A Locally Tuned Neural Network for Ground Truth Incorporation

Abstract

In most remote sensing applications, the forward probiem denotes the calculation of fields and waves from given parameters of the media. The inverse problem is to calculate the target or media parameters from measured fields and waves through relevant remote sensing electromagnetic theory. One of the most important steps of applying artificial neural networks (ANNs) to solve parameter inversion problems in remote sensing applications is to first establish a very reliable approximation of the true foryard mapping, y = Nx), based on an ANN approximation $, trained by data pairs of the media parameters x , and the measurements of the fields and waves y , generated through a rempte sensing electromagnetic model $' . While the trained A" $ can accurately approximate the electromagnetic model with negligible deviations, the degree of accurate ANN approximation of $' to the true mapping $ can only be verified by some available ground truth, which should be used to fine tune the trained ANN approximation $ . In this paper, we applied a minimum disturbance principle in fine tuning the approximated ANN by incorporating the small amount of available ground truth. More specifically, the ground truth is used to slightly modify the local vicinity of the mapping associated with this pair of training data without disturbing the whole mapping (i.e., without rocking the whole boat). This can be achieved by a locally tuned ANN formed by the radial basis functions, instead of the projection based ANN formed by the global logistic sigmoidal functions. FORWARD MODELS FOR INVERSE MODELS Remote sensing problems are of the general class of inverse problems, where we wish to infer the physical parameters which could cause a particular effect. Inverse problems admit of two lines of attack; creating a forward model of the process which then must be manipulated to yield an inverse [3], or creating an explicit inverse of the physical process [7]. An explicit inverse suffers from many-to-one problems when more than one cause could account for a particular effect. Forward models, on the other hand, can accurately model a causal relauonship. With a method of inverting the forward model, we can find a possible multiplicity of solutions from which we can select according to other information or additional constraints we wish to impose [ 11. The inversion of a forward model takes the form of a search in the input space of the model for an input which produces the desired output. With gradient information relating the input to some performance criteria, the search of the input space can proceed as a directed search, usually taken in the direction of this gradient. USE OF DATA DRIVEN MODELS WITH IMPLICIT FUNCTIONS There are three main types of forward models available: explicit functions, implicit functions, and data driven models. Explicit functions take the input and perform some direct functional mapping from input to output. To iteratively obtain an inverse. it is a simple matter of taking gradients of some performance criteria with respect to the inputs, using the known functional mapping. Implicit functions occur when we perform some operation on initial values, but do not have a direct functional relationship, as with initial conditions and their differential equations. Without a direct functional form for the implicit function, it is often infeasible to calculate gradients and perform a directed search in the input space. Data driven models come in two types: parametric and non-parametric. Parametric models assume some functional form for the process, while non-parametric models use flexible basis functions able to r e p resent a wide variety of functions. Data driven models are iteratively trained with representative data, adjusting the parameters of the model by minimizing the squared error between the data and the actual output of the model. Data driven models can be very useful for inverse problems, provided that the gradient of the output vector with respect to the input vector can be found. This is true not only of data driven appLdions; when we have an implicit functional relationship between the input and output, a data driven model can provide us with the means of inverting that implicit functional relationship. By training a data driven model with data produced by our implicit model, we can make a copy of our implicit model that we can invert. USING DIFFERENT TYPES OF DATA There are a few issues in connection with training a data driven model. One of these issues is the accuracy of the model. It may happen that the model is not uniformly accurate through its input space. We would then desire a method to line tune this area without retraining the entire model. Another issue is the type of data we use, and the domain we derive it from. We may use an analytic model to obtain our data, or we may be using measured data. Our measured data may come from a changing environment, or from a changing sample population. It would be advantageous to be able to use this data to update our model, with a minimum of training and a minimum of disturbance. Similarly, when using an analytic model, we may sometimes have a few points of measured data, and it would again be desirable to add the information this data represents to our model without having tu train a complete model. These problems can be solved by use of locally tuned functions about our regions of interest [5]. These functions would Serve as perturbations of our t~ained representation. Moreover, the different type of data and different types of error can be provided for simultaneously with our locally tuned functions. For example, we may receive more data in a particular region of the input space, and our model may have been ill trained in that region. A locally tuned function can be added to our model which alleviates both sources of error. We propose using radial 91-728?0/92$03.00 0 IEE 1992 1064 basis functions as locally tuned functions to add to trained neural networks, composed of sigmoidal activations. This provides a powerful combination of the features of sigmoid networks and radial basis functions, using the best fames of each