Qualitative Interpretation of Spectral Images: Reasoning with Uncertain Evidence

1 Introduction

Interpreting spectral images is a special problem of diag­ nosis. The problem requires comparing known patterns with input spectral images to identify which patterns may be contained by the images [Anand et al., 1991; Sadtler, 1988]. Because spectral data are always inaccu­ rate, one difficult task is to deal with uncertain evidence.

Currently known methods and systems of spectral im­ age interpretation are primarily based on quantitative analysis [Culthup et al., 1990; Sadtler, 1988]. The es­ sential principle of quantitative analysis is to determine the possibility that a pattern may be contained by an image by calculating the difference between the pattern and the parts of the image. In practice, however, a crit­ ical problem of applying quantitative analysis is that it is hard to identify any pattern when the inaccuracy of input data is not slight. Fuzzy logic and probability the­ ory can partially solve the problem [Duda and Nilsson,

1976; Zadeh, 1978], but they can not consider qualita­ tive correlations among data 1 which are very important and effective in spectral image interpretation.

We present a novel method for interpreting spectral images by using qualitative reasoning. The method draws inferences on the basis of qualitative features of spectral images, and uses qualitative correlations among data as evidence when input data are inaccurate.

We put forward a new concept called support coef­ ficient function (SCF). SCF can be used to extract, represent, and calculate qualitative correlations among data. On the basis of qualitative correlations and dy­ namically obtained information, shift intervals of inaccu­ rate data can be dynamically determined. When input data are inaccurate, qualitative correlations can provide evidence to enhance or depress hypotheses for inaccurate data [Zhao and Nishida, 1994].

We have developed a practical system on infrared spec­ tral image interpretation by using the method, and have fully tested the system against several hundred real spec­ tral images. The experimental results with the system are excellent. Both the rate of correctness (RC) and the rate of identification (RI) increase significantly by using the method 2 ·

In the following sections, we first describe the problem of spectral image interpretation in section 2. Then in sec­ tion 3, we address the essentials of our method including some useful definitions. In section 4, we introduce our method for interpreting inaccurate spectral data by con­ sidering qualitative correlations among data. Section 5 demonstrates the application of the method to a system on infrared spectral image interpretation, and shows the experimental results with the system. Related work is discussed in section 6. Our ongoing research is briefly introduced in section 7. Conclusions are addressed in section 8.

2 Motivating Problem

The primary task of spectral image interpretation is to identify what patterns a spectral image contains by interpreting the spectral image. In the rest of the paper, we limit the problem to interpreting infrared spectral images of unknown compounds to identify what partial compo­ nents (PC) the unknown compounds contain without loss of generality.

1 Detailed definition will be given in section 3.1.

2 RC and RI are two important metrics for evaluating the solutions of infrared spectral image interpretation. We will give the detailed definitions of RC and RI, and show the experimental results of our system in section 5.2.

Briefly, the infrared spectral image of an unknown compound can be thresholded and represented as a set of peaks:

where every peak consists of its frequency position (!), strength ( s ), and width ( w) respectively:

The patterns of known partial components can be rep­ resented in the following form:

which means if PCj is contained by a compound, then on the spectral image of the compound, there will be peak Pj 1 , Ph, ... , and Pj.,.

Ideally, if all peaks on spectral images are accurate, the process of interpreting infrared spectral images of unknown compounds can be simply described in the fol­ lowing way.

1. Select a peak from Sp. Retrieve all partial compo­ nents whose patterns have the same peak, and put the partial components in a candidate list: C L;
2. Select a partial component, PCj, from CL. If PCj C Sp, then put the partial components in a solution list: SL; otherwise, delete the partial com­ ponent from CL;
3. Goto 2 until all partial components in CL are checked;
4. Goto 1 until all peaks on Sp are identified;
5. Delete conflicts (overlaps) among partial compo­ nents in SL, and output SL as the solution 3 ·

The overview of the process is shown in Figure 1.

In practice, however, spectral data are often inaccu­ rate due to various reasons most of which are unforeseen, or unknown at all . Therefore, the peaks created by a par­ tial component on real spectral images are often different from the peaks described by the pattern of the partial component.

3 We will not discuss conflict-resolving in this paper, but concentrate on the method for identifying inaccurat'e data.

Fuzzy logic provides mathematical fundamentals of representation and calculation of inaccurate data [Bowen et al., 1992; Negoita and Ralescu, 1987; Zadeh, 1978]. For example, peaks in the patterns of partial components may be described in a fuzzy fashion like f 01 , is around fo (not / 01 , is equal to / 0 ), and a fuzzy region may also be defined to represent the peaks like [/0 -.6-f, fo+.6.f]. Probabilistic reasoning provides a practical framework for reasoning under uncertainty and reasoning with in­ accurate data [Dempster, 1968; Pearl, 1987]. In many systems, subjective statements are used to take the place of statistics of data or evidence when statistical sam­ ples are insufficient or absent , such as certainty factors in MYCIN [Shortliffe, 1976] and prior probabilities in PROSPECT [Duda and Nilsson, 1976].

However, both fuzzy logic and probability theory are based on quantitative features, but in many cases, espe­ cially when data are seriously inaccurate, qualitative fea­ tures are much more important than quantitative ones. Figure 2 shows two simple cases. The quantitative dif­ ference between the two spectral images in (a) is smaller than that between the two spectral images in (b). By conventional fuzzy or probabilistic methods, the two im­ ages in (a) should be closer than those in (b) . Actually, the two spectral images in (b) may be created by the same partial component in some cases, while the two spectral images in (a) are definitely created by two dif­ ferent partial components because the number of peaks is different. Here, the number of peaks on spectral im­ ages is a critical qualitative feature.

We propose a novel method to interpret infrared spec­ tral images by using qualitative reasoning. The method draws inferences on the basis of qualitative features of spectral images, and uses qualitative correlations among data as confirmatory or disconfirmatory evidence when input data are inaccurate.

Suppose P(a) represents the possibility that two im­ ages in (a) are created by the same partial component , and P(b) represents the possibili-ty that two images in (b) are created by the same partial component. At the beginning, P(a) > P(b), but with the obtaining of qual­ itative correlations among data, P(b) will increase, and P( a) will decrease.

By our method, the above ideal process can be briefly expressed as the following predicate calculus:

tive predicates in our method which represent that Pi qualitatively belongs to PCi (i.e., ? Pi E PCi ), and PCi qualitatively belongs to Sp (i.e., ? PCi C Sp ), respec­ tively. Realizing A@B is based on qualitative correla­ tions among data and dynamic information obtained in interpretation.

3 Preliminaries

3.1 Qualitative Correlations among Data

Definition 3.1 Related data: If data d1 , d2, ... , and dm describe a common phenomenon altogether, or they refer to the same behavior simultaneously, then they can be treated as related data.

In infrared spectral image interpretation, there are two types of related data: (1) as far as a single peak is con­ cerned, the frequency f, strength s, and width w of the peak are related data; ( 2) if the pattern of a partial com­ ponent is considered, all the peaks in the pattern are related data.

Definition 3.2 Qualitative correlations among related data: If di and di are two related data, then the presence of di somewhat (qualitatively) enhances the presence of di, and the absence of di somewhat (qualitatively) de­ presses the presence of di. The above effects are called qualitative correlations among related data.

In infrared spectral image interpretation, if data gath­ ered from spectral images look like, but are not exactly the same as, the data described in the patterns of partial components, considering qualitative correlations among related data can obtain qualitative evidence. For exam­ ple, suppose the frequency position of a peak is different from that in a pattern, but both the strength and width of the peak are the same as described in the pattern, then the frequency position of the peak may still be identified because both of its related data support it. Similarly, if peaks at low frequency sections are inaccurate, consid­ ering their related peaks at high frequency sections may help identify these peaks, and vice versa.

3.2 Support Coefficient Function

Definition 3.3 Support coefficient function {SCF): If there are m- 1 data related to di, then the support co­ efficient function of d; calculates the total effects from the related data by considering the qualitative correla­ tions between di and each of its related data.

Suppose 1r(di, di) represents the qualitative correla­ tion between di and di, then the support coefficient func­ tion of di can be defined as:

SCFi should directly depend on how many and how much related data support di. When SCFi is greater than a certain value given by domain experts, the related data tend to support di; otherwise, the related data tend to depress di.

3.3 Evidence Based on SCF

In section 2, we used Pi@PCi to express that Pi can be qualitatively identified from PCi . Realizing Pi@PCi requires to define a shift interval for PCj like:

and then to determine the possibility of Pi E PCi ± t::..

The above formula is similar to that in fuzzy logic, but contains completely different meanings. The primary difference is that the shift intervals are dynamically de­ termined by SCFi, while in fuzzy logic, the fuzzy regions are usually provided by domain experts in advance.

Definition 3.4 Shift interval: Shift interval is a dy­ namic region for inaccurate data. Given a standard fuzzy region for inaccurate di, the shift interval of di varies around the standard fuzzy region on the basis of SCFi . When SCFi shows that the related data support di , the shift interval of di becomes wider than the stan­ dard fuzzy region. On the other hand, when SCFi shows that the related data do not support di, the shift interval of di becomes narrower than the standard fuzzy region. Definition 3.5 Evidence based on SCF: SCFi deter­ mines the shift interval of di, that is, SCFi determines how widely di is allowed to shift. The wider the shift in­ terval, the more easily di is identified. Therefore, SCFi provides confirmatory or disconfirmatory evidence for identifying di.

4 Qualitatively Interpreting Spectral Images

As described in section 3, there are two types of related data in infrared spectral image interpretation. First, all the features of a peak are related data. Second, all the peaks created by the same partial component are also related data. For making our introduction brief, in this section, we omit the process of considering the former type of related data, and directly introduce the process of considering the latter one.

Suppose m peaks (p1 , P2, ... , Pm) may be created by the same partial component, and d0 is the standard fuzzy region for real peaks, then our method can be described as the following steps.

[Step 1] Defining support coefficient function

First, we define the qualitative correlation between two related peaks, Pi and Pi, as:

where ci(Pi) expresses the qualitative effect of Pi on Pi : Ci (Pi)= 1 means that Pi is enhanced since its related data item Pi can be identified; ci(Pi)=O means that Pi is de­ presses since its related data item Pi can not be iden­ tified. Ci (Pi) provides dynamic information for making interpretation. The definition of Ci (Pi) is simply based on the consideration that if a peak of a partial compo­ nent has been identified, then the peak will support the coexisting peaks (related peaks) that the partial compo­ nent may create at the same time.

Based on ci(Pi) (j = 1, 2, ... , m, j #i), we define the support coefficient function of Pi in the following way.

where 0 < SCFi ::=; 1, and SCFi expresses the qualitative correlations between Pi and all of its related peaks.

If m = 1, then SCFi = 1. When m > 1, SCFi is in the direct ratio to the number of related peaks which may be identified.

[Step 2] Defining Shift interval

where 0 < t::..d; < 2d 0 , and t::..d; is in the direct ratio to SCF;.

If m = 1, then SCF; = 1, and t::..d; = d0 · When m is fixed, the more the related peaks are identified, the greater SCF; is, therefore the greater t::..d; is. When SCF; is fixed, t::..d; depends on the number of related peaks.

[Step 3] Making qualitative hypotheses

On the basis of SCF; and t::..d;, we can make qualita­ tive interpretation for inaccurate data. A triple is used to represent the interpretation:

where Pi; is a peak described in the pattern of a partial component (or in other words, Pi; is a peak that the partial component should create); Pi is a real peak on the input spectral image which is suspected to be Pi;i and J.Li is the possibility of p; being accepted as Pi; (i.e., Pi@PCi) which is defined as:

where 0 :::; J.Li ::=; 1.

The following properties can be drawn from the above formulas. These properties clearly explain how qualita­ tive correlations among related data provide evidence of making interpretation for inaccurate data.

Property 4.1: With the same m, the more the related peaks are identified, the greater SCF; is; otherwise, the smaller SCF; is.

Property 4.2: With the same m, the greater the SCF;, the greater is t::..d;, which implies that the more the re­ lated peaks support p;, the more widely Pi is allowed to shift 4 ·

Property 4.3: With the same p;, the greater the t::..d;, the greater is J.Li, which implies that the wider the dy­ namic shift interval, the greater is the possibility of Pi being identified as Pi;. Formally, if l:::,d~' ~ l::,.d~ ~ l::,.d;, then J.Li'~J.Li~J.Li·

4 Related properties can be drawn as: Property 4.2a: With the same SCF;, the greater them, the less l::,.d; varies along with m, which implies that the greater the number of related peaks, the less a single peak can affect p;. Property 4.2b: l::,.d; is in linear relation to SCF; with a slope that is equal to, or greater than 1.5, which implies that l::,.d; heavily depends on SCF;. Property 4.2c: Along with the increase of m, the slope increases very slightly, which implies that l::,.d; depends on the number of the related peaks which support p;, rather than the total number of related peaks.

Property 4.4: (Based on Property 4.2 and Property 4.3) SCF; provides evidence for accepting or rejecting inaccurate Pi because J.Li is in the direct ratio to t::..di, and t::..d; is in the direct ratio to SCF;.

Actually, IN represents a qualitative hypothesis that partial component PCi (PCi = {Pin Pi» ... ,pi,..}) is contained by the input image (i.e., PCi@Sp). The pos­ sibility of the hypothesis can be easily calculated from J.Ll, J.L2, ... , and J.Lm:

where constant s1, s2 , ... , and Sm represent the priorities of the related data. For example, the peaks of C H3 in frequency section of 2800 em - 1 to 3000 em - 1 are more distinct than its peaks elsewhere, as a result, the peaks in this section are prior to those located in other sections.

5 A System on Spectral Image Interpretation

We have developed a practical system on infrared spec­ tral image interpretation by applying the method, and have fully tested the system against several hundred real spectral images. The experimental results with the sys­ tem are excellent.

5.1 Overview of the System

The system is implemented with C and MS-WINDOWS. Figure 3 shows the data flow diagram of the system. The input data of the system are gathered from given spectral images, and the solutions are partial components that the input images may contain.

Because inferences are based on qualitative features and qualitative correlations among related data, the sys­ tem can obtain very high correct interpretation perfor­ mance with noisy images or images of mixed compounds.

5.2 Experimental Results

We have compared two methods in the experiments. The first method (called "AF") is a conventional fuzzy method which is used by most similar systems [Clerc et al., 1986). AF adopts empirical fuzzy regions, and does not use SCF. The second method (called "AF*") is the method proposed in this paper which uses SCF to pro­ vide qualitative evidence for identifying inaccurate data.

We have tested AF and AF* against several hundred real infrared spectral images of organic compounds. The experimental results show that AF* is significantly bet­ ter than AF.

Table 1 lists part of the experimental results in which the first column shows the correct solutions; the second column indicates the solutions obtained by AF; and the third column shows the solutions obtained by AF*.

@) : identified PC set i s the same es that in the correct solution (!n this cese, RI=ll

0 : identified PC set b not the Male as that in the correct solution (the nuatler indicetes Rl)

Table 1: Experimental results with AF & AF*

There are two important standard metrics for evaluat­ ing the solutions of infrared spectral image interpretation for unknown compounds:

Definition 5.1 Rate of correctness (RC ): the rate that the identified partial component set is exactly the same as the partial component set in the correct solutions.

Definition 5.2 Rate of identification (RI ): the rate that the partial components in the correct solutions are iden­ tified.

Table 2 shows the comparison between AF and AF* with the two standard metrics. The comparison demon­ strates that both the RC and the RI increase by inte­ grating SCF, and the RC increases more significantly. The reason is that although AF can identify most patterns on spectral images, the rate that it can identify all patterns on spectral images is low because in prac­ tice there are always some partial components whose real peaks seriously shift from the expected values.

So far as we know, the RC of our system is the highest among the similar systems, and the RI of our system is higher than that of many similar systems [Clerc et al., 1986; Hasenoehrl et al., 1992; Puskar and Levine, 1986; Sadtler, 1988).

5.3 An Example

It is interesting to discuss the following example shown in Figure 4.

Figure 4 shows a spectral image of an unknown com­ pound (Anisole). The peak with an arrow is created by partial component CH3 · However, the peak can not be identified by conventional methods because the peak of CH3 at this frequency position should be a strong peak (i.e., s > 1.000), but in this example, the strength of this peak is very weak ( s < 0.130). So the membership degree of this peak is very close to zero, if not equal to. By using our method, the peak is identified as the peak of CH3 because the support coefficient of this peak is very high (SCF = 0.93).

6 Comparison with Related Work

Traditional methods and systems of spectral image in­ terpretation are based on numerical analysis which iden­ tify partial components by calculating the quantitative similarity or closeness between the patterns of known partial components and the input spectral images [Clerc et al. , 1986; Culthup et al., 1990; Sadtler, 1988). As we have discussed in section 2, in most cases especially when the inaccuracy of data is not slight, qualitative fea­ tures of spectral images are much more important and effective than quantitative ones. In other words, the similarity /difference between two spectral images does not strictly reflect the similarity/ difference between two structures. Therefore, in general, the solutions of the systems based on numerical analysis are only a series of candidates from which users have to finally decide the possible ones by themselves.

Many recently known methods and systems are based on AI techniques (Anand et al., 1991; Hasenoehrl et al., 1992; Puskar and Levines, 1986). Common techniques mainly include production systems, fuzzy logic and neu­ ral networks. Although various reasoning and interpret­ ing methods have been studied, the approaches to deal­ ing with inaccurate data in this kind of methods and systems are almost the same, i.e., reference values and fuzzy regions of inaccurate data are empirically provided in advance. Thus, qualitative correlations among data and dynamic information can not be used properly. As a result, compared with our system, the limitation of this kind of methods and systems is that, generally, they are only effective for a class of compounds, or pure com­ pounds because when image data are seriously inaccu­ rate, many useful inferences can not be drawn.

7 Ongoing Research

The reasons causing image data inaccurate are vari­ ous. However, the interaction among partial components themselves is an important factor or, in other words, the influence from coexisting partial components may be a chief reason to cause the peaks of a partial component to shift from their theoretical .values. In practice, spec­ troscopists frequently use the knowledge like: "if C6H6 coexists with CH3, then the peaks of CH3 around 2900 cm- 1 may shift.", or "if -C-0-C- has been identified, then the pattern of CH3 may change." Therefore, it is possible to update the possibilities of identified partial components by considering the interaction among them after qualitative interpretation has been made.

Take the spectral image shown in Figure 4. The image contains partial component C6H5 - (benzene-ring), -C-0C-, CH3 and others. On the spectral image, the peak of CH3 around 2900 em - 1 is seriously inaccurate due to the influence of C6H5 - and -C-0-C-. Although CH3 is identified based on qualitative correlations among re­ lated data, the possibility of CH3 being contained by the image is not high (P(CH3) =55%).

By Bayes rules associated with subjective statements, we can get P(CH3 I C6H5-) and P(CH3 1-C-0-C-) re­ spectively. And finally, P(CH3) may be updated with P(CH3 I S), where S stands for all relevant observations.

Theoretical and experimental work on the issue is in progress.

8 Conclusions

We have presented a novel method for interpreting spec­ tral images on the basis of qualitative features of spectral images and qualitative correlations among related data. We first put forward a new concept called support coef­ ficient functi9n (SCF). Then we proposed an approach to determining dynamic shift intervals, and an approach to calculating possibility of identifying inaccurate data, respectively. A triple, (Pi, Pi" P,i), is used to represent the hypothesis that Pi can be accepted as Pj, based on qualitative evidence provided by SCF.

We have also introduced a knowledge-based system on infrared spectral image interpretation which is developed by applying the proposed method. The system can suc­ cessfully identify partial components that unknown im­ ages may contain. Several hundred real spectral images have been identified, and the results of implementation are quite encouraging (the RI and the RC are about 0.90 and 0.74 respectively).