# Improving Wishart Classification of Polarimetric SAR Data Using the Hopfield Neural Network Optimization Approach

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## Abstract

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## 1. Introduction

## 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 ${k}_{p}={2}^{-1/2}{\left[{S}_{hh}+{S}_{vv}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{S}_{hh}-{S}_{vv}\hspace{0.17em}\hspace{0.17em}2{S}_{hv}\right]}^{T}$. Hence, ${k}_{i}{k}_{i}^{T}$ is the Hermitian product of the target vector of the one-look i
^{th}pixel. PolSAR data need to be multilook processed for speckle reduction by averaging n neighboring pixels. The coherency matrix is then obtained as,$$\langle T\rangle =\frac{1}{n}\sum _{i=1}^{n}{k}_{i}{k}_{i}^{*t}$$ - 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 w
_{j}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 w_{j}) according to the number of pixels n_{j}belonging to the class w_{j}as follows,$${V}_{j}^{t}=\frac{1}{{n}_{j}}\sum _{i=1}^{{n}_{j}}{\langle T\rangle}_{i}$$ - Compute the distance measure for each pixel i characterized by its coherence matrix 〈T〉
_{i}to the cluster center as follows,$$d\left({\langle T\rangle}_{i},{V}_{j}^{t}\right)=\text{ln}\left|{V}_{j}^{t}\right|+Tr\left({\left({V}_{j}^{t}\right)}^{-1}{\langle T\rangle}_{i}\right)$$ - Assign the pixel to the class with the minimum distance,$$i\in {w}_{j}\hspace{0.17em}\hspace{0.17em}\mathit{iff}\hspace{0.17em}\hspace{0.17em}d\left({\langle T\rangle}_{i},{V}_{j}^{t}\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}<d\left({\langle T\rangle}_{i},{V}_{m}^{t}\right)\hspace{0.17em}\hspace{0.17em}\forall {w}_{j}\ne {w}_{m}$$
- 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 t
_{max}. Nevertheless, the criteria that we adopt are the following: assuming that at each iteration t we have pixels belonging to the class w_{j}and at the next iteration t + 1 the pixels belonging to the same class are ${n}_{j}^{t+1}$, 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 (D
_{ii}): The D_{ii}is defined as the averaged distance between all the pixels within the cluster w_{i}to the cluster center V_{i}. It measures the compactness of cluster w_{i}and is given by,$${D}_{ii}=\frac{1}{{n}_{i}}\sum _{k=1}^{{n}_{i}}d\left({\langle T\rangle}_{k},{V}_{i}\right)=ln\left(\left|{V}_{i}\right|\right)+Tr\left({V}_{i}^{-1}{V}_{i}\right)$$_{ii}indicates the dispersion of the pixels into the cluster. - The distance between two clusters (D
_{ij}) is defined as,$${D}_{ij}=\frac{1}{2}\left\{ln\left(\left|{V}_{i}\right|\right)+ln\left(\left|{V}_{j}\right|\right)+Tr\left({V}_{i}^{-1}{V}_{j}+{V}_{j}^{-1}{V}_{i}\right)\right\}$$_{ij}values indicate the high separation of these two clusters. - The cluster separability (R
_{ij}) involves two clusters and is defined as,$${R}_{ij}=\left({D}_{ii}+{D}_{jj}\right)/{D}_{ij}$$_{ij}value indicates that these two clusters are well separated; R_{ij}is the Davies-Bouldin index [16] in classical clustering approaches, here adapted to PolSAR data classification. This quantity measures the quality of the partition, i.e., the clustering quality.

_{ij}value. To quantitatively verify the performance of the proposed methodology, and also in order to meet the termination criterion, we compute the global averaged cluster separability with the following equation,

_{w}is the number of R

_{ij}combinations with i ≠ j, i.e., n

_{w}= 36.

#### 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?

_{j}and k to the cluster w

_{m}. This means that the more similar the distances to the same cluster center for both pixels, the more probable that the pixel i changes its label to look like that of the pixel k. Instead of using distances directly, we can map them to a defined range, as defined in Equation (9), obtaining the support that a pixel receives with respect to a cluster and the same reasoning applies. The second question can be answered by considering that if a pixel i has a label identical to those of its neighbors, it does not need to change its current label. The third question requires that these changes must be oriented to preserve maximum cluster separability. Based on the above three considerations, we will define two contextual coefficients, called regularization and separation. The contextual term refers to the consideration of the central pixel and its neighbors. The regularization coefficient controls the changes based on the supports received from the pixels belonging to the clusters. The separation coefficient controls the cluster separability, hence justifying its name. Finally, the fourth question may be answered by taking into account that no more changes are required if we achieve the maximum degree of stability. This is equivalent to achieving a minimum value in the energy function defined in the HNN process.

_{j}if the distance to the corresponding cluster center is the minimum of all distances to the remaining clusters. Based on these distances, we define the support received by the pixel i for belonging to the cluster w

_{j}as follows:

_{1}to h = m = w

_{9}. As we observe, the support ${\mu}_{i}^{j}(t)$ varies in the range (–1, +1]. Indeed, if $d\left({\langle T\rangle}_{i},{V}_{j}^{t}\right)=0$ then ${\mu}_{i}^{j}(t)=+1$ and if $d\left({\langle T\rangle}_{i},{V}_{j}^{t}\right)\to 0$ then ${\mu}_{i}^{j}(t)\to -1$. Under the above transformation, the decision rule in Equation (4) can be expressed as a function of the support at the iteration t, according to Equation (9), which expresses that the pixel i belongs to the cluster w

_{j},

_{j}because the support received by the pixel for this cluster is the greatest of all supports received for the remaining clusters.

_{j}, we build a network of q nodes, net

_{j}, where the topology of this network is established by the spatial distribution of the pixels in the M × N-pixel image to be classified. Each node i in the net

_{j}is associated with the pixel location (x,y) in the image, i.e., i ≡ (x, y) and q = M × N. The node i in the net

_{j}is initialized with the support provided by the Wishart classifier through Equation (9) at the last iteration. These initial support values are also the initial network states associated with the nodes in the networks. Through the HNN, the state of each node is reinforced or punished iteratively based on the influences exerted by their neighbors and also through its self-influence. With this, we are trying to make better decisions based on more stable state values through Equation (10). Figure 1 shows the flowchart for the overall procedure and illustrates the architecture and the set of networks built to implement the HNN paradigm.

_{j}and k to w

_{m}; from the Wishart classification we build the j nets (j = 1 to 9). Every node with its state value or support ${\mu}_{i}^{j}(t)$ on each net

_{j}is associated with a pixel i on the original image, both with identical locations (x,y). The states at each network are updated according to the number of iterations t. After the HNN convergence, a re-classification is obtained, where each pixel could change its label, this fact is indicated by the super-index in ${w}_{j}^{*}$ and ${w}_{m}^{*}$ for pixels i and k respectively.

#### 2.3.2. Dynamics of the Hopfield Neural Network

_{j}may be considered to form a matrix Q

^{j}. To illustrate the Hopfield networks in more detail, consider the special case of a Hopfield network with a symmetric matrix. The input to the i

^{th}node comes from two sources: external inputs and inputs from the other nodes. The total input ${u}_{i}^{j}$ to node i is then given by Equation (11).

^{th}node; ${Q}_{ik}^{j}$ is the weight of the connection between nodes i and k; and ${\theta}_{i}^{j}$ represents an external input bias value which is used to set the general level of excitability to the network. There are two types of Hopfield networks, namely [19,20]: (a) The analog ones in which the states of the neurons are allowed to vary continuously in an interval, such as [−1, +1] and; (b) the discrete ones in which these states are restricted to the binary values −1 and +1. The drawback of these binary networks is that they oscillate between different binary states, and settle down into one of many locally stable states. Hopfield has shown that analog networks perform better since they have the ability to smooth the surface of the energy function, which prevents the system from being stocked in local minima [14,15].

_{i}is a time constant which can be set to one for simplicity [20,22]. We have considered the sigmoid activation function to be the hyperbolic tangent function [20], $g({u}_{k}^{j})=\text{tanh}({u}_{k}^{j}/\beta )$ as this function is differentiable, smooth and monotonic. A detailed discussion about the settings of the time step dt and gain β

^{−1}can be found in [19]. As dt increases, the probability that the energy falls into a local minimum increases also. According to some experiments carried out by Joya et al.[19], where this parameter has been set to values in the range 1 to 10

^{−2}, the best performance is achieved with the minimum value (i.e., 10

^{−2}), hence we have fixed it to 10

^{−3}, which is an order of magnitude smaller than the one experimented in [19].

_{0}as follows [25]: (a) given the original data with dimensions x and y coming from the Wishart classifier, they are down-sampled by a factor of $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$32$}\right.$ in both dimensions, and then we compute the energy as in Equation (19) after the initialization of the networks net

_{j}; (b) we choose an initial β

_{0}, that permits about 80% of all transitions to be accepted, i.e., transitions that decrease the energy function, and it is changed until this percentage is achieved; (c) we compute the M transitions ΔE

_{k}and we look for a value for β for which $\frac{1}{M}{\sum}_{k=1}^{M}\text{exp\hspace{0.17em}}(-\mathrm{\Delta}E/\beta )\hspace{0.17em}$ after rejecting the higher order terms of the Taylor expansion of the exponential, β = 8〈ΔE

_{k}〉, where 〈·〉 is the mean value. In our experiments, for the set of images we have employed, we have obtained 〈ΔE

_{k}〉 = 5.83, resulting in β

_{0}= 46.64, with a similar order of magnitude as reported in [26]. Taking into account that β(t) = 0, t →+∞ and considering t = 10

^{6}, we obtain β = 3.38, i.e., β

^{−1}= 0.30. In our image classification approach, we have carried out different experiments by applying the above scheduling and also assuming a fixed gain, without apparent improvement in the final results. Hence, we set the gain to 0.30 during the complete process.

#### 2.3.3. Energy Definition

_{j}, called energy, is defined as follows,

^{j}ln 2 ≈ 0.19 when ${\mu}_{i}^{j}(t)$ is +1 or −1 and is null when ${\mu}_{i}^{j}(t)$ is zero. In our experiments, we have verified that this term does not contribute to the network stability and only the energy is increased in a small amount with respect to the other two terms in Equation (13). Hence, for simplicity, we have removed it from Equation (13).

_{j}; (b) a separation coefficient which computes the consistency between the clusters in terms of separability, where high separability values are suitable. The neighborhood ${N}_{i}^{n}$ is defined as the n-connected spatial region in the network around the node i, taking into account the mapping between the pixels in the image and the nodes in the networks. The regularization coefficient is computed at the iteration t as follows,

_{r}and w

_{s}, respectively, i.e., labeled as r and s. Because we are trying to achieve maximum separability between clusters, we compute the averaged cluster separability according to Equation (16). We compute the separabilities between the pixel i and its k neighbors in ${N}_{i}^{n}$ A low R

_{rs}value, equivalently a high ${R}_{rs}^{-1}$, expresses that the clusters w

_{r}and w

_{s}are well separated. Based on this assumption, the separation coefficient is defined as,

_{r}and w

_{s}and the averaged cluster separabilities between w

_{r}and the clusters w

_{u}in ${N}_{i}^{n}$. The Coefficients 2 and 1 in Equation (16) are introduced so that c

_{ik}(t) is in the range (−1, +1]. This mapping is made to achieve the same range as ${r}_{ik}^{j}(t)$. Note that the separation coefficient is independent of j, i.e., of the net

_{j}, as the labeling to calculate this coefficient involves the states of all networks. This implies that it is the same for all networks.

_{j}with the maximum support ${\mu}_{i}^{j}$ must be labeled as belonging to the cluster w

_{j}and vice versa. This implies that a node with high/low support must have a high/low state value simultaneously. Under this assumption, the self-consistency is mapped as an energy function as follows,

_{B}at each iteration is minimal, as expected.

#### 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 net
_{j}for each cluster w_{j}. For each net_{j}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 ${\mu}_{i}^{j}$, i.e. the support provided by the Wishart-based classifier, Equation (9); compute ${Q}_{ik}^{j}\left(t\right)$ and ${\theta}_{i}^{j}\left(t\right)$ through Equation (20); set ε = 0.01 (a constant to accelerate the convergence); t_{max}= 4 (maximum number of iterations allowed, see Section 3.2); set the constant values as follows: L_{i}= 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 net
_{j}compute ${u}_{i}^{j}\left(t\right)$ using the Runge-Kutta method and update ${\mu}_{i}^{j}\left(t\right)$, both according to Equation (12) and if$\left|{\mu}_{i}^{j}\left(t\right)-{\mu}_{i}^{j}\left(t-1\right)\right|>\epsilon $then nc = nc + 1; when all nodes i have been updated, if nc ≠ 0 and t < t_{max}then go to Step 2 (new iteration), else stop. - Outputs: ${\mu}_{i}^{j}\left(t\right)$ updated for each node; it is the degree of support for the cluster w
_{j}, 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

_{max}, fixed to eight in our experiments: this value is set based on the observation that for a number of iterations greater than eight, the averaged separability values always get worse. For each iteration, we compute the averaged separability value according to Equation (8) for the classification obtained at this iteration, and we select the number of iterations k

_{w}with the minimum averaged separability coefficient value R̄. The classification results obtained for k

_{w}are the inputs for our HNN optimization approach.

_{max}, for the HNN process. We have set t

_{max}to four because after experimentation, we verified that more iterations were not suitable due to an over-smoothing of the textured regions.

#### 3.2. Results

_{max}= 7.

_{w}= 2.

_{max}and for the specified neighborhoods.

_{i}, w

_{k}) = −1 if w

_{i}= w

_{k}and δ(w

_{i}, w

_{k}) = 0 if w

_{i}≠ w

_{k}, w

_{i}and w

_{k}are the corresponding clusters. The goal is to maximize iteratively the a posteriori probability expressed as p

_{i}(t) = exp{−U

_{i}/β(t)} or equivalently, to minimize the total energy function U(t) = ∑

_{i}U

_{i}/β(t), where β(t) is the scheduling defined as in our HNN approach, as it is inspired in [32]. For comparison purposes, we always apply the same number of iterations in the HNN and ICM case. For the DSA, we prefer to consider two iterations, because this is the number of iterations where it achieves its best performance. The MAJ is not an iterative approach.

_{i}for ${\mathit{N}}_{i}^{8}$ ranges in [0,1] for maximum and minimum homogeneities. Finally, we compute the homogeneity for the whole image as the average of the individual homogeneities for the image with size q = M × N

#### 3.3. Discussion

- The low entropy vegetation consisting of grass and bushes belonging to the cluster w
_{2}has been clearly homogenized, this is because there are many pixels belonging to w_{4}in these areas re-classified as belonging to w_{2}. - Also, in accordance with [13], 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 w
_{1}and w_{2}, so that they are re-classified as belonging to w_{4}and w_{5}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 [33].
- 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.

## 4. Conclusions

## Acknowledgments

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**Figure 1.**Flowchart of the overall procedure and architecture for the Hopfield Neural Network (HNN) paradigm.

**Figure 2.**Averaged separability values for the first two iterations of the Wishart classifier and the best performance achieved with HNN and DSA during the four iterations according to Table.

**Figure 4.**(

**a**) Classification by the Wishart approach after two iterations. (

**b**) Classification by the proposed HNN optimization approach after the first iteration with a window size of 3 × 3. (

**c**) Classification results for DSA after the second iteration. (

**d**) Classification results by the iterated conditional mode (ICM) after the first iteration. (

**e**) Classification results by Majority.

**Figure 5.**Expanded area corresponding to mountains extracted from Figure 4. (

**a**) by the Wishart approach classification after two iterations, (

**b**) by HNN optimization approach after the first iteration, (

**c**) by ICM after the first iteration, (

**d**) Majority.

**Figure 6.**Expanded area corresponding to a city extracted from Figure 4. (

**a**) Wishart with two iterations. (

**b)**HNN with one iteration. (

**c**) ICM with one iteration. (

**d**) Majority.

**Figure 7.**(

**a**) Classification by Wishart after three iterations. (

**b**) Classification by the proposed HNN optimization approach after the first iteration with a window size of 3×3. (

**c**) Classification results for DSA after two iterations. (

**d**) Classification results by the ICM after the first iteration. (

**e**) Classification results by Majority.

Wishart | |||||||
---|---|---|---|---|---|---|---|

# of iterations | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

R̄ | 91.8 | 78.3 | 116.1 | 145.2 | 112.8 | 93.0 | 106.6 |

**Table 2.**Averaged separability values R̄ for the HNN and Deterministic Simulated Annealing (DSA) against the number of iterations.

R̄ | # of iterations | ||||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||

${\mathit{N}}_{i}^{8}$ | HNN | 65.5 | 69.9 | 71.2 | 74.3 | 74.3 | 74.6 | 74.8 | 74.9 |

DSA | 76.6 | 67.8 | 69.1 | 70.0 | 70.0 | 70.3 | 70.4 | 70.5 | |

${\mathit{N}}_{i}^{24}$ | HNN | 135.8 | 183.7 | 189.2 | 192.1 | 192.9 | 193.7 | 194.6 | 196.2 |

DSA | 177.3 | 158.9 | 182.2 | 188.1 | 190.5 | 197.1 | 199.4 | 202.6 | |

${\mathit{N}}_{i}^{48}$ | HNN | 301.2 | 444.1 | 445.3 | 449.1 | 449.9 | 452.2 | 455.3 | 458.1 |

DSA | 354.3 | 321.4 | 432.6 | 488.4 | 489.9 | 493.2 | 495.1 | 497.3 |

**Table 3.**Averaged Cluster Separability values (R̄), Homogeneities (H̄) and central processing unit (CPU) times for Wishart, HNN, DSA, ICM and Majority (MAJ) for San Francisco Bay (SFB) image.

Wishart | HNN | DSA | ICM | MAJ | |
---|---|---|---|---|---|

R̄ | 78.3 | 67.7 | 67.8 | 71.5 | 93.1 |

H̄ | 0.354 | 0.286 | 0.291 | 0.192 | 0.184 |

Average CPU times (minutes/iteration) | 3.01 | 2.02 | 2.31 | 2.28 | 2.35 |

**Table 4.**Averaged Cluster Separability values (R̄), Homogeneities (H̄) and CPU times for Wishart, HNN, DSA, ICM and MAJ for Baltic Sea Lakes (BSL) image.

Wishart | HNN | DSA | ICM | MAJ | |
---|---|---|---|---|---|

R̄ | 24.3 | 22.4 | 22.4 | 24.2 | 25.9 |

H̄ | 0.121 | 0.088 | 0.089 | 0.081 | 0.083 |

Average CPU times (minutes/iteration) | 1.10 | 0.76 | 0.77 | 0.75 | 0.81 |

## Share and Cite

**MDPI and ACS Style**

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

**AMA Style**

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/rs4113571

**Chicago/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