matrices and respectively. The unique solution

for the estimate of parameter vector

is given by

the Eq. (13).

(13)

To avoid overfitting situations where the ratio of

model complexity to the training set size is too high,

Elastic Net regularization method is chosen (Hastie

et al., 2001). Elastic Net regularization method

solves the cost function represented in Eqs. (14) –

(16).

min

∈

1

2

(14)

1

2

.

‖

‖

‖

‖

(15)

0, 01

(16)

where is the complexity parameter. The publicly

available package glmnet (Qian et al., 2013) is

applied for estimating the parameter vector

.

Assuming N images are available with the training

set. Let’s say N-1 images are used in the training

phase for computing l number of solutions in λ for

estimating

and k

th

image which is left out in the

VISAPP 2016 - International Conference on Computer Vision Theory and Applications

352

training phase is used for validation phase (Friedman

et al., 2010). During the validation phase, a measure

risk (i.e. absolute error) is computed for each value

of . Then, the where the measure risk is minimal

will be selected and its resulting parameter vector

is chosen. In the subsequent section, synthesizing of

images for new feature vectors will be described.

2.3 Synthesis

Synthesizing an image for a given new feature

vector

is generated using the estimated parameter

matrix

from Section 2.2 and the desired number of

eigenimages EI

rs

from Section 2.1. Initially, a

dependent variable is derived for generating new

coefficient vector and estimating the size of the

synthesized image. The new dependent variable

and the resultant coefficient vector

are

represented by Eq. (17) and Eq. (18).

(17)

,

1

(18)

Finally, the synthesized image I

syn

is obtained from

the matrix multiplication of EI

rs

and c

w

.

(19)

The synthesized image is reshaped back to an image

of n x n size. This reshaped image is then resized

using the estimated width and height which can be

retrieved as the last two entities of the new

dependent variable

.

3 EVALUATION

In this section, the evaluation procedure of the

proposed methodology will be explained in detail. A

few details about the dataset on which the evaluation

was done will be mentioned. The simulation setup

and subsequent results will also be presented here.

3.1 Sheaves Dataset

The inspected sheaves with a diameter of around

100mm are applied in wire rope-based elevators. It

comprises a steel wheel body coated with

polyurethane on the circumference that carries the

rope notch. Coating is exercised through a casting

process with the final notch shape produced in a

turning lathe. The following are the event types of

the inspected sheaves:

Type 1: Surface spots that occurred during

lathing also known as lathing damages

Type 2: Events detected by the segmentation

processes that do not correspond to surface faults

also known as false positives from segmentation

Type 3: Delamination of the coatings

Type 4: Closed cavities

Type 5: Open cavities

Type 6: Swarf (turnings) that adhere to the

circumference

Figure 1: Examples of event types on elevator sheaves.

Row-wise, left-to-right: closed cavities, open cavities,

lathing damages, delamination, swarf, false positives from

segmentation.

Typical examples of aforementioned event types

are shown in Figure 1 (Weigl et al., 2014).

Inspection of the sheaves is confined to the coated

circumference which is scanned by a line-scan

camera with a resolution of 20µm per pixel.

Classification of event types which were detected on

the elevator sheaves is performed using a random

forest classifier. The classifier is trained on a fully

expert-annotated database. The output of the

classifier consists of the extracted feature vectors

and class probability vectors of the respective

images. The features computed for the images which

correspond to the event types describe the shape,

size, intensity level, histograms of the events.

3.2 Simulation Setup

Given 6 classes of Sheaves dataset, there will be 15

unique possible class pairs. For a given class pair

(CP), images are selected in such a way that the first

two highest class probabilities present in their

respective class probability vector are close to each

other. The steps for acquiring the training set images

for all class pairs are the following:

Step 1: Class probabilities present in the class

probability vector of all the images of Sheaves

Structural Synthesis based on PCA: Methodology and Evaluation

353

dataset are sorted in ascending order

Step 2: Gradient between the first two highest

class probabilities using the sorted class

probability vector from step 1 is computed

Step 3: Five percent of the maximum class

probability is assigned as a threshold value. This

threshold value is then compared with the

gradient from step 2 for selecting specific images

from the dataset which belong to the 15 class

pairs

Step 4: Assuming the evaluation procedure has

been initiated for a given class pair. Then, the

images resulting from step 3 for the given class

pair are sorted in such a way that the gradient

(computed from step 2) values are in ascending

order

Step 5: The sorted out images from step 4 are

then used as the training set for the given class

pair

Except for ‘delamination/open cavities’,

‘delamination/swarf’ and ‘closed cavities/swarf’

class pairs, images close to the decision boundary

for the remaining 12 class pairs were found with

respect to the above selection procedure.

Let’s assume ‘N’ test samples (i.e. images) are

contained in the training set of j

th

class pair and each

test sample is of different size. For evaluation, we

are holding out the i

th

instance (test sample) from the

training set and this one instance is then synthesized.

Likewise, this process is repeated for all N number

of instances in the training set.

Synthesized form of the test samples resulted

from the evaluation are then fed to feature extraction

software for computing their respective feature

vectors.

3.3 Results

In this section, the evaluation results will be

discussed. Figure 2 represents the Euclidean distance

‘ED

r,s

’ computed between feature vectors of the test

samples (i.e. training set of j

th

class pair) and that of

the synthesized form of test samples. Here, the

distance between the feature vectors of the k

th

test

sample with the highest probability indicating that it

belongs to the first class and l

th

test sample with the

highest probability indicating that it belongs to the

second class of the training set of j

th

class pair is

considered as the performance metric. This

performance metric is denoted by ‘Ed

om

’ in Figure 2.

Assuming an ideal scenario where the computed

euclidean distance is close to zero, this will give an

indication that the synthesized form of test samples

are a close approximation of the test samples in

terms of feature vectors and visual appearance. Here,

the evaluation result is far from the ideal scenario.

However, euclidean distance of majority of the

synthesized form of test samples of ‘lathing

damages/delamination’, ‘lathing damages/closed

cavities’, ‘lathing damages/swarf’, ‘false positives

from segmentation/delamination’, ‘false positives

from segmentation/closed cavities’, ‘false positives

Figure 2: Euclidean distance versus index of synthesized test samples of feasible 12 class pairs.

0 5 10 15

0

5

10

15

index of synthesized image

Euclidean distance

CP:lathing damages

/false positives from segmentation

ED

r,s

ED

om

0 10 20

0

5

10

15

index of synthesized image

Euclidean distance

CP:lathing damages

/delamination

ED

r,s

ED

om

0 10 20

0

5

10

15

index of synthesized image

Euclidean distance

CP:lathing damages

/closed cavities

ED

r,s

ED

om

0 10 20

0

5

10

15

index of synthesized image

Euclidean distance

CP:lathing damages

/open cavities

ED

r,s

ED

om

0 10 20

0

5

10

15

index of synthesized image

Euclidean distance

CP:lathing damages

/swarf

ED

r,s

ED

om

0 10 20 30

0

5

10

15

index of synthesized image

Euclidean distance

CP:false positives from segmentation

/delamination

ED

r,s

ED

om

0 5 10 15

0

5

10

15

index of synthesized image

Euclidean distance

CP:false positives from segmentation

/closed cavities

ED

r,s

ED

om

0 10 20 30

0

5

10

15

index of synthesized image

Euclidean distance

CP:false positives from segmentation

/open cavities

ED

r,s

ED

om

0 10 20 30

0

5

10

15

index of synthesized image

Euclidean distance

CP:false positives from segmentation

/swarf

ED

r,s

ED

om

0 5 10 15

0

5

10

15

index of synthesized image

Euclidean distance

CP:delamination

/closed cavities

ED

r,s

ED

om

0 5 10 15

0

5

10

15

index of synthesized image

Euclidean distance

CP:closed cavities

/open cavities

ED

r,s

ED

om

0 2 4 6

0

5

10

15

index of synthesized image

Euclidean distance

CP:open cavities

/swarf

ED

r,s

ED

om

VISAPP 2016 - International Conference on Computer Vision Theory and Applications

354

Figure 3: Test samples of ‘lathing damages/open cavities’ class pair.

Figure 4: Synthesized form of test samples of the ‘lathing damages/open cavities’ class pair.

from segmentation/open cavities’,

‘delamination/closed cavities’ and

‘delamination/open cavities’ class pairs are within

their respective performance metric value. This

gives an indication that those synthesized form of

test samples lie within the hypersphere (formed by

Im: 1

Im: 2

Im: 3 Im: 4 Im: 5

Im: 6 Im: 7 Im: 8 Im: 9

Im: 10

Im: 11 Im: 12 Im: 13 Im: 14 Im: 15

Im: 16 Im: 17

Im: 18

Im: 19 Im: 20

Im: 21 Im: 22 Im: 23 Im: 24 Im: 25

Im: 26 Im: 27 Im: 28 Im: 29

Syn Im: 1

Syn Im: 2

Syn Im: 3 Syn Im: 4 Syn Im: 5

Syn Im: 6 Syn Im: 7 Syn Im: 8 Syn Im: 9

Syn Im: 10

Syn Im: 11 Syn Im: 12 Syn Im: 13 Syn Im: 14 Syn Im: 15

Syn Im: 16 Syn Im: 17

Syn Im: 18

Syn Im: 19 Syn Im: 20

Syn Im: 21 Syn Im: 22 Syn Im: 23 Syn Im: 24 Syn Im: 25

Syn Im: 26 Syn Im: 27 Syn Im: 28 Syn Im: 29

Structural Synthesis based on PCA: Methodology and Evaluation

355

the test samples located as points in n-dimensional

feature space) in which the corresponding test

samples are contained.

For the remaining class pairs, the synthesized

images are located away from the hypersphere in

which the corresponding test samples are comprised.

This is due to the fact that, each and every image

present in the training set of these class pairs is very

distinct in shape and structure. Nevertheless, an

average of 58.5% of the synthesized images resulted

from all 12 class pairs are within the average of the

performance metrics of all 12 class pairs.

Figures 3 and 4 represent the test samples and

the synthesized form of test samples of ‘lathing

damages/open cavities’ class pair respectively. From

Figures 3 and 4, it can be seen that the methodology

performs the synthesis of specific test samples

(which are shown in the red rectangular boxes)

reasonably well. The cause behind such result is the

structure and shape present in the i

th

specific test

sample is still visible from the remaining samples of

‘lathing damages/open cavities’ class pair and are

contained as the prominent principal components in

the eigenimage matrix at PCA stage. Therefore,

more than half of the synthesized samples (i.e.

synthesized form of test samples) are close to the

test samples in terms of visual appearance. Similarly

for the remaining class pairs, the number of

synthesized samples which were visually close with

their respective test samples was found to be approx.

in between 50% to 60%.

4 CONCLUSION

A methodology for performing structural synthesis

based on principal components analysis is

formulated here. Also, a framework for the feature

based synthesis evaluation procedure has been

presented. The methodology has been evaluated on a

dataset from elevator sheaves inspection. Test

samples on which the methodology performs the

synthesis quite good is shown with respect to a

certain class pair. Although satisfactory result was

obtained for the remaining class pairs where test

samples were fewer, the methodology might have

worked better if the class pairs consisted of high

number of test samples. In contract to state of the art

texture and structural synthesis methods which

assumes the availability of a test sample, the

proposed methodology is able to generate

synthesized form of the test sample using the

neighbouring samples which lie close to the decision

boundary. This enables an inspection system’s

human operator to visualize the decision boundary

where availability of test samples lying close to the

decision boundary is rare. The next step in this

research work is to enhance the PCA with kernel

functions to compute principal components in high-

dimensional space, related to the input space by

some nonlinear map. In addition, we intend to factor

the dominant fault orientation out of the principal

component decomposition in a similar way varying

fault sizes are handled now. Using a subspace of

these principal components, the linear regression and

synthesis steps from the methodology could be

repeated to acquire much better visual closeness of

the synthesized samples with that of test samples.

ACKNOWLEDGEMENTS

This research work was funded via the projects

‘Improving the usability of machine learning in

industrial inspection systems’ (useML, Grant

number: 840202) and ‘Generating process feedback

from heterogeneous data sources in quality control’

(Grant number: 849962) by the Austrian Research

Promotion Agency (FFG) under the scope of the

‘Information and communication technology of the

future’ program. This program is promoted by the

Austrian Federal Ministry of Transport, Innovation

and Technology (BMVIT) and the Federal Ministry

of Science, Research and Economy (BMWFW). It

reflects only the authors’ views.

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