28.3 Kalman Filter-Based Spectral Characterization Signal-Processing Techniques
Three KFSCSP techniques are derived and presented in this section, called Kalman filter-based spectral signature estimator (KFSSE), Kalman filter-based spectral signature identifier (KFSSI), and Kalman filter-based spectral signature quantifier (KFSSQ). In KFSSE, the input and output of its measurement equation are specified by a noise-corrupted signature vector and its true signature vector to be estimated, respectively. KFSSE then uses a state equation to predict spectral values of the true signature vector across its spectral coverage via a signal model such as the Gaussian–Markov model. So, the role of KFSSE is to capture spectral signature changes between adjacent spectral bands compared with KFLU, which is developed to capture changes in abundance fractions between two adjacent pixel vectors. Most importantly, as noted previously, KFSSE does not need a linear mixture model as required by KFLU. Therefore, there is no need for KFSSE to find image endmembers to form a linear mixture model. On the other hand, KFSSI is developed to identify a signature vector via a matching signature vector chosen from a known database or spectral library. In doing so, KFSSI is derived from KFSSE by replacing the true signature vector used in KFSSE with an auxiliary signature vector that enables it to capture the matching signature vector in identifying the unknown signature vector. According to functionality, both KFSSE and KFSSI are developed as signature vector estimators, but not abundance vector estimators as the way that KFLU is originally designed. This is because no linear mixture model is used to unmix abundance fractions of image endmembers. Unlike KFSSE or KFSSI, KFSSQ can be considered as a follow-up signature filter. It models its state equation as a zero-holder interpolator by taking a KFSSE-estimated signature vector, that is, spectral estimate in its measurement equation as a system gain vector to achieve spectral quantification of the estimated signature vector. As a result, the quantification of the spectral signature value at the current lth band is used as a prediction of the spectral value at the next adjacent band, the (l + 1)st band. Its ability in spectral quantification makes KFSSQ particularly useful in applications of chemical/biological (CB) defense where the lethal level of a CB agent is determined by its concentration (Wang et al., 2004), and the collected samples are not necessarily correlated as image pixel vectors in an image cube. Therefore, the quantification of a CB agent is crucial in damage control assessment. Finally, in order to demonstrate the utility of the three KFSCSP techniques in spectral estimation, identification, and abundance quantification, experiments including computer simulations and real data are conducted for performance analysis.
28.3.1 Kalman Filter-based Spectral Signature Estimator
We assume that 
is a true signature vector to be estimated and 
is an observable signature vector from which the ture signature vector t can be estimated. Since the measurement equation and the state equation described by (28.1) and (28.2) are developed for image classification, they do not directly meet our current requirements and must be modified as follows:
where the index l denotes the lth band, 
is a system gain vector, 
is a transition parameter from the lth band to the l + 1st band, and 
and 
are white noise vectors, all of which must be determined a priori. According to Chang and Brumbley (1999a, 1999b), in order to implement (28.5) and (28.6) recursively, the initial condition to start the recursive algorithm between measurement and state equations is set to 
by assuming that the estimate of true spectral signature at the first band is 0. The KF is then implemented recursively until it reaches the last band (see recursive formulas (6–10) in Chang and Brumbley (1999a)). Comparing (28.5) and (28.6) to (28.1) and (28.2), there are several salient differences between KFLU and KFSSE. First of all, KFLU uses (28.1) and (28.2) to model the same image vector. That is, (28.1) represents the status of the image pixel r, which is linearly mixed by a set of p known image endmembers, 
and (28.2) specifies its corresponding abundance vector 
that is unknown but needs to be estimated via (28.1). Therefore, KFLU requires the prior knowledge of image endmembers to be used in the linear mixture model (28.1). As a rather different approach, KFSSE is developed to estimate a target signature vector t rather than an abundance vector α. It consideres a true signature vector 
as a state vector in (28.6) and estimates the state vector band by band via an observable vector 
specified by the measurement equation (28.5). As a result, the variables in both equations (28.5) and (28.6) are scalars and represent spectral signature values of the lth band of the target signature vector t within the observed signature vector r. In this case, the measurement equation (28.5) is simply a true signature vector corrupted by noise 
not a linear mixture model specified by (28.1). Another major difference is that both (28.5) and (28.6) are linear functions of the same lth band of the true spectral signature t, tl. This is different from (28.1) to (28.2), which are linear functions of two distinct vectors, image vector r(k) and abundance vector α(k) with different dimensions L and p, respectively. Therefore, compared with KFLU, in which input and output are an image pixel vector r(k) and an estimate of the abundance vector α(k) and 
, respectively, the input of KFSSE is simply the lth band spectral value of the observable spectral signature vector r, rl, and its output is an estimate of tl, which is the lth band spectral value of true signature vector t, denoted by 
. Furthermore, the state in (28.6) is designed to predict the spectral value of the (l + 1)st band of the true signature vector t by updating the currently estimated spectral value of its lth band accoridng to the spectral variation between two adjacent bands. This is quite different from the state in (28.2), which is included to keep track of changes in abundance vectors between two adjacent image pixel vectors. Besides, (28.6) models the relationship of the spectral signature values between one wavelength and the next adjacent wavelength as a Gaussian–Markov process specified by the state transition parameter 
and the additive Gaussian noise, 
. If the variance 
or standard deviation of v is set too low, then the state equation may not be effective enough to capture real variations in spectral values of one material. Accordingly, throughout the chapter, the standard deviation of the state noise is always assumed to be high.
28.3.2 Kalman Filter-Based Spectral Signature Identifier
In the previous section, KFSSE is developed to estimate a true signature vector t through an observed signature vector r. By remodeling its measurement and state equations, KFSSE can be re-interpretated as a spectral signature identifier, which is referred to as KFSSI. In order to perform spectral signature idenitification, it always assumes that there is a database or spectral signature library available for this purpose, denoted by 
, which is made up of K spectral signature vectors, 
.
We suppose that the observable spectral signature vector is 
, which needs to be identified via a data base or spectral library, 
. The idea is to use the measurement equation to describe the observable spectral signature vector r as a noise-corrupted matching spectral signature vector 
selected from Δ. An auxiliary spectral signature vector t is then introduced in the state equation to model the state that describes the ability of identifying a given matching signature vector s in the observable signature vector r. In other words, the identification of the observable spectral signature vector r is performed by finding a particular spectral signature vector sk from the Δ that best matches r. In this case, the measurement equation (28.5) and the state equation (28.6) can be re-modeled as
and
respectively, where the scalar system gain cl is generally considered as 1. Unfortunately, (28.7) and (28.8) are uncorrelated in the sense that there is no state parameter tl in the measurement equation. In order to resolve this dilemma, we re-express (28.7) as
where the matching spectral signature vector 
is included in the measurement equation (28.9) and therefore is the signature vector to be used to match the observable signature vector 
. The auxiliary spectral signature vector t in (28.9) then serves as a bridge between the observable signature vector r and the matching signature vector 
to dictate the ability of a given signature vector 
to match the observable signature vector r. As a result of introducing the target signature vector t in (28.9) the system gain parameter cl in (28.7) is re-expressed in (28.9) and becomes a spectral-varying parameter specified by 
. Interestingly, the use of such a spectral-varying system gain parameter by 
takes care of the effects resulting from a matching signature vector 
in (28.9). This is a key difference between KFSSE and KFSSI. Therefore, the KF using the pair of (28.9) and (28.8) to perform spectral signature identification is called Kalman filter-based spectral signature identifier (KFSSI). In this case, the input is provided by the observable spectral signature vector 
to be identified by a particular signature vector in 
via KFSSI. And the output of KFSSI is the estimate of the auxiliary signature vector t, 
. To implement (28.8) and (28.9) recursively, the initial condition used to start the recursive algoithm between measruement and state equations is set to 
by assuming that the estimate of the auxiliary signature vector t at the first band is 0. Then KF is implemented recursively until it reaches the last band. Since KF is an MSE-based estimation technique, the least squares error (LSE) between 
and the observable spectral signature 
is used instead of MSE as a quantitative measure defined by
Using (28.10), we define the observable spectral signature vector 
as a particular spectral signature vector 
in Δ that yields the least value of LSE(r, s).
It is worth noting that in addition to the auxiliary spectral signature vector t introduced in (28.9), another major distinction between KFSSE and KFSSI is that the latter identifies a signature vector via 
, whereas the former estimates the true signature vector without the need of Δ. As a result, their measurement equations are implemented differently where the signature vector used in the measurement equation (28.5) for KFSSE is the true signature vector t, whereas the signature vector used in the measurement equation (28.7) or (28.9) for KFSSI is a matching signature vector 
chosen from the database or spectral library, 
. A third major distinction is that the standard deviation of the state noise v, σv specified in (28.8), generally has a significant effect on LSE compared to σv used by KFSSE, which does not have such effect on the estimation performance of KFSSE. This is because KFSSI performs identification via a matching signature vector in Δ. Consequently, a small variation caused by σv may result in an incorrect identification. This is particularly true when the spectral signature vectors in the Δ are similar. Additionally, a fourth major distinction is that the two equations (28.8) and (28.9) implemented in KFSSI must process two different signature vectors, observable signature vector r and matching signature vector 
, compared with KFSSE, which only has the observable signature vector r processed in (28.5) and (28.6). Accordingly, KFSSI is sensitive to the measurement noise σu and the state noise σv, both of which are correlated. However, this is not the case for KFSSE. So, in KFSSI, both measurement noise and state noise must be appropriately addressed as will be demonstrated by the experiments in the following sections.
The relationship between KFSSE and KFSSI, which can be illustrated by the relationship between constrained energy minimization (CEM) and orthogonal subspace projection (OSP) in Chapters 2 and 12, is worth commenting on. CEM assumes that there is a desired target signature vector d to be detected. This d actually corresponds to the target signature vector t assumed in KFSSE to be estimated. It then uses d to generate detected abundance frcations of d present in all the pixels in an image cube to perform target detection, whereas KFSSE estimates 
to approxmiate the lth band rl in the observed signature r. In other words, CEM performs target detection in an image cube, whereas KFSSE estimates target abundance in a single signature vector. On the other hand, OSP assumes that there are p image endmembers that are used to unmix pixel vectors in an image cube by estimating abundance fractions of each of p image enedmembers present in each of the image pixel vectors. This is exactly what KFSSI does for a single signature vector where it uses a database formed by K signatures 
that correspond to p image endmembers used by OSP for unmixing and then identifies a signature vector from the database by finding a best matching signature vector. This functionality is equivalent to that of OSP, which uses its p estimated abundance fractions for each of image pxiel vectors to perform linear spectral unmixing. The key difference is that CEM and OSP operate on image cubes, whereas both KFSSE and KFSSI operate on single signature vector only.
28.3.3 Kalman Filter-Based Spectral Signature Quantifier
So far, KFSSE and KFSSI developed in the two previous subsections make attempts to perform signature vector estimation and identification from an observable signature vector r. This subsection presents another new application of KFSSE in spectral quantification to quantify a signature vector r, which is referred to as KFSSQ.
One of great challenges in hyperspectral data exploitation is spectral quantification. This is particularly important for CB defense where the concentrations of targets are of major interest rather than their shapes and sizes generally encountered in image processing. The concentration of an agent is a key element in the assessment of threat (Kwan et al., 2006). Over the past years, many algorithms have been developed for quantification (Goetz and Boardman, 1989; Shimabukuro and Smith, 1991; Settle and Drake, 1993; Smith et al., 1994; Tompkins et al., 1997; Heinz and Chang, 2001; Kwan et al., 2006). However, most of them are image analysis-based linear spectral unmixing methods, which estimate abundance fractions of image endmembers assumed to be present in a linear mixture model such as fully constrained least squares (FCLS) method (Heinz and Chang, 2001). The technique developed in this subsection is based on signature vector and allows quantification of a particular material substance present in a single signature vector of interest band by band without appealing for a linear mixing model. Once again, a direct application of KFSSE in spectral quantification is not applicable, since the state equation described by (28.6) is the spectral value tl but not its abundance fraction αl, which is of interest in quantification. In order to estimate the abundance fraction αl in the state equation, the state tl in (28.6) is replaced with its abundance fraction αl and the system gain cl in (28.6) is replaced with the lth band of the target signature vector t, tl. The state transition parameter 
is set to 1 such that the state equation is actually a zero-holder interpolator. The resulting measurement equation and state equation become
By virtue of the pair of (28.11) and (28.12), a KFSSQ can be designed and implemented to quantify the target signature 
present in the data sample vector r in terms of its abundance fraction α. In this case, the inputs of KFSSQ are observable signature vector 
and target signature vector 
. Its output is the estimate of αl associated with target signature vector t, denoted by 
.
There are two scenarios that can be implemented for KFSSQ. One is that the target signature vector t in (28.11) is assumed to be known as a priori. However, in many applications, we may not have such prior knowledge. Under this circumstance, another scenario is that t can be estimated from the observable signature vector 
via KFSSE where the tl in (28.11) is then replaced by its KFSSE estimation, 
. The resulting measurement equation becomes
The pair of (28.12) and (28.13) specifies the implementation of KFSSQ for unknown target signature vector t where the input is KFSSE-estimated 
specified by (28.13) and the output is the estimate of αl in (28.12) corresponding to 
, denoted by 
.
KFSSQ described by (28.12) and (28.13) is quite different from KFLU, which assumes all target signature vectors, that is, image endmembers are known a priori and then quantifies all the known signatures present in the data vector r via a linear mixing model by FCLS. However, the second scenario of KFSSQ specified by (28.12) and (28.13) can be implemented without prior knowledge of the target signature vector t, a task that cannot be accomplished by KFLU.
In the following, the algorithmic steps of each of the three KFSCSP techniques, KFSSE, KFSSI, and KFSSQ, are described. We assume that 
and 
are the observable signature vector and the target signature vector to be estimated, respectively.
KFSSE
- Preset the values of σu and σv.
- Set
for all 
. - let
and 
.
:
- Calculate Kalman gain
- Update the state at
by
- Update variance of the state at
by
- Predict the state at
by
- Predict variance of the state at
by
by 1 and proceed rl with 
, repeat the five recursive steps outlined in step 2 until 
.
with 
for all 
obtained from steps 2 and 3.
KFSSI
- Assume that
is a given spectral library or database and 
is a matching signatures selocted from Δ. - Pre-set the values of σu and σv.
- Set
and 
. - Set
for all 
. - Set
for all 
.
:
- Calculate Kalman gain
- Update the state at
by
- Update variance of the state at
by
- Predict the state at
by
- Predict variance of the state at
by
by 1 and proceed rl with 
. Update 
and repeat the five recursive steps outlined in step 2 until 
.
with 
for all 
obtained from steps 2 and 3.
and 
over 
.
as the signature s∗.
KFSSQ
- Preset the values of σu.
- Set σv = 0.
- Assume that α(t) is the abundance vector corresponding to the target signature t. Set
for 
. - Let
and
. - Set
for all 
.
:
- Calculate Kalman gain
- Update the state at
by
- Update variance of the state at
by
- Predict the state at
by
- Predict variance of the state at
by
by 1 and proceed rl with 
, update 
and repeat the five recursive steps outlined in step 2 until 
.
with 
for all 
obtained from steps 2 and 3.