1 Introduction

Many data interpretation programs extract qualitative features from data and carry out further analysis using them. Scale-space filtering[Witkin1,2] is a technique that can produce qualitative descriptions of continuous curves. This paper describes an extension of this technique that has been applied to the analysis of Differential Thermal Analyzer (DTA) curves. DTA is an instrument that geologists use to analyze contents of a soil sample. An investigator uses DTA to heat a soil sample and an inert reference material and measures the difference in the temperature of the two at discrete temperature intervals. DTA output is a graph of difference in temperature verses temperature of the reference material. Figure xxx shows an example of DTA output. As a soil is heated in a DTA, minerals in the soil sample undergo phase transitions and reactions at various stages. An exothermic reaction produces heat and an endothermic reaction consumes heat. An exothermic reaction results in a hill in the output curve. In contrast, an endothermic reaction results in a valley in it. The temperature at which a reaction occurs is diagnostic of minerals present in the soil. e.g. an endothermic reaction at 570 would indicate the presence of quartz. A number of reactions may occur simultaneously and may correspond to small and large features superimposed on each other. The resultant curve is complex, and a domain expert can not identify features in the curve with certainty. For example, in figure xxx, experts interprete that there is some chance of an endotherm at 907 at A. We have developed an expert system that can first extract probabilistic qualitative features from the DTA output of a sample and use them in recognizing its contents. The system uses an extension of scale-space filtering suitable for handling data sampled at discrete intervals and for producing a probabilistic scale-space description. The system also has a Bayes tree classifier that can use probabilistic qualitative features in the curve to recognize the contents of the sample.

1.1 d. Pairing of lines into contours

In this subsection, we describe how scale lines are paired into contours and how probabilities are associated with the resultant contours. Scale-lines are of one of two types: maxtype which corresponds to a maxima in the derivative and mintype which is a minima in the derivative. We will now define scale-ratio of these lines. If two lines have maximum scales s1 and s2, their scale-ratio is the ratio of larger scale divided by the smaller scale.

Thus, if lines L1 and L2 have the maximum scales of 5 and 10, then their scale-ratio is 2. When a line of maxtype is paired with another line of mintype at a higher temperature, the two correspond to a hill. When a line of maxtype is paired with another line of mintype at a lower temperature, the two correspond to a valley. The system employs following procedure to identify partner lines for a line L that has the maximum scale s and is locaed at temperature T. It considers all lines, with type opposite to that of L, and with distance from L less than a threshold defined as a function of s, as potential candidates. It identifies a line that has the minimum scale ratio with L among the candidates at temperature greater than T, and defines it as the hill partner. Similarly, it identifies a line that has the minimum scale ratio with L among the candidates at temperature lower than T, and defines it as the valley partner. Line L is paired with its hill partner to form a hill. Line L is paired with its valley partner to form a valley. The system then associates a probability with every hill and valley it finds. To do this, it uses domain specific statistics that are correlations between the inflexion points of a contour, the second derivative at the peak and the probability of its being a real contour. For example: If the first inflexion point of an endotherm is increasing, the second inflexion point is increasing and the second derivative at the peak is high, then the probability of the peak being an endotherm is .9

2 Experimental results

In the previous section, we argued that an algorithm that uses domain specific correlations should perform better than one that does not. We will now describe an experiment that we carried out to test this claim. For the sake of clarity, we will refer to the algorithm described in this paper as A, and another algorithm that pairs lines with minimum Manhattan distance between upper end points in the scale-space graph as M. These two algorithms were tested on a random sample of 20 curves from the data bank in the DTA laboratory in NASA Ames research center. Algorithm M detected 49A detected 890.5. In this evaluation we compared the performance of the algorithms with that of an expert. In a number of domains, the features identified by an expert differ from those identified by a non-expert. While the curve will be given an unique interpretation by a non-expert, it can have different interpretation in different domains. Algorithms described in this paper can produce different qualitative descriptions of a curve from correlations specific for different domains.

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