Several defuzzification methods for non-nominally scaled data have been proposed in published literature [
12,
13]. However, in remote sensing, crisp classification results are nominally scaled [
14]. Defuzzification in remote sensing therefore means that
for each entity is converted from
into
with
, where
indicates that the entity of concern is not a member of class
A, and
indicates that it is a member of class
A. Each entity is therefore usually assigned to the class for which it has the highest membership degree, that is, where
. This class is often referred to as the Best Classification Result (
BCR) [
11,
15], with
. A very simple but often applied method to defuzzify nominally scaled entities is to set a threshold t for
: entities with
remain unclassified, those with
are assigned to
BCR [
6]. However, it is obvious that doubtful crisp classification results can be produced with this simple decision rule for the following reasons: (1) even those entities whose fuzzy memberships indicate little clarity of their class assignment can be crisp assigned to their
BCR, that is, entities with
(uncertainty); (2) entities whose
is similar to any of the remaining class memberships of
(ambiguity) might be defuzzified; (3) entities whose
and all other class memberships indicate a high classification fuzziness (
) might be defuzzified.
2.2.1. Classification Uncertainty and Ambiguity
For each entity being fuzzy-classified using a classification scheme
M, with
m (
L-)classes, the elements of its classification vector
can be sorted following a “≥” relationship, beginning with
, where
holds the membership degree of the second-best class and so on until the
mth-best class. For better readability an index will be used here, to indicate the membership degree of an entity to its
ith-best class with
:
. Since the best possible membership degree an entity can have for an arbitrary class is
, the entity’s classification uncertainty can be expressed by:
. An entity’s classification is ambiguous as soon as it has membership degrees of
for any of the other classes in the classification scheme
M [
16,
17]. Additionally, the ambiguity of an entity is considered higher, the closer all its
values are to each other. That is, in a “≥” order of membership degrees per entity, an entity with
is less ambiguously classified than an entity with
. Consequently, quantifying and analysing the ambiguity and uncertainty for each fuzzy classified entity and setting meaningful thresholds to decide whether to defuzzify its fuzzy classification result or not, can make the crisp classification result as reliable as necessary.
2.2.2. Fuzziness
According to [
5], fuzziness can be expressed by the separability of a fuzzy set and its complement. For fuzzy classifications in remote sensing this means: the clearer an arbitrary class
A can be separated from its complementary class
, the less fuzzy the class is. Siler & Buckley [
4] transfer this to evaluate an entity’s classification fuzziness as follows: an entity is the less fuzzy assigned to a class or its complement, the closer its membership degree µ
A to this particular class is either to 1.0 or to 0.0. That is, an entity is the fuzzier assigned to
A, the closer
and vice versa. When applying a fuzzy classification scheme
M with several classes, as outlined before, this means an entity is the fuzzier classified, the more class memberships of
it has and it is fuzziest classified if all of the
m memberships are
. Besides minimizing an entity’s ambiguity and uncertainty, its fuzziness should be minimized too, in order to define sensible decision rules for the defuzzification of an entity’s fuzzy classification. Note: an entity with a membership degree of
and
simultaneously has the highest possible certainty and the lowest possible ambiguity and fuzziness.
2.2.3. Quantifying Classification Uncertainty, Ambiguity and Fuzziness per Entity
When determining the classification ambiguity, it is common in both published literature [
6,
15,
18] and existing software (for example eCognition), for only the best and second-best class memberships to be evaluated. This is because for entities with ordinally scaled
vectors, as soon as
that entity’s classification is already ambiguous. However, measurement of the classification ambiguity becomes more precise if all membership degrees are taken into account but in this case, the degree of ambiguity is dependent on the number of classes
m of a given classification scheme and can therefore be less easily compared with other classification schemes. In general, measures expressing an entity’s uncertainty, ambiguity and fuzziness should ideally be independent from
m and easy to interpret. Some measures of uncertainty, ambiguity and fuzziness are discussed below. These measures were implemented using the Cognition Network Language (
CNL) [
19] and can be applied as a so-called “Customized Algorithm” in eCognition (see the relevant file, together with a short description of the “Customized Algorithm” in
supplementary materials).
The Classification Stability Index
CSI, which is implemented in eCognition software as “Classification Stability” [
11], expresses the difference between
and
for each entity. If
is ordinally scaled [
15] the
CSI quantifies the entity’s ambiguity:
where the value range of
CSI is given by
. The lower the
CSI, the more ambiguous (less firm) an entity’s classification is. It takes into account
only and none of the remaining
µi of a classification. If all
m class memberships of a given classification scheme are to be taken into account, the
CSI extends to
:
The value range of
CSI* is given by
, which means that the
CSI* can have negative values. Burrough [
18] suggests the Confusion Index (CI) to express the ambiguity of an entity’s classification result, which is simply the compliment of the CSI. It can be calculated by:
with the value range of
. That is, an entity is an increasingly distinct member of its
BCR the lower the
CI is. Analogous to the
CSI, the
CI can be extended to a more precise index by taking into account all
m memberships of an entity to the classes of a given scheme:
The value range of the is then . Thus, it needs to be interpreted differently: the closer the of an entity’s classification is to m, the less distinctly it is assigned to its BCR.
- 2.
Ambiguity Index
There have been different definitions proposed for the Ambiguity Index (
AI). Burrough [
18] defined it as the difference between the best possible classification result
and the best classification result actually achieved (
):
where the value range for
is given by
. This means: the less certain it is that an entity has been assigned to the best class, the more ambiguous its class assignment is. This parameter therefore measures the classification uncertainty of an entity, rather than its ambiguity. Siler & Buckley [
4] instead suggested adding together all membership degree values achieved by an entity, divided by its best membership degree:
where the value range for
is given by
.
takes into account an entity’s membership degree for all classes in a given classification scheme. However, as for the
CS* and
CI*, under this definition the index is dependent on
m, while
is independent of
m. In contrast to
,
truly measures the classification ambiguity: even if
for an entity is low, but the entity has only one single class assignment
. That is, the classification result for this particular entity might be uncertain but not ambiguous. Vice versa, the maximum ambiguity is achieved if all of the entity’s membership degrees are equal, independent of their grade, that is, if
. In case the entity of concern remains unclassified
and
remains undefined.
- 3.
Fuzziness
Siler & Buckley [
4] suggested quantifying the fuzziness of an entity’s classification by evaluating its number of class assignments with the highest possible fuzziness, that is, with a membership degree of
. The more class assignments with
an entity has, the fuzzier its classification is. Consequently, the more class memberships with
or
an entity has, the less fuzzy it is classified. Membership degrees of
and
impact the accumulated fuzziness, respectively. They suggested two methods: a less precise method, with:
where the value range for
is given by
, and a more precise method, which is given and discussed in
Appendix A. The latter is similar to the method suggested by de Luca & Termini [
20]. However, although it is more precise, it is more sensitive when applying complex classification schemes with many classes: for the entity of concern a single membership to one of the scheme’s classes with
or
is already enough for this measure to equal its maximum or minimum value. In contrast,
behaves continuously: it achieves its maximum if all class memberships yield
, otherwise it decreases with the number of memberships
per entity, whereas the closer the memberships are to 0.0 or 1.0 (
or
) per entity the more
decreases. Nevertheless, none of the measures of fuzziness are capable of expressing an entity’s classification certainty or ambiguity. A detailed overview of fuzzy uncertainty and related discussions, has been provided by Pal & Bezdek [
21].
2.2.4. Decision Rules for Defuzzification
Defuzzifying a fuzzy classification result of a given entity means to crisply assign it to its BCR. However, as already stated above, fuzzy classification results should only be defuzzified if the entity of concern is undoubtedly assignable to its BCR. In this context “undoubtedly” translates to: least uncertain, least ambiguous and least fuzzy. Since uncertainty, ambiguity and fuzziness can be measured as outlined before, these measurements can support the user in deciding when a particular fuzzy classification result counts as being defuzzified. That is, when “doubts” about an entity’s BCR are low enough for it to be crisply assigned to that class. Consequently, the user needs to set thresholds for the measured classification uncertainty, ambiguity and fuzziness per entity, above which he or she allows the fuzzy classification result to be defuzzified. Since entities below the set thresholds remain unclassified after defuzzification, the user also needs to consider the amount of classified and unclassified entities. In remote sensing this means the amount of area being classified or unclassified. Combining all (or some) of the presented measures means that several conditions need to be fulfilled simultaneously before an entity is allowed to be crisply assigned to its BCR. The latter means setting a threshold for each measure.
- 1.
Uncertainty
The uncertainty of a fuzzy classification result is expressed either by
(the closer
is to 1.0, the more certain the classification result, and vice versa), or inversely by Burrough’s Ambiguity Index
(the closer
to 0.0, the more certain the classification result and vice versa). Both measures indicate to what degree an entity fulfils the classification criteria for its
BCR. For simplicity reasons, only
is regarded in this manuscript. As stated earlier, setting an arbitrary threshold for
is common practice, and the simplest decision rule for defuzzification. However, according to Siler & Buckley [
4], entities with
must be regarded as a member of the
BCR’s complementary class
. Consequently, defuzzifying such entities would be a contradiction in terms. Additionally, only defuzzifying entities with
avoids the defuzzification of entities with maximum fuzziness. Consequently, a defuzzification threshold of
is sensible. The closer the threshold for
is set to 1.0, the more certain and—to a certain degree—the less fuzzy the classification can be regarded.
- 2.
Fuzziness
A classified entity with a membership of
to its
BCR must be considered as fuzzy and uncertain. According to
Section 2.2.2 it is classified with the highest possible fuzziness if all of its
, that is, if
. Thus, if fuzziness measured with
is applied as a defuzzification criterion, the decision rule should be
. The latter is achieved already if
. However, even then, and even if an entity’s classification is certain (
), it still might be highly fuzzy if all remaining
. Consequently, if only entities classified with the least possible fuzziness should be defuzzified, a threshold for fuzziness with
should be selected.
- 3.
Ambiguity
Ambiguity describes how distinctly an entity is assigned to its
BCR. As outlined in
Section 2.2.1, an entity’s fuzzy classification ambiguity increases the more of its class memberships
are equal, and it can be measured as depicted in Equations (4)–(7) and (9), whereby
CI,
CI*,
CSI and
CSI* can be below 1.0 if
and all remaining
. In contrast,
equals its maximum only if all
have exactly the same value. Since its value range is:
, a fuzzy classification result is the less ambiguous, the closer the threshold for
is set to
and the more ambiguous, the closer it is set to
m.
- 4.
Compound decision rule for defuzzification
A fuzzy classified entity is the less doubtfully a member of its
BCR the more certain, the less fuzzy and the less ambiguous its classification is simultaneously. Consequently, an entity’s defuzzification should be based on a compound decision rule, which simultaneously demands all the defuzzification criteria be fulfilled, which roughly means.
In this configuration a least doubtfully classified entity is given if its
, its
, and its
, which is given if
and
. Vice versa, if an entity’s
, its
and its
, “doubts” about its class assignment to its
BCR are at a maximum (see
Section 2.2.3). Nevertheless, the precise thresholds should be determined by the user’s requirements concerning the classification’s reliability after defuzzification. Applying a defuzzification rule as described here means that entities fulfilling these criteria are crisp-assigned to their
BCR, while the rest remain crisp-unclassified.