Our aim in this project was to develop theory for the use of neural network predictions in spatially-distributed problems, and to implement this theory in application areas. Two application areas have been investigated intensively, namely the reconstruction of wind-fields from radar scatterometer data, and image segmentation.
In general we wish to carry out inference for some variables 
given some data 
. For example 
may represent wind vectors at a grid of locations, and
would be satellite
observations at the same locations. At each spatial location i
neural networks can be trained to output an estimate of 
, and these can be converted into
scaled likelihoods using 
. Under the assumption that 
these scaled likelihoods can be combined with a prior
model 
to obtain the posterior
distribution 
, see [1].
We have applied this method to two main problems. The first of these is an image segmentation task, where we wish to assign labels (e.g. ``road'', ``sky'', etc) to each pixel in an image. Here we built a tree-structured belief network (TSBN) model of the label images. This was trained using both maximum likelihood methods (using the EM algorithm) and conditional maximum likelihood methods. These TSBN models were evaluated as coding models of label images, and gave superior performance compared to block coding methods. The TSBN prior was combined with scaled likelihoods from neural networks to make predictions at each pixel. See refs [5] [6] [7] [8] for more details. This research also led to the development of novel ``dynamic trees'' models of images, which are TSBN models which can adapt their architecture in response to an input image [9] [10].
The second application was in wind-field modelling. We developed Gaussian-process prior models of wind fields. Because these models can be too smooth if there are frontal features in the wind field, we developed Gaussian process models with constrained discontinuities to model these situations. The prior models were fused with neural network predictions to obtain posterior estimates of the entire wind field. This method has been evaluated extensively on operational data (over 300 scenes), and produced significantly more accurate results approximately six times faster than the current operational methods. Our method is now being considered for use in day-to-day operations [2] [3] [4]. For more information on wind-field modelling and fusion with scatterometer data, follow this link, and see especially the work on ambiguity removal.
[2] NCRG/98/023 Bayesian Inference for Wind Field Retrieval I T Nabney, D Cornford, and C K I Williams. Neurocomputing Letters, 26-27:1013--1018, 1999.
[3] NCRG/98/025 Adding Constrained Discontinuities to Gaussian Process Models of Wind Fields D. Cornford, I. T. Nabney, and C. K. I. Williams. In M. J. Kearns, S. A. Solla, and D. A. Cohn, editors, Advances in Neural Information Processing Systems 11. MIT Press, 1999.
[4] NCRG/99/001 Modelling frontal discontinuities in wind fields. D. Cornford, I. T. Nabney, and C. K. I. Williams. Technical Report, Neural Computing Research Group, Aston University, 1999.
[5] NCRG/98/013 Combining neural networks and belief networks for image segmentation C. K. I. Williams and X. Feng. In T. Constantinides, S-Y. Kung, M. Niranjan, and E. Wilson, editors, Neural Networks for Signal Processing VIII. IEEE, 1998.
[6] NCRG/98/014 Training Bayesian networks for image segmentation Xiaojuan Feng and C. K. I. Williams. In F. Preteux, J. L. Davidson, and E. R. Dougherty, editors, Mathematical Modeling and Estimation Techniques in Computer Vision , volume 3457, pages 82--92. SPIE, 1998.
[7] Combining belief networks and neural networks for image segmentation gzipped postscript Xiaojuan Feng and C. K. I. Williams. 1999 Submitted to IEEE Trans. PAMI.
[8] Tree-structured Belief Networks as Models of Images. gzipped postscript Xiaojuan Feng and C. K. I. Williams. 1999 To appear in Proc. ICANN'99. IEE, 1999.
[9] DTs: Dynamic Trees gzipped postscript C. K. I. Williams and N. J. Adams. In M. J. Kearns, S. A. Solla, and D. A. Cohn, editors, Advances in Neural Information Processing Systems 11. MIT Press, 1999.
[10] SDTs: Sparse Dynamic Trees. N. J. Adams and C. K. I. Williams. To appear in Proc. ICANN'99. IEE, 1999.