FUZZY EVALUATION OF VISUAL CONNECTEDNESS IN THERMOGRAPHY IMAGES OF CYLINDRICAL SURFACE

. This article presents an idea of visual connectedness measure between components in thermography images of rotating steel roller, cylindrically shaped. Both definition and method of measure calculus is given along with a short discussion.


Introduction
Computational intelligence, able to imitate human perception along with decision making, expansively applies to many research fields. From the most basic, such as industry/production through data analysis and monitoring, to defense systems, simulating human expert with a computer software gains advantages, such as repeatability, fatigue-freedom or low-cost operation.
One of the most important fields mentioned is image description and understanding. Classical image is a set of pixels, RGB or monochromatic, amount of which highly extends 10 6 . Any inference or data analysis on such amount of information is hard to investigate or simply time consuming (critical in real-time systems).
In this paper the authors propose a method that, for a given type of thermal image, generates a description on level of abstraction higher than pixels, based on so called visual connectedness. The obtained results can be used to develop advanced inferring methods of such images or to improve the existing ones, e.g. temperature regulators. The presented method is designed to analyze temperature distribution images of rotating steel cylinder's surface intended for paper milling industry.
The important feature of the problem under consideration is that the thermal processes that take place on the surface of the heated object result in thermographs that are particularly fuzzy and irregular. Those images can be relatively easily analyzed by humans, however they are not very manageable to the typical procedures of automated image processing. Moreover, the noncontact temperature field imaging of a rotating cylindrical roller, requires the use of specific methods at the data acquisition stage. It is also important to mention that additional sources of expertise in the discussed problem are: 1) a partial knowledge of the physical cause-effect relations that are observed in the obtained thermal images; 2) a qualitative description of the observation method for such images given by an experienced operatora process expert. However, both the knowledge about physical phenomena and the description of human expert behavior are incomplete and imprecise. Hence a formal tool of choice, naturally suited to use in this type of task, is the fuzzy sets and fuzzy logic theory. Fig. 1. A temperature distribution on a paper sheet as a consequence of moisture inhomogeneity [8] We believe that proposed evaluation method will be useful in moisture distribution control problems [8], as they are closely related with paper sheet's temperature (see Fig. 1).

Description of a thermal image
Every thermal image of a plain surface can be considered as a sum (superposition) of responses to heat pulses, generated over the surface [9,23]. Heating source depends on technological process and expected outcome. Most common are laser point [15] and inductive heating [5,6,13]. Due to physical heat flux distribution of a given source (often denoted as Q/dS), the set of pulses is not directly given. To bypass this inconvenience, the authors propose a definition of a temperature component along with a method to represent an input image in terms of those components.
Moreover, recognizing the thermal image representation as a thermal pulse responses superposition of inductively heated surface with further clustering of such type of basic temperature components (information granules) based on proposed visual connectedness fuzzy evaluation, allows to identify coherent areas of the heated object that have certain thermal state. This leads to a image processing method that is sensitive to the physical phenomena responsible for shape of observed images.

Definition of a temperature component
A temperature component describes a temperature distribution on a plain surface and can be considered as a pulse response to a point heat source. Such component can be denoted as an ordered four (1), with its representation in spatial-temperature domain given by (2  The reason behind the bell-shape of (2) is the nature of heat transfer problems; it results from the solution of differential heat conduction equation for point source heat pulse [4,9,12,31]. The sum of two exp functions allows to take the observed object's cylindrical geometry into account.  where: Nnumber of components in set K Having a set K, one can calculate an approximation of input image in terms of components K. It can be defined as an image with characteristic function given by (4).
where: |K| -number of components in set K, ppixel coordinates on image surface

Image source
Test images were obtained with use of a model developed by the authors [13], see Fig. 2. It is a coated steel cylinder of length 1,2m and 20cm radius, driven by a motor-inverter module.

Fig. 2. The physical model of a roller
The heat source is provided by a set of 6 movable inductive heaters [7], with 1kW of power dispatch per each. Heaters are driven horizontally and can be placed or moved anywhere along the horizontal axis. Images are acquired by a Flir A615 infrared camera synchronized with cylinder's angular position by a quadrature incremental encoder, connected to its shaft.

Fig. 3. Typical image seen by the IR camera
Since the camera can see only a part of the heated surface (see Fig. 3), proper algorithms for image concatenation were developed [11]. The processing begins with obtaining a sequence of images (e.g. 8 of them). Images are equally spaced along the perimeter with their acquisition point fixed by the encoder.
In order to retrieve the visible part of the surface, a 3D cylindrical geometry model (5) is used. It is expressed in terms of coordinates on cylinder's surface.
where: u, v [px]coordinates of a surface point; C L , C R [m]cylinder's length and radius; T H , T W [px]height and width, of the texture; By means of reverse mapping (inverted perspective projection) given by (6), one can unwrap a texture from each image of the sequence. This can be seen as an opposite to well-known texturing process in 3D modeling or simply as rectification of curved image. It is worth to note, that only visible part can be unwrapped, as shown in Fig. 4 by hatched area.
where: M camcamera parameters. Camera parameters M cam are selected manually by user or automatically by an algorithm [10]. They allow to mount the camera freely with respect to the object. Such flexibility can be convenient in industrial applications. Unwrapped set of textures (from a given sequence) have to be concatenated into full temperature map, as show in Fig. 5.
In case of areas, where images are not overlapping, one can simply copy the content into the output temperature image. However in other case, two source of image distortion have to be taken into account. First one is the decreasing resolution caused by curved part of the surface, located near image top and bottom edge. This can be simply overcome by ignoring most distorted areas. Second one is caused by directional emissivity  [16,30] and other misuses of infrared sensors [3,17,18,22,26,29].
If surface has low values of , temperature measurements have to be corrected on basis of angle between surface normal vector and camera optical vector. The resulting part of the output image can be then calculated by means of fuzzy theory [11]. To illustrate the results of such unwrapping-concatenation procedure, an experiment with setup Fig. 6 was conducted. The roller was wound with a resistive wire, connected to an external power supply. A sequence of images was acquired, rectified and concatenated. Obtained result is shown in Fig. 7.  Fig. 7 shows a temperature map from a real experiment involving inductive heaters instead of resistive wire. A sample image of the source sequence was previously shown in Fig. 3.

Image decomposition
In order to proceed with further analysis of visual connectedness, temperature maps have to be transformed into domain of components (2). To obtain a set of components K, denoted by K, one has to find its cardinality and values of (1). This process is a non-linear optimization (7) where the difference between input image and components' superposition is taken into account.
The initial values of K are to be found by means of an iterative algorithm, inspired by Mountain Method [2,14,20,32]. During each iteration, the algorithm searches the input image for a point with highest temperature . When found, a new component is generated and stored into K with the r value of (1) set to r start . The input image is then destroyed by removing new component from the input image by means of algebraic subtract. The value of r start corresponds to the physical size of the inductive heater and has to be chosen manually. It's value is constant regardless processed temperature map. The number of components can be limited by their count or difference/distance between original input image and its approximation (4). Although the optimization (7) does not change the number of components, it helps to obtain better approximation (4) by adjusting the r parameter of each component K.
The obtained set K is called a representation of an input image in terms of temperature components. A proper example was shown in fig. 8, where image I is represented by its profile (arbitrary chosen row or column). Three obtained components are the representationthey are approximation of input profile in terms of eq. (4).
Overall process can be compared to a well-known vectorization of raster images. More details on decomposition can be found in [11].

Visual connectedness of two components
The proposed visual connectedness evaluation is based on similarity measure [24]. Its goal is to assign a unit degree to a pair of components in similar way that a human expert states "these two areas of higher temperature are visually connected". The difference between typical similarity measure and the proposed one is the nature of underlying object. To assess similarity one takes two physically separable objects or states and measures a difference between them. In contrast, the proposed visual connectedness is applicable to objects that can superpose, which is justified by the heat transfer physics intrinsic to component definition given.
Proposed method combines three separate evaluations by means of fuzzy inference, shown in Fig. 9.

Fig. 9. Fuzzy based visual connectedness evaluation
To obtain full information of set K in terms of visual connectedness, one has to calculate its value for each pair of components (K i , K j ). Such result can be arranged in matrix, alike similarity matrix. A visualization of a sample matrix is shown in Fig. 10.

Fig. 10. An example of connectedness matrix. Numbers on both axes denote an index of a component K
It is important to note that both temperature and spatial connectedness are binary relations of each pair of components (K i , K j ), resulting in fuzzy variables: SpP, SpT however visual importance is unary with outcome denoted as ZnW.

Visual importance
The assessment of visual importance allows to promote components that are well visible (high ) since barely visible ones are canceled out. It is done by calculating membership value of component n to a fuzzy set SMALL (8). Set BIG is given intrinsically.

 
Final memberships of fuzzy variable ZnW are given by (9). Since ZnW is the only unary relation in the proposed evaluation method, its value has to be calculated for both components in pair (K i , K j ) resulting in two fuzzy variables: ZnW 1 and ZnW 2 .

Spatial connectedness
Spatial connectedness of a pair tells to what degree relations "K i is near K j " and "K i is far form K j " hold. It is based on three Euclidean distances (10) from which the lowest is selected (11). This procedure allows to take into account cylindrical surface of the cylinderwhen two components are located near top and bottom edge of the image space, they are actually near each other in cylindrical surface space.
The authors have chosen exponential function (12) for the  big membership, as its values are closest to the human perception [1,19,25]. The selected membership function is shown in Fig. 11b.

Temperature connectedness
The temperature connectedness is based on temperature analysis along shortest path between two assessed components, Fig. 12. The shortest path is based on the same principle as the distance (11). For each pixel along such path, a temperature is calculated by means of (4). Obtained values construct a set . Conducted experiments show, that this method can be based on temperature ratio of extremes along the path. This assessment is given by (14). Both extremes ( min and  max ) are calculated from superposition of (2) functions over whole set K.
Final assessment is given by (15).

Fuzzy inference
In order to obtain a numerical degree of visual connectedness, the following set of 16 rules has to be applied to (9), (13) and (15). It is a complete, non-contradictory and non-redundant [21] set of rulesany possible combination of input values has its reflection in corresponding rule. They were formulated by an expert and provide the core of the visual connectedness definition.
Unlike parameters (e.g. c SV ) previously introduced, the proposed rule base stays constant independently of those parameters. Moreover, not shape of inductive heaters nor dimensions of the roller would affect the rules. This is caused by the fact that rules are separated from the input data by the physical meaning of the fuzzy variables involved.
The output of the rule block used is denoted as fuzzy variable S and described by three values: high, medium, low.
In every rule of the given set, a consequence uses one of three fuzzy values (h, m, l) assigned to the fuzzy output variable S. Those values are represented by singletons with values respectively 1, 0.5, 0. The inferring model used is TSK (Takagi-Sugeno-Kang) [27,28], hence final value is calculated with (16).
where: p n [-]n-th rule activation degree, k n [-]value of corresponding conclusion.
As for logical conjunction operator, a classical function min or prod can be used.

Conclusion
The proposed method allows to calculate the visual connectedness of a pair of components that represent temperature distribution on a cylindrical surface. Extending this calculus through to every pair in K leads to a connectedness matrixa form of similarity matrix.
The values obtained by means of (16) can be further processed via standard approach, e.g. graph-based clusterization methods in order to arrive at image described by components (primitives) and objects (disjoint sets of one or more components).
Constant values used in presented method: c SV , r SV , m SV , c 1 , r start have to be chosen manually according to process expert's intuition and knowledge. Assuring proper values of those parameters affects the performance of overall assessment.
In further work, the development of infrared image segmentation algorithm based on human perception of a cylindrical surface, introduced by the described visual connectedness, is planned. It will allow to obtain surface description as a set of consistent spatial and temperature areas. It may for e.g. be a source of information for control decisions. The starting point for such algorithm is taking into account the physical side of the phenomena occurring in the thermal system, in particular the shape of the impulse response generated on the surface of the cylinder. Such approach allows for a significant decrease in information granularity in component-represented image compared to the conventional "pixel" images. Simultaneously it is possible to obtain segmentation results similar to the perception of such images by human.