Improving Wishart Classification of Polarimetric SAR Data Using the Hopfield Neural Network Optimization Approach
2. Hopfield Neural Network Optimization Process
2.1. Decomposition and Wishart Classifier
- The polarimetric scattering information may be represented for each image pixel by the Pauli scattering vector . Hence, is the Hermitian product of the target vector of the one-look ith pixel. PolSAR data need to be multilook processed for speckle reduction by averaging n neighboring pixels. The coherency matrix is then obtained as,
- From the coherency matrices, we apply the H/ᾱ decomposition process as a refined scheme to parameterize polarimetric scattering problems. The scattering entropy, H, is a key parameter in determining the degree of statistical disorder, in such a way that H = 0 indicates the presence of a single scattering mechanism and H = 1 results when three scattering mechanisms with the same power are present in the resolution cell. The angle ᾱ characterizes the scattering mechanism as proposed in [8–10].
- The next step is to classify the PolSAR data into nine classes in the H/ᾱ plane, although zone three never contains pixels. These classes include different types of scattering mechanisms present in the scene, such as vegetation (grass, bushes), water surface (ocean or lakes) or city block areas. Section 3.3 includes a description and discussion about the content of these classes.
- Hence, the classification process results in eight valid zones or clusters, where each class is identified as wj or j, i.e., in our approach j varies from one to nine. Then, we compute the initial cluster center of coherency matrices for all pixels belonging to each zone (class wj) according to the number of pixels nj belonging to the class wj as follows,
- Compute the distance measure for each pixel i characterized by its coherence matrix 〈T〉i to the cluster center as follows,
- Assign the pixel to the class with the minimum distance,
- Verify if the termination criterion is met, otherwise set t = t + 1 and return to Step 1. The termination criterion is set to a prefixed number of iterations tmax. Nevertheless, the criteria that we adopt are the following: assuming that at each iteration t we have pixels belonging to the class wj and at the next iteration t + 1 the pixels belonging to the same class are , if the relative difference between both quantities is below a certain percentage, then the process also stops. We experimented with thresholds between ±0.5% and ±5%.
2.2. Cluster Separation Measures
- The dispersion within clusters (Dii): The Dii is defined as the averaged distance between all the pixels within the cluster wi to the cluster center Vi. It measures the compactness of cluster wi and is given by,
- The distance between two clusters (Dij) is defined as,
- The cluster separability (Rij) involves two clusters and is defined as,
2.3. The Hopfield Neural Network for Improving the Wishart Classification
2.3.1. Preliminary Considerations and Network Architecture
- How can we achieve that a pixel changes its current label so that it is classified as belonging to a different class?
- How can we achieve that a pixel does not change its label when its neighbors have identical labels as the label of the pixel under analysis?
- How can we achieve maximum cluster separability?
- When can we consider that no more changes are required?
2.3.2. Dynamics of the Hopfield Neural Network
2.3.3. Energy Definition
2.3.4. Derivation of the Connection Weights and External Inputs for the HNN
2.3.5. Summary of the HNN-Based Image Classifier
- Initialization: create a network netj for each cluster wj. For each netj create a node i at each pixel location (x,y) from the image to be classified; t = 0 (iteration number); load each node with the state value , i.e. the support provided by the Wishart-based classifier, Equation (9); compute and through Equation (20); set ε = 0.01 (a constant to accelerate the convergence); tmax = 4 (maximum number of iterations allowed, see Section 3.2); set the constant values as follows: Li = 1; β = 3.38; dt = 10−3. Define nc as the number of nodes that change their state values at each iteration. The iterations in this discrete approach represent the time evolution involved in Equation (12).
- HNN process: set t = t + 1 and nc = 0; for each node i in netj compute using the Runge-Kutta method and update , both according to Equation (12) and ifthen nc = nc + 1; when all nodes i have been updated, if nc ≠ 0 and t < tmax then go to Step 2 (new iteration), else stop.
- Outputs: updated for each node; it is the degree of support for the cluster wj, see Figure 1. The node i is classified as belonging to the cluster with the greatest degree.
3. Experimental Results
3.1. Design of a Test Strategy
- The low entropy vegetation consisting of grass and bushes belonging to the cluster w2 has been clearly homogenized, this is because there are many pixels belonging to w4 in these areas re-classified as belonging to w2.
- Also, in accordance with , the areas with abundant city blocks display medium entropy scattering. We have homogenized the city block areas removing pixels in areas that belong to clusters w1 and w2, so that they are re-classified as belonging to w4 and w5 as expected.
- Some structures inside other broader regions are correctly isolated. This occurs in the rectangular area corresponding to a park, where the internal structures with high entropy are clearly visible .
- Additionally, the homogenization effect can be considered as a mechanism for speckle noise reduction during the classification phase, avoiding the early filtering for classification tasks.
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|# of iterations||1||2||3||4||5||6||7|
|R̄||# of iterations|
|Average CPU times (minutes/iteration)||3.01||2.02||2.31||2.28||2.35|
|Average CPU times (minutes/iteration)||1.10||0.76||0.77||0.75||0.81|
Pajares, G.; López-Martínez, C.; Sánchez-Lladó, F.J.; Molina, Í. Improving Wishart Classification of Polarimetric SAR Data Using the Hopfield Neural Network Optimization Approach. Remote Sens. 2012, 4, 3571-3595. https://doi.org/10.3390/rs4113571
Pajares G, López-Martínez C, Sánchez-Lladó FJ, Molina Í. Improving Wishart Classification of Polarimetric SAR Data Using the Hopfield Neural Network Optimization Approach. Remote Sensing. 2012; 4(11):3571-3595. https://doi.org/10.3390/rs4113571Chicago/Turabian Style
Pajares, Gonzalo, Carlos López-Martínez, F. Javier Sánchez-Lladó, and Íñigo Molina. 2012. "Improving Wishart Classification of Polarimetric SAR Data Using the Hopfield Neural Network Optimization Approach" Remote Sensing 4, no. 11: 3571-3595. https://doi.org/10.3390/rs4113571