Texture classification via conditional histograms
Introduction
Previous works have shown that histograms can be used as powerful descriptions for non-parametric classification (Unser, 1986a, Valkealahti and Oja, 1998, Ojala et al., 1996, Ojala et al., 2000, Hofmann et al., 1998, Puzicha et al., 1999). In contrast to parametric features (Haralick, 1979), histograms contain all the information of distributions avoiding the problem of feature selection. In general, the development of a method for automatic feature selection is not trivial since optimal performance requires a careful selection of features according to particular types of textures (Ohanian and Dubes, 1992, Jain and Zongker, 1997, Sullis, 1990, Ng et al., 1998). However, histograms cannot be directly used as texture descriptors; the computation and dimensionality impose prohibitive computational resources for applications (Rosenfeld et al., 1982, Augusteijn et al., 1995). The computation of histograms of low-dimensionality is considered as an open problem (Unser, 1986a, Valkealahti and Oja, 1998, Rosenfeld et al., 1982, Ojala et al., 2001).
In addition to make texture descriptors useful for applications, the reduction of histogram’s dimensionality has two important implications. First, it avoids sparse histograms due to insufficient training data. Secondly, if histograms are compact, it is possible to consider more complex pixels interdependencies increasing the discrimination power. It is important to notice that if the histogram reduction is effective, we should expect good discrimination with similar features than the ones used to codify the high dimension description. As such, it is important to distinguish between the effectiveness of the reduced process and the additional power obtained by including more complex pixel interdependencies.
A powerful approach to histogram reduction is to perform a quantisation to adapt the histogram bins according to the distribution (Puzicha et al., 1999). In (Valkealahti and Oja, 1998, Ojala et al., 2001) the adaptation is defined by using techniques of vector quantisation (Gersho and Gray, 1992). Results on texture discrimination have shown that this is a very powerful technique to reduce multiplex (i.e., >2) co-occurrences. However, although a tree structure can be used to handle the complexity required by the encoder, yet the encoder requires a significant number of operations and sample data. The search uses more memory that a full search vector quantisation and the process can lead to sub-optimal solutions (Gersho and Gray, 1992).
In this paper, we simplify histograms by considering combinations of the random variables defining the joint probability function (i.e., grey tone or colour dependence) (Unser, 1986a, Rosenfeld et al., 1982, Ojala et al., 2001). In (Rosenfeld et al., 1982), and later (Unser, 1986a), histograms define the probability of the differences of grey levels for pixel pairs. The motivation is that these operations define the principal axes of the second order joint probability function (Unser, 1986a). However, they have had an unpredictable success in applications (Schulerud and Carstensen, 1995, Chetverikov, 1994). The main caveat of this representation is that, in general, random variables in a texture do not define Gaussian independent distributions. Additionally, although the average error of the difference is well approximated by the factorised probability (Ojala et al., 2001), the difference operation loses spatial information and histograms can become bad approximations of joint probability functions. In order to maintain spatial information, we propose to encode the texture’s random structure by computing joint probabilities defining the dependence between pixels forming regions sharing intensity or colour properties. To reduce dimensionality, statistical distributions are computed for binary images. We combine joint distributions of binary values and the probability of intensity values to define a collection of histograms. These histograms are normalised, thus they can be used for non-parametric texture discrimination independently of the size of the sampled region (Puzicha et al., 1999).
We use the non-parametric classification to implement a segmentation application based on a hierarchical quadtree scheme. Hierarchical strategies have been very effective for image segmentation. The hierarchical approach has two main advantages. First, it performs a fast partition by considering regions rather than individual pixels in fixed overlapping windows. The partition is defined by considering regions of different sizes at different levels of the hierarchy. The second advantage is that it reduces classification errors due to mixture of features computed in a fixed window size. This is convenient to delineate accurate region borders (Ojala et al., 2000, Hsu, 1978, Dutra and Mascarenhas, 1984, Marceau et al., 1990, Briggs and Nellis, 1991, Ma and Manjunath, 1997). These properties have been exploited in algorithms of segmentation based on intensity (Horowitz and Pavlidis, 1976, Wu, 1992), motion (Szeliski and Shum, 1996, Lee, 1998) and texture information (Ojala et al., 2000, Chen and Pavlidis, 1979). Our segmentation is based on the technique presented in (Chen and Pavlidis, 1979), but it divides an image using non-parametric classification.
Section snippets
Statistical characterisation of textures
The interdependence of pixels in a texture can be defined by the joint probabilities computed for random variables associated to pixel intensities. Given n discrete random variables c(x1), c(x2), …, c(xn) at positions x1, x2, …, xn, the nth-order density function defines the probability that the variables take the values c1, c2, …, cn, respectively (Papoulis, 1991). That is, f(c1, …, cn; x1, …, xn) = P{c(x1) = c1, …, c(xn) = cn}. Here, c takes values within the range of possible grey levels or colours. Since
Conditional histograms
We can describe the interdependence of pixels in a texture by considering ideas of binary feature selection. Previous works have shown that binarisation can be very effective to characterise the spatial dependence of pixels in images (Wang and He, 1990, Chen et al., 1995, Hepplewhite and Stonham, 1997, Ojala and Pietikäinen, 1999). As suggested in (Ojala et al., 2000), we use the joint distribution of binary patterns. However, in order to be able to locate a texture embedded in different
Classification
In general, the classification performance depends on the discrimination approach. Numerous discrimination approaches are possible (Devijver and Kittler, 1982, Schalkoff, 1992) and classifiers can improve the results at the expense of complexity, computational resources and requirements in the size and quality of the training data. However, it is beyond the scope of this work to evaluate classification schemes. We are interested in evaluating the discrimination properties of conditional density
Segmentation
We used the classification to implement a segmentation application based on a top-down hierarchical subdivision. This approach searches for an optimum partition by dividing the image in a quad-tree homogenous decomposition. This comprises three steps. First, a region is classified. Secondly, it is partitioned and each partition is classified. Finally, it is necessary to measure the homogeneity of the partition. If the region is homogenous, then the whole region is assigned to the same class and
Experimental data
In order to assess the discrimination capabilities, we have performed two experimental tests based on the data presented in (Valkealahti and Oja, 1998, Ojala et al., 1996, Ojala et al., 2001, Ohanian and Dubes, 1992). The first test (Valkealahti and Oja, 1998, Ojala et al., 2001) defines 32 texture categories from selected images of the Brodatz collection. The second test (Ojala et al., 1996, Ohanian and Dubes, 1992) defines 16 texture categories from four types of images.
For the 32-category
Conclusions and further work
We have proposed a characterisation of textures based on a mixture of colour and contextual information obtained from binary features. The characterisation defines one-dimensional histograms that represent the conditional probability of intensity values given the joint probabilities of pixels in image regions. Experimental results show that a non-parametric classification based on conditional histograms produces a compact and powerful set of features. High classification performance is obtained
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