Implementación de los mecanismos alternativos de resolución de conflictos

A time series, T = (t1, t2, . . . , tm), is an ordered set of m real-valued variables [112], where the variables are indexed according to the order they are occur in time [42]. There has been much reported work on Time Series Analysis (TSA) [176, 176, 188]. TSA was traditionally directed at forecasting and event comparison [188]. Exemplar application domains include stock market and weather forecasting. With respect to classification, TSA has been successfully used to extract patterns from data [71]. According to [200] time series classification is concerned with the mapping of unlabelled time series onto some predefined set of classes. The basic concept of time series classification is to represent data in the form of curves or sequences and then attempt to match these

Images Image colour histograms Image database

Figure 3.6: An example of grey-scale retinal images represented as time series

curves (using similarity measure such as Euclidean distance) so as to classify the new data.

TSA does not necessarily imply that the data to be considered must have some tem- poral dimension. It is often useful to represent data, that does not naturally represent a time series (such as images), using curves. The most basic approach whereby images can be represented as time series is using colour histograms [8, 126]. Histograms can be conceptualised as time series where the X-axis represents the sequential histogram “bin” numbers, and the Y-axis the size of the bins (number of pixels contained in each bin). An example of this can be observed in Figure 3.6. The histograms (based on the grey-scale values) are generated from each image and the resulting histograms are kept in an image database. Further analysis (such as classification or image retrieval) can then be applied on the generated histograms (instead of the actual images). There are many examples where images have been represented using the notion of histograms. In [8], a similar approach to extracting time series was utilised, but in this case the time series were represented using Symbolic Aggregate approXimation (SAX) [125] in order to reduce the size of the feature space. However, for many applications the usage of colour descriptors alone is not sufficient to capture all the salient characteristics of an image and thus in [126] texture and colour histograms were combined. This was achieved by first extracting colour histograms from the RGB colour channels, using 256 bins for each, and concatenated them to form a single sequence of length 768 (256 ×

3) bins. Then, Gabor wavelet filters, of different scales and orientations, were applied before generating a texture histogram of 256 bins for each image. The colour and tex- ture histograms were then concatenated into a single 1024 bin histogram to form the desired image “signature”.

extracted from the shapes contained in the input image sets [3, 51, 112, 115]. In [115] shape modelling using the centroid-radii model was adopted [190]. Using this model, the distances between several points on each shape to its centroid, at regular radii inter- vals, was measured and interpreted as the Y-axis (time-line) of a time series [115]. This model was found to be able to differentiate distinct shapes provided a sufficient number of radii were used [190]. One particular issue of using shapes as features is the need to represent the shapes in a manner that is rotation invariant [3, 115]. To overcome this problem in [3] a multi-scale shape representation was proposed for a single closed contour, where for each contour point, the contour convexity/ concavity information was captured. The approach was invariant with respect to some transformations. How- ever, the computational cost was found to be expensive at O(N3), where N was the number of contour points, for each shape comparison. In [115] a different strategy was proposed to achieve rotation invariance. In this work, time series were extracted from shapes using the method described in [190]. To achieve rotation invariant matching of two shapes, the second shape was rotated, and then the minimum distance of all possible rotations computed (the first shape was held in a fixed position) to identify the best matching shape. Region Of Interest (ROI) based image classification to classify brain Magnetic Resonance Images (MRIs), based on the shape of an object of interest, was presented in [51] where the time series was derived based on the boundary line delineating the shape. Here, a Minimum Bounding Rectangle (MBR) was employed to circumscribe the ROI. Each point on the Y-axis represented the length of the intersec- tion of the identified ROI pixels with the radii line projected from the midpoint of the lower edge of the MBR.

3.5.1.1 Colour Histograms as Image Features

As noted in Section 1.2, the nature of the images of interest (retinal images) does not permit the accurate identification and extraction of image features based on shapes. Thus colour information in the form of colour histograms was considered, because (i) it has been successfully applied in image classification problems, and (ii) the colour histograms themselves can be immediately interpreted as time series so that TSA can be applied.

Colour histograms are considered to be the simplest way of representing the charac- teristics of an image in terms of colour distribution, and to be an effective representa- tion for identifying objects in images [187]. Colour histograms are robust and invariant against object changes in terms of shape and position [187], although they are not good at capturing spatial relationships [198]. The main advantage of colour histograms is their uncomplicated nature [25, 187] and that for many applications they provide for effective discrimination [41]. Colour histograms are basically created by dividing the colour space into equal sized bins (or colour values), and then counting the number

of pixels that fall into each bin [73]. Much work on using colour histograms, with respect to image classification and various domains and problems, have been reported [28, 41, 73, 96, 105].

A simple and computationally cheap image retrieval method using colour histograms has been described in [28]. Here each image was assigned three histograms correspond- ing to the R, Gand B streams in the RGB colour model. Each histogram comprised 48 bins and each was normalised. In [105] a vector quantisation method was embed- ded in the generation of colour histograms. Here the HSV colour model was used and transformed into Gaussian components. This was achieved by quantising each of the colour components (H, S and V) into 16, 8 and 4 colours respectively, based on the “codebook” generated by a Gaussian mixture vector quantisation. Histograms of the Gaussian components were then extracted for each image and formed into a feature space to provide support for image retrieval. In [41], the S and V components (of the HSV colour model) were both divided into three sections. The H component was further quantised into two: (i) 18 sections of 20◦ each, and (ii) 24 sections of 15◦ each. These were then formed two HSV histograms of 162 (18×3×3) and 216 (24×3×3) dimensions respectively. Each image in the image dataset was represented by these histograms. A different colour model, Hue, Value and Chroma (HVC) was utilised in [73]. Here the colours were reduced into only 11 colours, based on human perceptual natures (e.g. red, yellow and cyan), before the generation of the histograms.

As noted above one particular disadvantage of the colour histograms representation occurs when two (or more) images with different appearances having similar colour histograms [198, 202, 215]. To address this issue the use of spatial-colour histogram techniques has been proposed [96, 146] so as to add a spatial element into the colour histogram generation process. For example [96] used two colour histograms to select colour attributes. The first represented the colour distribution of the image background, while the second described a specific image object. Colours were then selected accord- ing to whether they occupied a significant percentage of either the image background or the object. Then, using the identified colours, the regions containing the colours were extracted through an image decomposition based approach. Another approach, that applied a similar image representation technique, was reported in [146]. However, instead of using two different histograms, a single colour histogram that represented an image colour distribution was utilised to identify the dominant colours (measured according to the number of pixels per colour). In [73] a region based colour histogram approach was also proposed where each image was partitioned into nine equal sized regions and a local histogram generated for each region. Similar approaches to ex- tract colour histograms by regions have been proposed in [197, 202]. An alternative approach is described in [198] where Local Feature Regions (LFRs) were first identified (using a Harris-Laplace detector) before constructing colour histograms for each LFR.

Attribute₁ Attribute₂ Attribute₃ Image1 Image2 Image3 Image4 Image5 ImageM-1 ImageM Attribute_b-1 Attribute_b • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

Figure 3.7: Example of images represented in a tabular form

A template based method, that preserved object texture and shape, was described in [215] where k-th order spatial histograms (or spatiograms) were generated. Another approach that has utilised colour histograms coupled with a morphological operator to retrieve objects that exists in the image (or video) is described in [218]. The morpho- logical operation was used to neutralise the rotation angle of the queried object in the input image. Segmentation of the object of interest was however required before the matching process could be performed.

One of the proposed solution presented in this thesis (see Chapter 5 for details) employs a strategy that is founded on ideas described in [73, 197, 202] due to its efficacy and low computational complexity.