Fuzzy Invariant Indexing (FII)

3.2 Fuzzy Invariant Indexing (FII)

This section describes in great detail the general framework of the fuzzy invariant indexing technique, which is based on fuzzy invariant object descriptions and fuzzy if-then classifica-tion rules (for a good introducclassifica-tion to fuzzy sets and fuzzy logic, see [KLIRand YUAN1995]).

The section shows how fuzzy sets can be utilised to model the uncertainty of invariant ob-ject descriptions and how to perform the generation of obob-ject hypotheses by applying fuzzy classification rules.

3.2.1 Fuzzy Classification Rules

In the FII-technique object hypotheses are generated by evaluating fuzzy if-then classifica-tion rules, which take the invariant object descripclassifica-tions observed in an image as input and produce object classes with corresponding credibilities as output. These fuzzy classification rules have the general form:

IF ^†

9

%óò

\

9

AND ^ô¯ô¯ô AND ^†

»

%õò

\

»

THEN ^{ö %aò÷} (3.1)

with

Ñ % L

)ø€)¯ù¯ù4ù¯)?ú

(object classes)

¸ % L

)ø€)¯ù¯ù4ù¯)*û

(sub-rules for class^Ñ )

ü % L

)ø€)¯ù¯ù4ù¯)

ñ

(conditions of a sub-rule) where^† denotes the^ü -th input variable of sub-rule

¸

for the ^Ñ -th object, ^\ò the cor-responding fuzzy invariant object description (see Sect. 3.2.2), ^ö the output variable of sub-rule

¸

and^÷ò the^Ñ -th object class modelled as a fuzzy singleton. The total amount of conditions^ñ of a rule depends on the number of functional independent invariants of the underlying geometric configuration (see Sect. 2.2) of sub-rule

¸

for object ^Ñ and the total amount of sub-rules ^û depends on the number of different geometric configurations for object^Ñ .

In order to generate object hypotheses these fuzzy if-then classification rules are inferred in the following way:

mý

i% þ­ÿ

9

ÙdðÙ

»

†

! (3.2)

where

†

! denotes the membership function of the fuzzified invariant object descrip-tion ^\ò , which is evaluated for the invariant description ^† observed in an image. The resulting membership values ^ý of the sub-rules of object^Ñ are combined disjunctively:

ý % þ

9

Ùd«Ù

ý (3.3)

The final result is the indexed^Ñ -th object model with the measured credibility ^ý .

3.2.2 Fuzzy Invariant Object Descriptions

The main problem in generating the fuzzy classification rules (3.1) is to find appropriate fuzzy membership functions to model the fuzzy invariant object descriptions^\ò for a given object. Generally, two different types of descriptions must be distinguished: fuzzy invariant values and fuzzy invariant signatures.

Fuzzy Invariant Values

The most commonly used invariant object descriptions are invariant values. These values result from evaluating invariant functions based on geometric structures which can be ex-tracted for an object to be recognised (see Sect. 2.3).

Although systematic errors might occur during the feature extraction stage of the recognition system, the investigation of invariant values measured in different perspective views (see also Fig. 3.2) has indicated that the fluctuations can be adequately approximated by bell-shaped membership functions

m

:

c

$æ

!%zÞ

Œ

m

×

×

× ) æ

(3.4)

where the parameters ⁾ determine the shape of the functions. The advantage of employing these membership functions to model fuzzy invariant values is that they can be efficiently evaluated. Furthermore, the shape is determined by only two independent pa-rameters which can be easily adjusted or learned in an acquisition process by choosing the parameters in the following way:

The parameter^c determines the position of the maximum of the bell-shaped func-tion (3.4). Therefore, this parameter should be the mean of the fluctuating invariant values: ^{c %}

9

»

²š\

, where ^{\ ) Ln…Ú… ñ} are the invariant values for an object taken in^ñ different images.

The parameter determines the position of the inflexions of (3.4), which are located at . This parameter should be the standard deviation of the invariant values:

%  9

» ² <$\

w

! ; ‘ ×

.

For example, Fig. 3.1 shows the distribution of invariant values for the object nut of the Baufix domain (Appendix B) observed in 30 different images. These invariant values have been measured using the three functional independent invariants of the geometric structure of one ellipse and three straight lines, where the invariant values on the^V -axis are calculated using Eq. (2.8), the invariant values on the ^X -axis are calculated using Eq. (2.9) and the invariant values on the -axis are calculated using Eq. (2.10).

The histograms of these invariant values are depicted at the left hand side of Fig. 3.2.

The use of the aforementioned construction method leads to the parameters ^{9?9 % L ù‰} ,

3.2 Fuzzy Invariant Indexing (FII)

1 1.2 1.4 1.6

1.8 2 2 2.5

3 3.5

4 2

2.5 3 3.5 4

z (Measured values of invariant 3)

x (Measured values of invariant 1)

y (Measured values of invariant 2)

Figure 3.1: Invariant values of a conic and three lines for a nut observed in 30 images

9?9

% I ù

I

Æ for the first fuzzy invariant value,

9>;

% Ç ùø

,

9>;

% I ù

L@‰.§

for the second fuzzy invariant value, and

9>=

% ø€ù

§

,

9>=

% I ù

L¯‰

for the third fuzzy invariant value, respectively.

The corresponding fuzzy invariant values are shown on the right hand side of Fig. 3.2.

Thus, the resulting fuzzy if-then classification rule for the object nut can be written in a human readable form as:

IF (inv1 1.4) AND (inv2 3.2) AND (inv3 2.5) (3.5) THEN (object IS nut)

Further classification rules for several objects of the Baufix domain and their corresponding parameters that have been used to obtain the experimental results in Sect. 3.4 are listed in Appendix C.

Fuzzy Invariant Signatures

The other type of invariant object descriptions are invariant signatures which are not only single invariant values but rather complex invariant curves. Generally, these invariants are constructed by employing differential invariants or by mapping an extracted image curve into a distinguished coordinate frame using e.g. the canonical frame construction method (Sec. 2.3.1).

Invariant signatures are rarely employed in object recognition. However, in situations where object structures cannot be described adequately or reliably by simple geometric primitives, invariant signatures are more likely to be used. An example is the jigsaw puzzle mentioned in [ROTHWELLet al. 1992].

Since most of the invariant indexing techniques cannot utilise complex curves as indices, invariant signatures are reduced to single invariant values. This is done by evaluating the

0 5 10 15 20

1 1.5 2 2.5 3 3.5

Number of measured values

Invariant value

0 0.2 0.4 0.6 0.8 1

1 1.5 2 2.5 3 3.5

Membership

Invariant value (a) Histogram and resulting fuzzy invariant value for invariant 1

0 0.2 0.4 0.6 0.8 1

1 1.5 2 2.5 3 3.5

Membership

Invariant value 0

5 10 15 20

1 1.5 2 2.5 3 3.5

Number of measured values

Invariant value

(b) Histogram and resulting fuzzy invariant value for invariant 2

0 5 10 15 20

1 1.5 2 2.5 3 3.5

Number of measured values

Invariant value

0 0.2 0.4 0.6 0.8 1

1 1.5 2 2.5 3 3.5

Membership

Invariant value (c) Histogram and resulting fuzzy invariant value for invariant 3

Figure 3.2: Generation of fuzzy invariant values using the measured data of Fig. 3.1

Im Dokument Flexible object recognition based on invariant theory and agent technology (Seite 43-47)