Image Processing & Communication, vol. 19, no. 2-3, pp.59-70
DOI: 10.1515/ipc-2015-0011
59
IMAGE SEGMENTATION AND OBJECT COUNTING AS AN INSTANCE
OF TRAVELLING SALESMAN PROBLEM
PAWEŁ W OŁOSZYN
AGH University of Science and Technology
[email protected]
Abstract. The goal of this paper is to evalu-
wide spectrum of approaches including for example sim-
ate possible application of travelling salesman
ple thresholding, edge-detection, clustering and partition-
problem and its solving methods to image seg-
ing, active contours, neural networks or scale-space seg-
mentation and object counting. This approach
mentation. The majority of these methods is definitely de-
is inspired both by human skill of subitization
signed as general-purpose stage of more complex image
and by ability of biological systems to solve op-
analysis processes. The aim of segmentation in general
timization problems. Subitizing, or the ability
is to provide regions relevant for further processing, for
to determine a number of objects without count-
example objects distinguished from other objects or their
ing them, can be treated as a special case of
background. Thus various methods are developed to be
image processing focused on extracting objects
possibly robust and effective regardless of visual charac-
from background and enumerating them. This
teristics of processed scene, mimicking human ability to
paper describes a method of generating linear
distinguish objects by their perceptual differences.
image profile based on simple graph representa-
It is also possible to propose different approach which
tion in which an optimal or suboptimal cycle is
concentrates on specific task involving image segmenta-
sought. Segmentation and counting can be then
tion, for example object counting. For such a task some
carried out by dividing the profile into coherent
constraints may be imposed on processed scene, for in-
segments.
stance objects must be clearly discernible on the background or their size must not vary too much, as in the
1
Introduction
case of counting pips visible on cards or gambling dice.
Specific task can be achieved with the use of less general
Image segmentation is one of commonly encountered segmentation and counting method, and the output of the
research problem in computer vision which inspires to process is also supposed to be very specific. For example
search for robust and effective solutions. There are nu- segmentation involved in counting dice dots can lead to
merous algorithms and methods published spanning a determining their number but at the same time provide no
Unauthenticated
Download Date | 7/31/17 10:08 PM
60
P. Wołoszyn
information about shape or spatial relations between dots. This process is characteristic for small numbers of objects
It can be argued that general methods are more valu- only: it is observed in the range from 1 to 3-4 objects
able as they provide consistent solutions in variety of ap- and the response time and probability of error are lowest
plications. On the other hand more specific and narrower and almost constant for up to 4 objects. If the number of
approaches also can be interesting and inspiring as they objects grows beyond 4 the response time increases and
can adapt and bring together solutions already developed errors are introduced. Two other processes of determinfor entirely different problems. In this paper an exam- ing object numerosity are referred to as estimation and
ple of such kind of approach is discussed. The task cho- counting. Both of them are applicable to higher number
sen as the background of image segmentation is the pro- of objects and are much slower than subitizing. Estimacess known as subitizing [7], or deciding the number of tion produces only approximate results although it is still
objects in visual scene without counting them systemat- faster than counting which in contrast gives precise anically. This ability of humans and perhaps some other swers [15].
animal species [8] is an example of very specific image
There is long scientific debate on the mechanisms inprocessing task with significant limitations. According volved in subitizing and counting [11], and it still reto some research discussed further this ability could even mains unclear whether there is one common mechanism
be independent of sensory modality and include tactile or employed in both activities or two distinct processes or
acoustic perception as well.
perhaps both activities are overlapping and utilising the
The underlying problem from which a solving tech- same neural networks. Nevertheless subitizing and countnique was drawn is travelling salesman problem (TSP), ing differ in many aspects. Spatial arrangement of oba prominent graph-optimization problem with numerous jects does not influence subitizing but becomes imporpractical and scientific applications.
The solution of tant in counting [10]. On the other hand adding dis-
specially prepared TSP instance provides a path reflect- tractor objects, for example additional shapes with modiing connectivity of image regions and leading alternately fied features, which require conscious differentiation from
through counted objects and their background. By follow- counted objects prevents subitizing [17].
ing that path and observing image intensity or some other
These properties of subitizing process bring some askind of spatially distributed stimulus one can obtain a spe- sociation of image segmentation or at least its simpler
cific signature of the scene being analysed. The signa- forms designed for images with clearly discernible objects
ture contains segments characteristic to subitized objects placed on neutral background, like for example thresholdseparated by background fragments. Counting that seg- ing methods. Counting extracted objects can be seen as
ments (or gaps between them) produces the number being somewhat similar to connected-component labelling persubitized without extracting and labelling objects.
formed on binary thresholded image consisting of similar or identical objects distributed over a uniform back-
2
Subitizing as a special case of segmentation
ground. It is worth noting that in typical experiments on
subitization exactly that kind of images, for instance a
pattern of dark or bright circular dots on opposite back-
The ability to quickly, confidently and accurately deter- ground, is most frequently used.
mine numerosity of objects seen in visual scene without
There are also some works researching subitization of
enumerating them systematically is known as subitizing. differently coloured objects, for example red and green
Unauthenticated
Download Date | 7/31/17 10:08 PM
61
Image Processing & Communication, vol. 19, no. 2-3, pp. 59-70
dots [12], however the shape of objects and their vi- or even as a transient stage of that process. In other words
sual characteristics still qualify for extraction with un- objects locations are used rather as a means of spatial dissophisticated segmentation methods. Moreover, further cernment related to the fact that objects are not occupying
works suggest that colour heterogeneity does not affect the same place in the scene.
subitizing [19] and, more generally, that subitization op-
One interesting property of subitization is that it is not
erates exclusively on object locations and not on other fea- restricted to visual modality of perception. Tactile subitiztures [20].
ing has been observed in numerous experiments including
Therefore it seems conceivable to treat subitizing as a single- and multisensory (both visual and tactile) stimulaspecial case of image segmentation aimed at partition- tion [5] which provide results suggesting that subitization
ing image into several regions representing objects ex- relies on more general mechanism. The same mechanism
tracted from their background, with subsequent counting is suspected to be involved in tactile subitizing in congenthose regions. The exact size, shape, spatial arrangement, itally blind persons [4]. It suggests that image processing
colour variability, texture and other features can be ig- occurring in subitization need not be limited to optical imnored not only at the counting stage, but also during seg- ages and it could operate on other spatial intensity maps
mentation, leaving one simple parameter like for example of much simpler stimuli as well. In that case the segmenluminance alone as the only criterion for differentiating tation approach presented here still remains applicable.
objects from their background.
It imposes some limitations on such process which
3
Motivations for TSP approach
therefore will be unable to properly count objects poorly
lit, camouflaged, nested within other objects, touching Digital images are often treated as graphs. This creates
themselves or connected with other objects and so on. a bridge between image processing and graph theory and
However it is unclear whether subitizing is possible at all allows creative adaptation of graph algorithms for solvin these problematic situations as the subject literature fo- ing image-related problems. There are several works on
cuses mainly on experiments performed in model condi- image segmentation based on graph representation of imtions.
ages, for example [2, 22, 16, 18] or [3], the latter offering
It should be noted here that in the approach described brief review of related research. These approaches embelow image processing involved in subitization model is ploy such concepts as minimum spanning tree or graph
not equivalent to object detection or blob detection be- partitioning and treat image segmentation as a specific
cause it does not provide object locations in the output. kind of optimization task. In this context travelling salesEnumerating objects without knowing where they are lo- man problem which belongs to graph optimization family
cated may seem self-contradictory, nevertheless subitiz- of problems seems to be at least conceptually related. Of
ing occurs exactly without systematically shifting atten- course TSP due to its complexity class is rather avoided in
tion from one object to another whereas attention shifting exact algorithmic methods unless time and efficiency are
takes place during counting higher numerosities of ob- not taken into consideration.
jects. Although cited research states that object locations
On the other hand there exist numerous approximate
are the only factors influencing subitization, it does not heuristic techniques for solving TSP quickly and effiimply nor is there any empirical evidence that those loca- ciently [14]. These methods often use nature-inspired
tions become explicitly known as a result of subitization ideas, such as self-organizing neural maps or ant colony
Unauthenticated
Download Date | 7/31/17 10:08 PM
62
systems. Interestingly the same methods are also used in
P. Wołoszyn
4
The method proposed
image processing tasks such as image segmentation [2,
24, 23, 6] or edge detection [9, 13]. If ant colonies or According to characteristics of subitization discussed earneural networks are capable of solving TSP, then it is also lier the proposed method operates on monochromatic impossible to use this ability for solving different problems ages with pixel values representing luminance. Based on
by reducing them to instances of TSP.
the image a graph representation is constructed with verAs ant colony or neural heuristics provide only approx- tices corresponding to pixels. In order to achieve the efimate solutions of TSP, the solutions of reduced problems fect described previously the graph edges has to be prewill also be approximate, although in general it is not a pared in such a manner that the cost of travelling within
case of approximation-preserving reduction in the strict object area or within background should be lower than
sense. Nevertheless it does not completely invalidate the the cost of crossing object-background boundary. The
use of TSP as a mechanism of object counting. Some er- space where vertices are located is therefore given addirors in numerosity judgements occur naturally in all kinds tional third dimension orthogonal to two-dimensional imof counting activities – subitizing, estimation and even ex- age plane and image luminance is treated as a height map
plicit counting.
determining vertices elevation. The graph thus becomes
The solution of TSP instance takes the form of cyclic three-dimensional and each vertex location is given by
path which is particular ordering of all graph vertices. The two planar pixel coordinates and one luminance coordiimage being processed can be treated as a grid graph con- nate.
sisting of image pixels and any Hamiltonian cycle in such
Given such a graph the most natural way of assigning
a graph arranges pixels in a sequence. In order to use that edge weights is to use Euclidean distance. The choice
sequence to determine the number of objects some useful of distance results in additional property of TSP conproperty of pixel ordering is required.
sidered in the graph which becomes Euclidean TSP. Al-
Perhaps the simplest example of such a property is though it does not influence many TSP-solving heurisgrouping pixels by their object membership: pixels be- tics it still gives some advantage over non-Euclidean or
longing to the same object occupy contiguous segment of non-metric TSP with respect to more strict solving techthe sequence separated from other similar segments with niques [1]. Other metrics are also possible for calculatgroups of pixels belonging to the background. In spite of ing edge weights as long as they retain proper relations
losing object locations and their spatial relations an order- between geometric and luminance distances of vertices
ing of pixels having such property can be regarded as a and prevent excessive travelling between objects and the
background. For example maximum metric can be probspecific image segmentation.
Travelling along cyclic path and enumerating consecu- lematic because it does not distinguish adjacent vertices
tive pixels produces cyclic profile of pixel values with as with the same luminance from adjacent vertices with lumany distinct segments as the number of objects present minance difference smaller than pixel-to-pixel distance.
in the image. The order of segments is in fact unimpor-
As the luminance becomes a dimension there arises the
tant because it is only sufficient to count segments without need for adopting a scaling factor responsible for balancidentifying them on the original image. For similar reason ing geometric and luminance distances between image
the path leading through all the pixels need not reflect any pixels. Assume that image pixels form a square grid, the
geometric properties of image content.
distance between two closest pixels serves as the unit and
Unauthenticated
Download Date | 7/31/17 10:08 PM
Image Processing & Communication, vol. 19, no. 2-3, pp. 59-70
63
equals 1 and the maximal luminance difference between plete graph by setting prohibitively large weights on uncounted objects and background also determine the unit desirable edges, although it makes the TSP instance nonvalue in the third dimension. If pixel values are no further metric.
scaled it leads to situation where the distance between two
After the graph is constructed the search for optimal or
pixels of the same value connected diagonally, namely suboptimal Hamiltonian cycle is performed. The choice
√
2, is the same as the distance between object pixel and for TSP solving method is of course inconsequential as
adjacent background pixel. Equal distances will not sup- long as the quality of solution does not fall too low repress optimal or suboptimal Hamiltonian cycle from re- sulting in disintegrating object segments and interleaving
peatedly crossing object-background boundary.
them with background. Actually in the experiments deTherefore the luminance has to be further scaled and scribed below a specific method was used for solving TSP
the scaling factor must be larger than the longest distance which has been described in [21]. The method involves
between any two connected pixels. This in turn raises an- an agent system consisting of two types of agents which
other issue concerning pixels connectivity as it must be generate and modify cyclic paths while selecting only the
decided whether the TSP graph should be complete or longest ones with the lowest cost. Although the method,
there should be some limited scope of vertex neighbour- being a distributed variant of greedy heuristic, does not
hood directly connected to it with edges. If the neighbour- surpass other methods in efficiency or solution quality, it
hood is too wide the TSP solution path can directly jump is interesting because of its design which does not rely
between pixels with similar luminance belonging to sep- upon geometric interpretation of TSP instance which need
arate objects. On the other hand if the neighbourhood is not be Euclidean nor even metric.
too narrow and the degree of vertices becomes too low it
The method utilizes only very basic operations like seis possible that TSP solving process will run out of edges quence concatenation or comparation and does not require
connecting pixels belonging to the same object and it will intelligent agents or sophisticated algorithms. It could be
force the solution to repeatedly leave and enter the same implemented in artificial neural network using time interobject thus fragmenting it into more than one segment.
vals as the representation of edge weights. This choice of
It follows therefore that in the projection on two- TSP solver is conscious and in author’s opinion it matches
dimensional image plane the maximal distance between the main goal of presented approach which is to model
connected vertices should be less than the minimal dis- the process of subitization that seems to rely upon evolutance between pixels discerned as belonging to two sepa- tionary old and simple mechanism implemented in neural
rate objects. The rules concerning luminance scaling and system.
connectivity range stated above apply also to non-square
The final step in the entire procedure is building
and irregular graphs where locations of vertices in two and analysing image profile obtained by travelling along
geometric dimensions do not correspond to pixel grid but Hamiltonian cycle and enumerating luminance values asinstead are distributed in some other, even random fashion sociated with vertices. Since the path is cyclic, it is not imand luminance values are interpolated or otherwise sam- portant at which vertex the profile will begin or end. The
pled from original image at arbitrary location. Resulting profile should consist of more or less homogeneous seggraph still can be treated as Euclidean if the complete- ments with values corresponding to image background or
ness of the graph is not required by TSP solver involved counted objects joined with rising or falling edges. Countin further processing. It can be also converted to com- ing one chosen type of edges in the full cycle will produce
Unauthenticated
Download Date | 7/31/17 10:08 PM
64
P. Wołoszyn
Fig. 2: Depending on TSP solution quality and symmetry
two visually distinct but narrowly spaced shapes can be
segmented as two separate objects (left) or connected into
one segment (right).
Fig. 1: Example graph showing pixel connectivity. Vertices are connected with 8 neighbours which are closest in
this two-dimensional projection. Boundary vertices have ter readability of figures and plots luminance is inverted
less neighbours but the same radius of neighbourhood. with higher values representing darker shades of grey. Of
Vertex colour corresponds to its third coordinate.
course it does not influence Euclidean distance between
vertices and the rest of process is identical regardless of
the number of objects subitized in the image. To make it
more resistant to noise and other fluctuations in luminance
some kind of hysteresis is needed. In the following section the results of several experiments are presented illustrating discussed problems and showing how the proposed
method performs in some artificial and real-life examples.
5
Experimental results
whether there are subitized black objects on white background or white objects on black background.
TSP solutions are then found with the use of aforementioned agent system which gradually generates closed
cycles containing growing number of vertices. Strictly
speaking these are suboptimal solutions and their quality
is deliberately reduced by shortening solving time which
results in various defects: one or more vertices can be
According to the procedure described above an ex- omitted from the solution and the path can be crossing
perimental system was developed to evaluate proposed itself several times.
method and investigate its properties. Image graphs are
Example solutions are shown on Fig. 2 which also il-
constructed from greyscale images which are resampled lustrates the problem of object spacing: two objects which
from higher resolution images to lower resolution match- are visually distinct can be situated so closely that some of
ing the number of graph vertices arranged in square grid. their vertices become directly connected with edge. The
Pixel neighbourhood is limited to 8 adjacent and diago- cycle found as the solution may or may not pass through
nally adjoining neighbours and thus maximal edge length that edge and in result the objects may be segmented as
√
allowed is 2 measured in two-dimensional projection one or two separate segments.
onto x-y image plane (Fig. 1).
The method does not rely on any kind of shape analysis
Vertices luminance, shown on figures as a shade of grey and it tries to model very basic process taking into conin each vertex, is converted to z coordinate and scaled by sideration only local luminance differences and avoiding
a factor of 2 for the earlier discussed reason of avoiding wider contexts. The only global aspect considered in the
cycles entering objects multiple times. To achieve bet- method is the total cost of cyclic path found in the graph.
Unauthenticated
Download Date | 7/31/17 10:08 PM
65
Image Processing & Communication, vol. 19, no. 2-3, pp. 59-70
minance profile obtained by travelling along solution cycle is depicted in the lower part of the figure. In the profile there are three segments corresponding to black dots,
each of them consisting of vertices visited in some order
not necessarily related to object shape.
The length of segments is naturally proportional to the
area of objects but the position of segments within the entire cycle is not corresponding in any way to objects spatial relations including their spacing. More or less regular
placement of segments within profile on Fig. 3 should be
Fig. 3: Results of processing sample binary image. Original image (upper left) is used as a height map for 20 × 20
graph. Suboptimal cycle found in the graph (upper right)
is then used to generate image profile (lower). Horizontal
axis corresponds to graph distance travelled along one full
cycle. For the sake of clarity vertices are not numbered on
horizontal axis and the vertical luminance scale has been
inverted with 0 corresponding to white and 1 to black.
interpreted only as a fortunate coincidence. If the TSP
solver tries to avoid self-crossing paths, either explicitly
by looking for intersections or indirectly by thoroughly
optimizing cycle cost, it will encounter difficulties in exploring narrow passages of background between objects.
In general if the path enters such a passage from one side,
it should leave it to the same side thus travelling in both
directions through the passage.
Therefore it will not be possible to prevent connecting ob-
If objects become more tightly spaced, it is probable
jects which are in fact already connected with an edge. that the lack of background vertices and creation of very
Excluding that specific edge requires excluding all other narrow passages in some cases will force the solution cyedges with the same weight measured in two-dimensional cle to travel from one object to another in very short path,
projection because the Euclidean distance in x-y plane is as can be seen on Fig. 4. It means that the spacing bethe only criterion of adding edge to the graph.
tween segments in image profile not only does not reflect
Fig. 2 also shows some artefacts in both solutions con- objects placement but also can become the more distorted
sisting of single vertices stuck inside a longer sequence of the closer the objects are to each other.
opposite colour points, for example single black vertex in
As the main goal of the entire procedure is to count
the middle of white segment. These artefacts result from segmented objects the profile can be converted to a numinsufficient quality of solutions but were left in presented ber for example by simply counting raising edges passresults on purpose to demonstrate what kind of problems ing through a threshold halfway between black and white
can be encountered if the TSP solving mechanism pro- level. However such a method can be inaccurate if false
vides inexact solutions. The artefacts are in fact premature edges are present due to TSP solving artefacts interruptcrossings of object-background boundary without visiting ing uniform segments as illustrated also on Fig. 4. These
all vertices belonging to the object.
artefacts take a form of white or black spikes on image
On Fig. 3 an example of image and results of its pro- profile. As a method of eliminating these spikes some
cessing are shown. This is a model example of segment- form of hysteresis can be added while processing the iming and subitizing binary image consisting only of white age profile.
background and three widely spaced black dots. The lu-
Hysteresis occurs naturally in biology, including neuUnauthenticated
Download Date | 7/31/17 10:08 PM
66
Fig. 4: Another example with three objects. Figure layout
is the same as on Fig. 3, but the objects are larger. Original
image is binary but interpolation during resampling introduced some intermediate shades of grey into the graph.
P. Wołoszyn
Fig. 5: Processing an image of two objects of contrasting
colours against a linear gradient background. Each end of
background gradient has the same luminance as the object
on the opposite side. Figure layout is the same as in Fig. 3.
ral systems and thus seems to be appropriate addition to lution will be forced to visit that confined island passing
a model of subitizing mechanism. In this case hysteresis at least twice through the coherent object thus breaking
can help to skip artefacts by requiring several same-level it into several segments. This leads to conclusion that a
vertices to occur in a row before counting them as an ob- margin of background must be preserved around objects
ject. Conversely, more than one background vertex must for the discussed method to work properly.
be encountered before assuming that cyclic path has left
an object.
An interesting observation can be made using full
grayscale image with non-uniform background presented
Inaccuracy of TSP solution results in inaccuracy of ob- on Fig. 5. A linear gradient as a background has been used
ject segmentation still present after applying hysteresis to to prepare an image of two objects of opposite colours
thresholding. By delaying background-object transition contrasting with their local surroundings. It is not a typsome object vertices are classified as background (false ical image used in research on subitization and the aunegative) and at the same time some background vertices thor could not find published articles reporting human efare still treated as belonging to an object (false positive). ficiency of subitizing similar scenes.
Fortunately these problematic vertices are close neighbours of correctly classified ones.
Nevertheless the results of processing gradient image
show that it is still possible to extract and count objects
It can also be observed that the distance between ob- although more advanced method of image profile analysis
jects and image boundaries is equally important as inter- is needed. The profile is mainly consisting of background
object spacing. If narrow passages are created by placing vertices grouped into intervals ordered more or less preobjects near the boundaries it can lead to similar problems cisely monotonically. Again, the monotonicity of these
as with tightly packed objects. In particular an object di- intervals depends on the quality of TSP solution.
rectly touching image edges can cut off an island of back-
On this background there are two segments correspond-
ground vertices from the rest of background by confining ing to objects, one added and one subtracted from the
it to image corner for example. In such a case TSP so- main luminance trend. To detect these objects simple
Unauthenticated
Download Date | 7/31/17 10:08 PM
67
Image Processing & Communication, vol. 19, no. 2-3, pp. 59-70
enough background surrounding the concerned objects.
For example in an image of a die only one of three potentially visible faces can be processed. This may be
achieved by cropping the image but it may be difficult to
satisfy all of above requirements in all cases by simply selecting a portion of the image. To secure enough margin
around counted objects it may be necessary to synthesize
false background for instance by blurring the image outside the ROI. This approach is similar to the concept of
foveated imaging.
Fig. 6: Processing a real image of gambling die. Image
has been cropped to contain only one face with six pips.
In the example shown on Fig. 6 the image was simply
Due to very close placement of objects there are difficulties in separating them from each other. Figure layout is cropped to exclude dots visible on other faces (it should
the same as in Fig. 3.
be noted that the number of dots on the image is higher
than human subitizing range). Then two processing atthresholding based on absolute values will not be sufficient and some sort of threshold adaptation seems to be
necessary. It is also possible to apply more sophisticated
filtering commonly found in signal processing but such
concepts as hysteresis or adaptation are naturally pertaining to biological and neural systems. As the gradient image is not a typical case of subitization, this example will
not be further elaborated, however it is presented here to
show possible extension of the discussed method.
tempts were made for comparison. In the first attempt the
resolution used for graph construction was the same as in
previous experiments (20 × 20 vertices). The results show
that the problem of narrow passages and tightly spaced
objects is further exacerbated by introducing more intermediate shades of grey which make direct travel between
objects less costly than in purely binary images. Objects
become easily merged into single fuzzy segments as can
be seen on resulting image profile.
Apart from artificially generated images the method
can be applied also to real life images in order to assess
In the second attempt illustrated on Fig. 7 a larger graph
its performance and look for other potential weaknesses. was created with twice as many vertices reducing plaAlthough quantitative evaluation has yet to be performed, nar distance between vertices and increasing resolution
tests carried out on selected images give encouraging re- by a factor of √2. This allowed for better object separasults. The images contained grayscale pictures of com- tion and more reliable segmentation but for obvious reamon objects with marked features used specifically for sons required longer processing. The results are satisfyencoding numbers, a typical example being gambling dice ing as the image profile contains distinct segments which
(Fig. 6).
can be easily counted with the use of simple thresholding
The most important problem encountered early in pro- with hysteresis. The bottom part of Fig. 7 shows effects
cessing such images was selecting appropriate region of of applying threshold at the midpoint of luminance scale
interest which should include only counted objects and and hysteresis delaying state transition for 4 vertices thus
exclude irrelevant features but at the same time provide eliminating all narrower spikes.
Unauthenticated
Download Date | 7/31/17 10:08 PM
68
P. Wołoszyn
Of course in computer implementation it is still orders
of magnitude slower than other well-developed methods
of image segmentation and counting. Therefore the proposed method is not meant as a competing solution in
practical applications. It should be treated rather as an
inspiration for research focused on developing simple,
biologically plausible, although not necessarily efficient
solutions without resorting to sophisticated mathematical
concepts and algorithms.
Characteristic property of the method is its low-level
design which does not take into account such concepts as
object, shape, texture or spatial relations. The only concept involved is the distance between pixels which is a
very local property reduced to differences of luminance
Fig. 7: The same image as in Fig. 6 processed with twice
in direct neighbourhood. The only global property conas many (800 instead of 400) graph vertices resulting in
higher resolution and better object separation. Figure lay- sidered is the total cost of cyclic path leading through pixout is similar to Fig. 3, the added plot (bottom) shows the els.
result of applying threshold with hysteresis to the image
This imposes a limitation on this method capabilities:
profile.
segmented objects have to be possibly uniform and con-
6
Conclusions
trasting in relation to their background. In future research
it is possible to use some attribute other than luminance
The method described in this paper is to be treated as a to determine pixel distances. Such attribute could depend
proposition of a model of subitization process character- not only on pixel luminance but also on its surroundings
istic for human cognition. Its main point is to linearize the and could come from some kind of image preprocessing.
Experimental results suggest that best results can be
processed image and generate image luminance profile by
ordering its elements in such a way that image segmenta- achieved for binary images or pictures with small numtion can be reduced to profile segmentation. To achieve bers of grey levels. Selecting appropriate region of interthat effect a path has to be found which will travel through est is significant as it is also important to provide enough
picture elements with tendency to preserve both location background margin around subitized objects. To obtain
and luminance similarity which are assumed as criterions satisfying object separation the resolution of image graph
should be sufficiently high to hold several background
of object membership.
The travelling salesman problem and its solving meth- vertices between objects.
ods may seem exotic as a means of generating said path.
These properties exhibit similarities to visual system
On the other hand there are many biology-inspired heuris- physiology which in turn provides some prospects of
tics for solving TSP and it is conceivable that actual neural further developing the proposed method by applying
networks are able to perform such optimization. The pre- foveated imaging approach. It would mean abandoning
sented method produces interesting results even for mod- regular grid topography of image graph and diversifying
erate quality TSP solutions.
its resolution by concentrating vertices in central fovea
Unauthenticated
Download Date | 7/31/17 10:08 PM
69
Image Processing & Communication, vol. 19, no. 2-3, pp. 59-70
where the gaps between objects are most probably lo-
[7] Kaufman, E.L., Lord, M.W., Reese, T.W., Volk-
cated. To take advantage of variable resolution the selec-
mann, J. (1949). The discrimination of visual num-
tion of ROI should include moving the central focus area
ber, American Journal of Psychology, 62, 498–525
to the centre of subitized scene. These propositions will
be examined in future research.
[8] Murofushi, K. (1997). Numerical Matching Behavior by a Chimpanzee (Pan troglodytes): Subitiz-
Acknowledgment
This work is supported by AGH grant no. 11.11.120.612.
ing and Analogue Magnitude Estimation, Japanese
Psychological Research, 39(3), 140–153
[9] Nezamabadi-pour, H., Saryazdi, S., Rashedi, E.
(2006). Edge detection using ant algorithms, Soft
References
Computing, 10, 623–628
[1] Arora, S. (1998). Polynomial time approximation
[10] van Oeffelen, M.P., Vos, P.G. (1983). Configura-
schemes for Euclidean traveling salesman and other
tional effects on the enumeration of dots: Counting
geometric problems, Journal of the ACM, 45(5),
by groups, Memory & Cognition, 10(4), 396–404
753–782
[11] Piazza, M., Mechelli, A., Butterworth, B., Price,
[2] Baraghimian, G.A. (1989). Connected component
C.J. (2002). Are Subitizing and Counting Imple-
labeling using self-organizing feature maps, in Pro-
mented as Separate or Functionally Overlapping
ceedings of the 13th Annual International Com-
Processes?, NeuroImage, 15, 435–446
puter Software and Applications Conference, Orlando
[12] Puts, M.J., de Weert, C.M. (1997). Does color influence subitization?, Acta Psychologica, 97, 71–78
[3] Felzenszwalb, P.F., Huttenlocher, D.P. (2004). Efficient Graph-Based Image Segmentation, International Journal of Computer Vision, 59(2), 167–181
[13] Rai, R., Pradhan, R., Ghose, M.K. (2013). AASC:
Advanced Ant Based Swarm Computing for Detection of Edges in Imagery, International Journal of
[4] Ferrand, L., Riggs, K., Castronovo, J. (2010).
Subitizing in congenitally blind adults, Psycho-
Emerging Technology and Advanced Engineering,
3(12), 107–115
nomic Bulletin & Review, 17(6), 840–845
[14] Rego, C., Gamboa, D., Glover, F., Osterman,
[5] Gallace, A., Tan, H., Spence, C. (2007). Multisen-
C. (2011). Traveling salesman problem heuristics:
sory numerosity judgments for visual and tactile
leading methods, implementations and latest ad-
stimuli, Perception & Psychophysics, 69(4), 487–
vances, European Journal of Operational Research,
501
211(3), 427–441
[6] Hung, C., Sun, M. (2010). Ant colony optimization
[15] Revkin, S.K., Piazza, M., Izard, V., Cohen, L., De-
for the K-means algorithm in image segmentation,
haene, S. (2008). Does Subitizing Reflect Numeri-
in Proceedings of the 48th Annual Southeast Re-
cal Estimation?, Psychological Science, 19(6), 607–
gional Conference, Oxford, USA
614
Unauthenticated
Download Date | 7/31/17 10:08 PM
70
P. Wołoszyn
[16] Shi, J., Malik, J. (2000). Normalized Cuts and Image Segmentation, IEEE Transactions on Pattern
Analysis and Machine Intelligence, 22(8), 888–905
[17] Trick, L.M., Pylyshyn, Z.W. (1993). What enumeration studies can show us about spatial attention: Evidence for limited capacity preattentive processes, Journal of Experimental Psychology: Human Perception and Performance, 19(2), 331–351
[18] Wassenberg, J., Middelmann, W., Sanders, P.
(2009). An Efficient Parallel Algorithm for GraphBased Image Segmentation, Lecture Notes in Computer Science, 5702, 1003–1010
[19] Watson, D.G., Maylor, E.A. (2006). Effects of color
heterogeneity on subitization, Perception & Psychophysics, 68(2), 319–326
[20] Watson, D.G., Maylor, E.A., Bruce, L.A. (2005).
The Efficiency of Feature-Based Subitization and
Counting, Journal of Experimental Psychology:
Human Perception and Performance, 31(6), 1449–
1462
[21] Wołoszyn, P. (2013). Distributed Greedy Approach
to Solving Travelling Salesman Problem, in Proceedings of the Ist International Virtual Scientific
Conference ScieConf, Zilina
[22] Xu, Y., Uberbacher, E.C. (1997). 2D image segmentation using minimum spanning trees, Image and Vision Computing, 15, 47–57
[23] Yu, Z., Yu, W., Zou, R., Yu, S. (2009). On ACOBased Fuzzy Clustering for Image Segmentation,
Lecture Notes in Computer Science, 5552, 717–726
[24] Zhao, B., Zhu, Z., Mao, E., Song, Z. (2007). Image
Segmentation Based on Ant Colony Optimization
and K-Means Clustering, in Proceedings of 2007
IEEE International Conference on Automation and
Logistics
Unauthenticated
Download Date | 7/31/17 10:08 PM