New ‘Spider’ Convex Hull Algorithm

For an Unknown Polygon in Object Recognition

Dmitriy Dubovitskiy

and Jeff McBride

Director, Oxford Recognition Ltd, Cambridge, U.K.

Medical Device Specialist, Liverpool, England and Cork, Ireland

Keywords:

Convex Hull, Pattern Analysis, Segmentation, Object Recognition, Image Morphology, Machine Vision,

Computational Geometry.

Abstract:

Object recognition in machine vision system and robotic applications has, and is still, an important aspect

in automation applications of our everyday life. Although there are a lot of machine vision algorithms there

are not always entirely clear and uniﬁed solutions for particular applications. This paper is concerned one

particular step in image interpretation connected with the convex hull algorithm. This new approach to the

process of convex hull step of object recognition offers a wide range of application and improves the accuracy

of decision making on later steps. The challenging fundamental problem of computational geometry is offering

the solution in this work to solve convex hull procedure for an unknown image polygon. The unique feature

of the offered new approach is the ﬂexible intersection of all convex set points of an object on a digital

image. The convex combination points remains unknown and allow us to get the real vector space. The image

segmentation algorithm and decision making procedure working in conjunction with this new convex hull

algorithm will take robotic applications to a higher level of ﬂexibility and automation. We present this unique

procedure for automating and a new model of image understanding.

1 INTRODUCTION

The modern development in digital camera manu-

factures brings new potential for digital image pro-

cessing. The fast growing image capturing CCD ar-

ray development is capable of generating ever larger

amounts of data for processing. Storage facilities

have developed to nearly unlimited capacity, to the

extent that a human is unable to process these vol-

umes of data manually/visually. With this increased

capability in image capture and storage all of industry

and science will, in future, have to relay on stable and

robust robotic potential to interpret the acquired data.

Throughthis innovationBioinformatic, visual naviga-

tion, quality control and medical diagnostics are com-

ing to the new stage of automatic image recognition.

If we consider that each image consist of one or

several objects each of which has certain features.

The set of features that are recognisable to the human

eye makes us to acknowledge the nature of the image

that is in front of a camera. In computer memory there

is fundamental connectivity problem for the compil-

ing of the identiﬁed features into a recognisable image

with the features contained within a deﬁned border

line. This process of dividing an image into meaning-

ful regions or segments called segmentation. Due to

the nature of the image/features and/or the light con-

dition under which the image is captured,a particular

object’s features could vary from one side of an object

to another. This paper offersa new universal approach

to the segmentation task and/or the selection of Re-

gions Of Interest (ROI) for accurate border matching.

In absence of no complete and unique theoretical

model available for simulating human object recog-

nition, we consider a uniﬁed concept in this paper for

convexhull algorithm as a componentof image analy-

sis which involves the use of digital image processing

methods in attempt to provide a machine interpreta-

tion. The colour, morphological or pattern identiﬁers

for automatic image recognition could vary and/or

combine in this approach. This includes the possi-

bility of combining the set of identiﬁers to provide a

new level of stability and robustness in automatic ob-

ject recognition and decision making procedures. The

offered novel computational geometry roots has the

appearance of a virtual spider net formation on an im-

age space, so we are calling the new algorithm ”Spi-

der convex hull”. The same approach could be easily

311

Dubovitskiy D. and McBride J..

New ‘Spider’ Convex Hull Algorithm - For an Unknown Polygon in Object Recognition.

DOI: 10.5220/0004368703110317

In Proceedings of the International Conference on Biomedical Electronics and Devices (MHGInterf-2013), pages 311-317

ISBN: 978-989-8565-34-1

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

extended to multidimensional space, but for simplic-

ity in this paper we present C++ code and graphical

illustration for 2D application. This suggested solu-

tion is universal and can be used in wide range of

machine vision applications. This new level of au-

tomation provides possibilities to improve the quality

of people life in a multitude of potential applications.

2 IMAGE RECOGNITION

In this section, we consider the applied procedures

that are necessary during object recognition. These

procedures are adaptive and have no binding to a par-

ticular range of applications. A typical colour im-

age consists of mixed RGB signals. A grey-tone im-

age appears as a normal black and white photograph.

However, on closer inspection it may be seen that it

is composed or a large number of individual picture

cells or pixels. In fact, a digital image is an [x, y]

array of pixels. One can get a better feel for the dig-

ital limitations of such a digitised image by zooming

into a section of the picture that has been enlarged so

that the pixels can be examined individually. It is then

easy to appreciate that each pixel contains z grey level

digitalisation. This level will include a certain amount

of noise and so it is seldom worth digitising more

accurately than 8 bits of information per pixel. The

number of these levels depends on the signal-to-noise

ratio of the image capture devise and the analogue-to-

digital converter. Modern digital cameras can store up

to 24 bits per pixel. Note, that if the human eye can

see an object in a digitised image of practical spatial

and grey-scale resolution, then, it is in principle pos-

sible to devise a computer algorithm to do the same

thing. In a human eye, image points are organised

into a photosensitive matrix or array where each point

can be enumerated in terms of coordinates x and y.

The value of each isolated point can be represented

by value of a function I with coordinates x and y.

Here, x can be taken to represent the horizontal axis

and y the vertical axis. Video information is stored

in the same way, i.e. in terms of the function I (x, y)

(Rosenfeld, 1982; R.O. Duba, 1973). The colour con-

tent(s) of an image is very important and contributes

signiﬁcantly to the image processing operations re-

quired and the object recognition methodologies ap-

plied (Freeman, 1988). In the case of medical imag-

ing in general, colour processing and colour interpre-

tation is critical to the diagnosis of many conditions

and the interpretation of the information content of an

image by man and machine. Colour image processing

is become more and more important in object analy-

sis and pattern recognition. The numerous and non-

related algorithms for understanding two- and three-

dimensional objects in a digital image have and con-

tinue to be researched in order to design systems that

can provide reliable automatic segmentation, object

detection, recognition and classiﬁcation in an inde-

pendent environment (e.g. (E.R.Davies, 1997), (Free-

man, 1988), (Louis and Galbiati, 1990) and (Snyder

and Qi, 2004)). In relation to an object’s shape, size,

morphological similarity, texture and continuity these

tasks can be very challenging.

Several conditions needto be considered in the de-

velopment of any machine vision system: 1) the tar-

get resolution or contrast 2) The structural algorith-

mic approach including object representation 3) What

type of hardware would be suitable to reach speed and

accuracy For example, optical microscopy involves

the use of image processing methods that are often

designed in an attempt to provide a machine inter-

pretation of a biological image, whereby some deci-

sion criterion to be applied, such that a pattern of bi-

ological signiﬁcance can be recognised (Russ, 1990),

(M.A.Hornish and R.A.Goulart, 2008).

Associations between the features and an object

pattern attributes forms automatic learning context for

knowledge data base (Dubovitskiy and Blackledge,

2009),(Dubovitskiy and Blackledge, 2008), whereby

the representation of the object is assembled into the

feature vector (Grimson, 1990), (Ripley, 1996). The

knowledge data base depends on establishing equiva-

lence relations that express a ﬁt of evaluated objects

to a class with independent semantic units. Where

by assigning a particular class to an object the pattern

recognition task is accomplished.

2.1 Practical Image Recognition

Implementation

Practical image recognition systems generally contain

several stages in addition to the recognition engine it-

self. Image pre-processing is used for adjusting the

artefacts after the operation of an image acquisition

system. We consider the following sub-tasks.

Low brightness and Contrast. The correction of

brightness and contrast is usually a pre-processing

procedure, after which, the image looks clearer and

more precise. Nevertheless, it is necessary to note,

that such a correction does not provide any additional

information of value to procedures such as feature ex-

traction or boundary detection for example. The exis-

tence or otherwise of spatial frequencies is indifferent

to whether the map of the image is contrast stretched

or not. In current applications, the brightness and

contrast of the images used is sufﬁciently good for

the system to exclude this pre-processing procedure

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although in other applications of the algorithm, this

may be a necessary requirement. Image graininess

Some types of images can have a grainy structure -

often due to the nature or features of the image acqui-

sition system. It is a typical problem in those cases

where it is necessary to acquire an image with max-

imal resolution. The main problem with processing

coarse-grained maps is related to the in-practicality of

detecting the boundaries, i.e. boundaries are detected

that are associated with grains instead of the contours

of objects. A typical solution consists of smoothing

the image using minor diffusion in which the bound-

aries of the grains become fuzzy and diffused with

each other, while the contours of object remain (al-

beit over a larger spatial extent). A similar effect can

be obtained using the median ﬁlter. However, use

of the median ﬁlter includes an inevitable loss of in-

formation characterised by shallow details (i.e. low

grey level variability). In this thesis, the Wiener ﬁl-

ter (Wiener, 1949)is used which is computational ef-

ﬁcient, robust and optimal with regard to grain diffu-

sion and information preservation. This ﬁlter elimi-

nates high-frequency noise and thus does not distort

the edge of objects. Other solutions include prelimi-

nary de-zooming for the purpose decreasing grit size

up to and including the size of a separate pixel. Such a

method involves loss of shallow details however, and

thus, the size of the map (and accordingly, the pro-

cessing time) decreases. The other advantage of such

a method concerns hardware implementation, e.g. ap-

plication of a nozzle to an optical system. In situations

where the methods described here are unacceptable,

it is necessary to use a more complex quality detector

for boundary estimation which is discussed below.

Geometrical Distortions. In practice, the most im-

portant geometrical distortions are directly related to

character of an image acquisition. In the majority of

cases is possible to use a standard video camera as

the image sensor. However, the majority of industrial

production speciﬁcations for video systems use an in-

terlaced scan technique for image capture. This leads

to ‘captured lines’ in the image of both even and odd

types which leads to a time delay between neighbour-

ing lines (equal to half the acquisition time frame). If

there is a moving object in the ﬁeld of view, then its

position on even and odd lines will be different - the

picture of the object will be ‘washed’ in a horizon-

tal direction. This is a particularly important problem

in the extraction of edges. In this case, it is impossi-

ble to bleed the verticals. The elementary solution to

this problem is to simply skip the even or odd frames

(preferably the even frames as the odd frames consist

of later information). Another way is to handle even

and odd frames separately providing the processing

speed allows for practical implementation. If this is

not possible, it is necessary to use a video system with

non interlaced scanning.Over the past few years, with

the development of digital video and engineering the

capability has emerged to use digital video cameras

with high resolution. A singular advantage of this is

the uniformity of the picture without the distortions

discussed above. However, the video RGB of matri-

ces need to be analysed to avoid inter-colour distor-

tions. These distortions are connected to the geomet-

rical distribution of the RGB cells on the surface of

a CCD matrix and can be seen when increases in the

size of the digital are introduced. Special ﬁlters need

to be designed that can be used in the prevention of

this kind of distortion

Edge Detection. has gone through an evolution span-

ning more then 20 years. Two main methods of edge

detection have been apparent over this period, the ﬁrst

of these being template matching and the second, be-

ing the differential gradient approach. In either case,

the aim is to ﬁnd where the intensity gradient mag-

nitude g is sufﬁciently large to be taken as a reliable

indicator of the edge of an object. Then, g can be

thresholded in a similar way to that in which the inten-

sity is thresholded in binary image estimation. Both

of these methods differ mainly in that they proceed to

estimate g locally. However, there are also important

differences in how they determine local edge orienta-

tion, which is an important variable in certain object

detection schemes.

Each operator estimates the local intensity gradi-

ents with the aid of suitable convolution masks. In

a template matching case, it is usual to employ up

to 12 convolution masks capable of estimating lo-

cal components of the gradient in the different direc-

tions. Common edge operators used are due to Sobel

(J.M.S.Perwitt, 1970), Roberts (L.G.Roberts, 1965),

Kirsch (R.A.Kirsh, 1971), Marr and Hildreth (Marr

and E.Hildreth, 1977), Haralick (R.M.Haralick, 1980;

R.M.Haralick, 1984), Nalwa and Binford (Nalwa and

Binford, 1986) and Abdou and Pratt (Abdou and

W.K.Pratt, 1979). In the approach considered here,

the local edge gradient magnitude or the edge magni-

tude is approximated by taking the maximum of the

responses for the component mask:

g = max(g

: i = 1ton)

where n is usually 8 or 12. The orientation of the

boundary is evaluated in terms of the number of a

mask giving maximal value of amplitude of a gradi-

ent.

The integration of local operands into convex hull

algorithm is the way forward to isolate ROI or in par-

ticular cases accuracy identify object location. The

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texture analysis is the next step of image recognition.

The combination of edge detection and texture analy-

sis into convex hull offer the new accurate and reliable

stage of image processing tool.

3 TECHNOLOGY OVERVIEW

In this section, we brieﬂy review the currently avail-

able convex hull algorithms and components associ-

ated with the application. The common practice in

segmentation and image recognition technics is to use

procedure call binarisation. The principal question is

what does comes comes ﬁrst segmentation or recog-

nition? The answer is a combination of both through

the use of the convex hull approach.

The ﬁrst task in common solution is to remove

points with small amplitude of a local gradient of

brightness with the purpose of separating points of

a contour from textures, shallow details and noise.

There are two cases to the segmentation algorithm:

(i) Pixels similarities based approach

(i) Surface discontinuities based approach

The ﬁrst way is to select some value of a threshold

binarisation TR and to remove points with amplitude

of a gradient |g| < TR. Some of the thresholding pro-

cessing needs to be considered a priori.

The main problem in deﬁning the value of a

threshold TR say, is that it should be different for dif-

ferent images. Moreover, if the objects on the image

have different brightness, the value TR should be dif-

ferent for different areas of the image. The solution is

usually employed through a method of adaptive bina-

rization, based on calculating the value of a threshold

TR for small areas of the image (size 88.. 1212) - so-

called block binarisation. For each area, the average

value I

of brightness amplitude is evaluated and then

the value of a threshold is calculated as follows:

TR = k ∗ I

where k = 1.2.. 1.8 - binarisation coefﬁcient. Change

in the value of k are invalid when considerable

changes in the quality of an extracted contour occur

(as against the level of a threshold TR), and the value

k = 1.5 can be adopted for the overwhelming majority

of the images.

Another modiﬁcation of this method involves the

calculation of the level of a threshold separately for

different boundary orientations. This prevents the

deletion of important details close to brighter objects.

However, this method requires more computing cost

(as it is necessary to compute the local histograms but

allow for an increased value of a threshold TR without

loss of essential details and, as a corollary, reduction

in the quantity of false points.

In addition to the thresholding methods one can

employ multi-region based segmentation. Regions,

which are continuous, are simply connected clusters

of pixels which are mutually exclusive and exhaustive

(i.e. a pixel can only belong to a single region and all

pixels have to belong to some region). A region may

support a set of predicates; however, an adjacent re-

gion cannot support the same set of predicates. The

advantages of using a region-based segmentation are:

(1) there are far fewer regions than pixels in an im-

age, thus allowing data compression; and (2) regions

are connected and unique. The disadvantages of the

method are: (1) assumptions are made about the uni-

formity of image features; (2) a region could be erro-

neously considered to be a single surface; (3) surface

properties or such as reﬂection can produce regions of

noise.

There are two principal approaches to region-

based segmentation which are discussed below

1. Region Growing. Initially each pixel can be con-

sidered to be a separate region. Adjacent regions

are merged if they have similar properties (such as

grey-level). This merging process continues until

no two adjacent regions are similar. The similarity

between two regions is often based upon simple

statistics such as the variance measure or the range

of grey-levels within the regions. Region-based

segmentations are described in (Brice and Fen-

nema, 1970), (Yakimovsky and Feldman, 1973),

(Feldman and Yakimovsky, 1974), (C.A.Harlow

and S.A.Eisenbeis, 1973).

2. Region Splitting. Initially the image is regarded

as being a single region. Each region is recur-

sively subdivided into subregions if the region is

not homogeneous. The measure for homogene-

ity is similar to that for region growing. Robert-

son et al. subdivided a region either horizontally

or vertically if the pixel variance in the region is

large (P.W.Swain and Fu, 1973). Others - e.g.

(J.M.S.Perwitt, 1970) - use the bimodality of his-

tograms to split regions. This is referred to as the

mode method and has been extended (Ohlander,

1975; Price and Reddy, 1978) to use multiple

thresholds from the histogram of the region. How-

ever, as discussed above, a histogram gives global

information about a region. Using this method,

some pixels can be assigned a wrong label and

therefore pre- and post-ﬁltering is required (Oh-

lander, 1975).

Growing regions is a more difﬁcult task then

region splitting. However, region splitting, using

the method described in (P.W.Swain and Fu, 1973)

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(J.M.S.Perwitt, 1970), can lead to the generation of

more regions than required. Some region merging

at the end of the splitting phase is required. A

large group of segmentation techniques is used tex-

ture analysis. In the previous section, we described

methods for image segmentation based on the grey-

level properties of objects. These methods generally

work well for man-made objects which usually have

a smooth grey-level surface. We observe a textured

region as being homogeneous, although the intensity

across the region may be non-uniform. This leads to

the intensity-based segmentation methods to produce

results which do not match with our perception of the

scene.

Texture is important not only for distinguishing

different objects but also because the texture gradient

contains information describing the objects depth and

orientation. Texture can be described by its statistical

or structural properties (B.Lipkin and A.Rosenfeld,

1970). A texture surface having no deﬁnite pattern

is said to be stochastic, while texture with a deﬁ-

nite array of sub-patterns is said to be determinis-

tic. These textured surfaces can then be described

by some placement rule for the pattern primitives. In

reality, deterministic texture is corrupted by noise so

that it is no longer ideal; this is referred to as the ob-

servable texture. If the pattern making up the deter-

ministic texture itself has subpatterns, then these are

called microtextures and the larger patterns are called

macrotextures. One of most powerful texture mea-

sures is the fractal geometry. The main idea is that

fractal properties can be used like individual features

of an image or part of an image. One of the ways

to achieve adaptive thresholding has been patented in

a previous publication (Dubovitskiy and Blackledge,

2012).

Segmentation should be invariant to indexing al-

gorithms. The indexing phase has until recently been

almost entirely ignored. In practice, emphasis of the

recognition procedure has been placed on producing

reliable correspondence algorithms (Grimson, 1990).

The results of all these methods combined are not suf-

ﬁcient to automatically segment complex image struc-

tures such as medical images e.g. a cervical smear.

This can be achieved by using a local estimation and

a special suite of algorithm(s) developed with convex

hull to allow us to produce a successful result

Currently available convex hull algorithms could

be found (Andrew, 1979), (Brown, 1979). Some

Quickhull algorithms and a randomised incremental

program is available (C. B. Barber and Huhdanpaa,

1993). Let consider a problem of the object form,

where Q

are ﬁnite sets and G

(x, y) represents a lin-

ear objective function. Then the optimal solution for

convex hull obtains the value of the following:

= max

∑

(x, y)

The decomposition optimisation is possible and it

is equivalent to ﬁlter primal objective of the convex

hulls of the individual feasible sets. Most of them are

based on the decomposition of linear cost function.

However, due to either the lighting of the object, ex-

posure condition or the object positioning the object

properties can vary, making the application of the al-

gorithm not possible. Therefore there is a strong need

for local micro decision making about the object bor-

der.

4 CONVEX HULL ALGORITHM

‘SPIDER’

Suppose we have an image which is given by a func-

tion f(x, y) and contains some object described by

a set (a feature vector that may be composed of in-

teger, ﬂoating point and strings) S = {s

, s

, ..., s

We consider the case when it is necessary to deﬁne a

sample which is somewhat ‘close’ to this object. As

in the previous algorithm described, after binarisation

of image I(r, c) we acquire a two dimensional binary

representation of an object on the index map I

bin

(r, c)

which has the same dimension as the initial image

(1 corresponds to the presence of an edge or body

of the object and 0 corresponds to the background of

the image). Let us consider the task of obtaining the

co-ordinates of a convex polygon. This task is given

in the MATLAB function ‘Qhull’. The algorithm ap-

plied in this paper differs from that of MathWorks Inc

one in terms of its simplicity, reliability and fast com-

putation. The reason is that a number of cycles per-

formed is limited and equal to the total border length

of the object.

The main idea can be presented in terms of a ‘Spi-

der’, which creeps on a wall of the map and pulling

behind itself a thread. This thread is attached to the

object. At the ‘point of curvature’ the thread stores

the co-ordinates of the outer polygonal point. Each

thread could be considered as a line on an image. We

can propagate along a line of any local function. Thus,

the path on the perimeter around the object provides

the co-ordinates of all the outer polygonal points as

shown in Figure 1

Let us consider the algorithmic solution of this

task. For initial conditions we should select a position

of a thread without bends. Clearly, this will be along

one of four boundaries of the image. The direction

of a detour and the selection of the initial conditions

New'Spider'ConvexHullAlgorithm-ForanUnknownPolygoninObjectRecognition

315

Figure 1: Obtaining co-ordinates for Convex hull.

{NListDotsX[02*((maxX-minX)+(maxY-minY))] //Object’s

NListDotsY[02*((maxX-minX)+(maxY-minY))]}//dots list

ListDotsX[0]=StartX; // Sets the initial co-ordinates

ListDotsY[0]=StartY; // for other end of a thread

int nc=0,x4,y4,Mx4,My4;

double fi,cs,sn,step,r,RR,bz,sz;

for(nt=0;nt<(2*((maxX-minX)+(maxY-minY)));nt++){//Begin

fi=atan2(NListDotsY[nt]-StartY,...// creeps around

NListDotsX[nt]-StartX); // object

RR=sqrt(pow((NListDotsX[nt]-StartX),2)+...

+pow((NListDotsY[nt]-StartY),2));

cs=cos(fi);

sn=sin(fi);

if (fabs(sn)>fabs(cs)){ //Calculation

bz=fabs(sn); //the step length

sz=fabs(cs);

}else{

bz=fabs(cs);

sz=fabs(sn);

}

step=sqrt(pow(((sz*(1-bz))/bz),2)+pow((1-bz),2))+1;

for (r=0;r<=RR;r+=step){ // Searching for objects

x4=round((double)StartX + r*cs);//in way of thread

y4=round((double)StartY + r*sn);

if (*(ppg + x4*h + y4) == 1){

Mx4=x4; // saving last coordinate

My4=y4; // in temporary variables

}

if (((Mx4!=StartX)&&(My4!=StartY)) || //Stop check

((Mx4==StartX)&&(Mx4==NListDotsX[nt])&&

(Mx4!=NListDotsX[nt+1]))||((My4==StartY)&&

(My4==NListDotsY[nt])&&(My4!=NListDotsY[nt+1]))){

StartX=Mx4; // Assign new start co-ordinates

StartY=My4;

nc=nc++;

ListDotsX[nc]=StartX; // Saving list Convex hull

ListDotsY[nc]=StartY; // coordinates

}

Figure 2: C++ algorithm for Convex Hull.

do not depend on to the aforementioned conditions.

In the example considered here, the detour is clock-

wise and starts along the left vertical boundary of the

image. The solution is shown in Figure 2.

The presented algorithm is also useful for deﬁn-

ing the geometrical location of separated points or

objects. The algorithm has been successfully used in

the the developed computer recognition system. The

example of computing of the outer co-ordinates of a

polygon and a detour over the object contours are pre-

sented in Figure 3.

Figure 3: Object with Contour and Convex Hull.

5 CONCLUSIONS

The work reported in this paper is part of a wider

investigation into automating the application of im-

age recognition algorithms. Authors have been con-

cerned with the creation of the uniﬁed approach to the

Convex hull algorithm for digital image processing.

The novel algorithm has been explained in section IV.

The two main tasks concerned in this paper: (1) the

partial image analysis in terms of textural and mor-

phological properties that characterise an object (2)

the possibility of using a recursive segmentation ap-

proach for object recognition based on local features.

Both of these tasks are fulﬁlled by the geometrical

properties of the suggested solution. Novel in this ap-

proach is that analysis line is not along cartesian or

polar co-ordinates but along the object border. Fo-

cussing the analysis line by the expected object’s bor-

der allows researchers achievable scanning time and

will optimise accurate edge detection in the correct

image region. This considered convex hull algorithm

is generic and could be used in various applications.

That is why we have not included any particular ap-

plication area in this paper. One potential application

could be in medical imaging, e.g. cancer screening,

others include surface inspection system, visual navi-

gation and many more.

ACKNOWLEDGEMENTS

The authors are grateful for the advice and help of Dr

P. P. Chernov (Novolipetsk Iron and Steel Corpora-

tion), and Professor J. Blackledge.

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