# Feature Selection for Edge Detection in PolSAR Images

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## Abstract

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## 1. Introduction

- (i)
- Computational Resources used in this research.
- (ii)
- The data (images) used in this research.
- (iii)
- The ground references (GR) for each image.
- (iv)
- The Gambini algorithm (GA).
- (v)
- The statistical models stipulated through their probability density functions. In this article, we add two models not used in the literature for edge detection.
- (vi)
- Information fusion. We propose a new approach named $\tau $S-ROC. We use the principal component analysis (PCA) and a threshold $\tau $ to compute each image’s weight in the fusion process.
- (vii)
- We conclude this section by discussing the Hausdorff distance (Hd) as a quality measure.

## 2. Materials and Methods

#### 2.1. Computational Resources

#### 2.2. Data

- Airborne AIRSAR sensor in L-Band data [15]:
- Airborne Uninhabited Aerial Vehicle Synthetic Aperture Radar (UAVSAR) sensor in L-Band available on [16]:
- OrbiSAR-2 sensor in P-Band image available on [17]:
- (a)
- Sub-scene 01 of Santos City, Brazil (S01) acquired with the airborne OrbiSAR-2 sensor in P-Band (Figure 3a).
- (b)
- Sub-scene 02 of Santos City, Brazil (S02) also acquired with the airborne sensor provides details about the S01 and S02 datasets (Figure 3b). Both images were acquired on 12 August 2015.

#### 2.3. Ground References and Images

#### 2.4. The Gambini Algorithm

#### 2.5. Distributions

- Gamma univariate PDF for intensities: the distribution of each intensity channel follows a gamma law with probability density function$${f}_{\mathbf{Z}}(z;\mu ,L)=\frac{{L}^{L}}{\mathrm{\Gamma}\left(L\right){\mu}^{L}}{z}^{L-1}exp\left\{-\frac{L}{\mu}z\right\}{{\displaystyle \mathrm{\U0001d7d9}}}_{{\mathbb{R}}_{+}}\left(z\right),$$$$\ell (L,\mu ;\mathit{Z})=n\left[Lln(L/\mu )-ln\mathrm{\Gamma}\left(L\right)\right]+L\sum _{k=1}^{n}ln{Z}_{k}-\frac{L}{\mu}\sum _{k=1}^{n}{Z}_{k}.$$
- The density that characterizes the ratio of any pair of intensities is$$f(r;\rho ,L)=\frac{\mathrm{\Gamma}\left(2L\right){(1-{\rho}^{2})}^{L}(1+r){r}^{L-1}}{\mathrm{\Gamma}\left(L\right)\mathrm{\Gamma}\left(L\right){\left[{(1+r)}^{2}-4{\rho}^{2}r\right]}^{\frac{2L+1}{2}}}{\mathrm{\U0001d7d9}}_{{\mathbb{R}}_{+}}\left(r\right),$$$$r=\frac{{\sum}_{k=1}^{n}\frac{|{S}_{i}{\left(k\right)|}^{2}}{{\sigma}_{i}}}{{\sum}_{k=1}^{n}\frac{|{S}_{j}{\left(k\right)|}^{2}}{{\sigma}_{j}}}=\frac{{\sum}_{k=1}^{n}{\left|{S}_{i}\left(k\right)\right|}^{2}}{\tau {\sum}_{k=1}^{n}{\left|{S}_{j}\left(k\right)\right|}^{2}}.$$If we define $z=\tau r$, with $\tau ={\sigma}_{i}/{\sigma}_{j}$ with $i\in \{\mathrm{HH},\mathrm{HV},\mathrm{VV}\}$ and $i\ne j$, where $\tau $ is the ratio intensity parameter, then (6) becomes$$f(z;\rho ,L,\tau )=\frac{{\tau}^{L}\mathrm{\Gamma}\left(2L\right){(1-{\rho}^{2})}^{L}(\tau +z){z}^{L-1}}{\mathrm{\Gamma}\left(L\right)\mathrm{\Gamma}\left(L\right){\left[{(\tau +z)}^{2}-4\tau {\rho}^{2}z\right]}^{\frac{2L+1}{2}}}.$$The reduced log-likelihood function under this model is$$\begin{array}{c}\ell (\rho ,L,\tau ;\mathit{Z})=n\left(Lln\tau +ln\mathrm{\Gamma}\left(2L\right)+Lln(1-{\rho}^{2})-2ln\mathrm{\Gamma}\left(L\right)\right)\hfill \\ \hfill +\sum _{k=1}^{n}ln(\tau +{Z}_{k})+L\sum _{k=1}^{n}ln{Z}_{k}-\frac{2L+1}{2}\sum _{k=1}^{n}ln\left[{(\tau +{Z}_{k})}^{2}-4\tau {\rho}^{2}{Z}_{k}\right].\end{array}$$
- Feng et al. [23] show that the gamma distribution models the span, i.e., the sum of the intensities:$${f}_{\mathbf{S}}(s;\mu ,L)=\frac{{L}^{L}}{\mathrm{\Gamma}\left(L\right){\mu}^{L}}{s}^{L-1}exp\left\{-\frac{L}{\mu}s\right\}{\mathrm{\U0001d7d9}}_{{\mathbb{R}}_{+}}\left(s\right),$$$$\ell (\mu ,L;\mathit{S})=n\left[Lln\frac{L}{\mu}-ln\mathrm{\Gamma}\left(L\right)\right]+L\sum _{k=1}^{n}ln{S}_{k}-\frac{L}{\mu}\sum _{k=1}^{n}{S}_{k}.$$

#### 2.6. Information Fusion Methods

#### 2.6.1. PCA Fusion Methods

- (i)
- Stack the binary images ${\widehat{\mathit{j}}}_{c}$ in column vectors to obtain the matrix ${\mathit{X}}_{\ell \times {n}_{c}}$.
- (ii)
- Calculate the covariance matrix ${\mathit{C}}_{{n}_{c}\times {n}_{c}}$ of ${\mathit{X}}_{\ell \times {n}_{c}}$.
- (iii)
- Compute the eigenvalues ($\mathrm{\Lambda}$) and corresponding eigenvectors ($\mathit{V}$) of the covariance matrix. Sort the eigenvalues and corresponding eigenvectors in decreasing order.
- (iv)
- Compute the vector $\mathit{P}=({P}_{1},{P}_{2},\dots ,{P}_{{n}_{c}})$, where ${P}_{i}={\left({\sum}_{c=1}^{{n}_{c}}{\lambda}_{c}\right)}^{-1}{\lambda}_{i}$, and ${\lambda}_{i}$ is the eigenvalue associated with the eigenvector ${V}_{i}$ of ${\mathit{C}}_{{n}_{c}\times {n}_{c}}$; notice that ${\sum}_{c=1}^{{n}_{c}}{P}_{c}=1$.

#### 2.6.2. S-ROC and $\tau $S-ROC Fusion Methods

- (i)
- Add the binary images ${\widehat{\mathit{j}}}_{c}$ to produce the frequency matrix ($\mathit{V}$).
- (ii)
- Use automatic optimal thresholds ranging from $t=1,\dots ,{n}_{c}$ on $\mathit{V}$ to generate matrices ${\mathit{M}}_{t}$.
- (iii)
- Compare each ${\mathit{M}}_{t}$ with all ${\widehat{\mathit{j}}}_{c}$, find the confusion matrix, and generate the ROC curve. The optimal threshold corresponds to the point of the ROC curve at the smallest Euclidean distance to the diagnostic line.
- (iv)
- Use PCA information to choose the channels that will be fused: only those above a threshold ($\tau $) will enter the S-ROC procedure. We used 10% of the sum of the PCA coefficients as the threshold. We named these methods by $\tau $S-ROC.
- (v)
- The fusion ${\mathit{I}}_{F}$ is the matrix ${\mathit{M}}_{t}$ which corresponds to the optimal threshold.

#### 2.7. Measures of Quality

## 3. Results

#### 3.1. Flower Simulated Image

#### 3.2. Flevoland Images

#### 3.3. San Francisco Image

#### 3.4. Sub-Scene 01 of Santos City

#### 3.5. Hd Metric Applied to Edge Evidence Estimates

#### 3.6. PCA Analysis

#### 3.7. S-ROC and $\tau $S-ROC Information Fusion

#### 3.8. Change Detection

## 4. Discussion

- (i)
- Although the estimation by maximization of the log-likelihood with the BFGS algorithm is stable, it is most sensitive to the initial point with the distribution of ratios. We used the first- and second-order moments estimates with good results.
- (ii)
- Table 2 shows that the PCA method recommends the fusion of the intensities or span channels, except in image S02; we used these channels to build the fusion methods $\tau $S-ROC.
- (iii)
- The $\tau $S-ROC method performs best concerning Hd.
- (iv)
- S02 (Figure 1) is a unique data set in which the ratio of intensities contributes to the $\tau $S-ROC fusion.
- (v)
- $\tau $S-ROC is better than S-ROC at discarding outliers.
- (vi)
- S-ROC and $\tau $S-ROC work well in images from various sensors.
- (vii)
- $\tau $S-ROC rejects false positive detection on homogeneous areas.
- (viii)
- Figure 16 and Figure 17 show the thresholding results by the likelihood using the HH, and Span channels, and edge detection by S-ROC and $\tau $S-ROC. These results motivated us to check the change detection in two images of Los Angeles taken from the same region at different times. The result is shown in Figure 18. The visual inspection of Figure 18a (LAA image) and Figure 18b (LAM image) shows the method’s ability to identify changes.
- (ix)
- We used images with different numbers of looks: images S01 and S02 are single-look, while the others have four; our methods present similar performance.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Probability Density Function | |

GR | Ground Reference |

SAR | Synthetic-Aperture Radar |

PolSAR | Polarimetric Synthetic-Aperture Radar |

AIRSAR | Airborne Synthetic-Aperture Radar |

OrbiSAR-2 | Orbital SAR |

JPL | Jet Propulsion Laboratory |

UAVSAR | Uninhabited Aerial Vehicle Synthetic-Aperture Radar |

SIM | Flower Simulated Image |

FLEV | Flevoland Image |

SF | San Francisco Image |

S01 | Sub-scene 01 of Santos City |

S02 | Sub-scene 02 of Santos City |

LAA | Los Angeles Image on April 2009 |

LAM | Los Angeles Image on May 2015 |

ROI | Region Of Interest |

MLE | Maximum-Likelihood Estimator |

BFGS | Broyden–Fletcher–Goldfarb–Shanno |

GenSA | Generalized Simulated Annealing |

S-ROC | Statistic Receiver Operating Characteristic |

$\tau $S-ROC | Statistic Receiver Operating Characteristic with a threshold |

PCA | Principal Component Analysis |

Hd | Hausdorff Distance |

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**Figure 9.**Edges evidence detected in different channels to FLEV image overlaid on polarimetric color composite.

**Figure 10.**Edges evidence detected in different channels to SF image overlaid on polarimetric color composite.

**Figure 12.**Edges evidence detected in different channels to S01 image overlaid on polarimetric color composite.

**Figure 13.**Ratios of intensities and edge estimates overlaid on polarimetric color composite. (

**a**) Channel HH/VV. (

**b**) Channel VV/HH. (

**c**) Detection in channel HH/VV. (

**d**) Detection in channel VV/HH.

Index | Channel (PDF) | FLEV | SF | S01 | S02 | SIM |
---|---|---|---|---|---|---|

1 | Gamma (HH) | 31.78 | 13.60 | 14.86 | 11.18 | 8.24 |

2 | Gamma (HV) | 14.00 | 40.04 | 33.37 | 28.00 | 7.61 |

3 | Gamma (VV) | 76.00 | 22.00 | 35.84 | 11.66 | 8.06 |

4 | Gamma for the span | 52.00 | 29.00 | 10.63 | 9.05 | 7.61 |

5 | PDF ratio (HH/HV) | 70.00 | 25.31 | 36.24 | 53.60 | 10.77 |

6 | PDF ratio (HH/VV) | 79.00 | 38.00 | 35.84 | 53.03 | 115.10 |

7 | PDF ratio (HV/VV) | 37.00 | 29.00 | 37.01 | 44.01 | 12.08 |

8 | PDF ratio (HV/HH) | 19.00 | 25.31 | 36.24 | 53.60 | 10.77 |

9 | PDF ratio (VV/HV) | 64.00 | 26.47 | 37.01 | 44.01 | 12.08 |

10 | PDF ratio (VV/HH) | 79.00 | 38.00 | 37.64 | 51.00 | 115.10 |

Channel (PDF) | FLEV | SF | S01 | S02 | SIM |
---|---|---|---|---|---|

Gamma (HH) | 29.04 | 28.34 | 24.00 | 22.71 | 53.28 |

Gamma (HV) | 18.86 | 19.95 | 18.78 | 18.21 | 19.90 |

Gamma (VV) | 17.43 | 18.20 | 15.52 | 15.77 | 12.09 |

Gamma for the span | 0.19 | 13.47 | 13.91 | 14.69 | 9.06 |

PDF ratio (HH/HV) | 2.36 | 7.75 | 9.09 | 10.02 | 0.36 |

PDF ratio (HH/VV) | 8.92 | 5.21 | 8.77 | 8.65 | 0.18 |

PDF ratio (HV/VV) | 4.55 | 2.66 | 4.54 | 5.07 | 0.00 |

PDF ratio (HV/HH) | 5.46 | 2.08 | 4.95 | 4.19 | 0.00 |

PDF ratio (VV/HV) | 5.60 | 1.46 | 4.00 | 0.44 | 0.00 |

PDF ratio (VV/HH) | 7.53 | 0.40 | 0.00 | 0.22 | 0.00 |

Fusion | S-ROC | $\mathit{\tau}$S-ROC | $\mathit{\tau}$S-ROC |
---|---|---|---|

(All Channels) | (Selected Channel) | Channels | |

FLEV | 32.00 | 23.70 | HH–HV–VV |

SF | 12.00 | 5.09 | HH–HV–VV–Span |

S01 | 35.84 | 10.63 | HH–HV–VV–Span |

S02 | 14.21 | 18.35 | HH–HV–VV–Span–HH/HV |

SIM | 13.03 | 20.09 | HH–HV–VV |

**Table 4.**Hd applied to the results of the fusion techniques to examine the sensitivity to $\tau $ variation.

Fusion Methods | ||
---|---|---|

S-ROC | $\mathit{\tau}$S-ROC | $\mathit{\tau}$S-ROC |

(All Channels) | (Selected Channel) | Channels |

7.61 ($\tau =5\%$) | HH–HV–VV–Span | |

13.03 | 20.09 ($\tau =10\%$) | HH–HV–VV |

8.24 ($\tau =20\%$) | HH |

Mean Time [s] | |||
---|---|---|---|

Edges Evidence | S-ROC | $\mathit{\tau}$S-ROC | |

Image | (All Channels) | (All Channels) | (Selected Channels) |

FLEV | 3136 | 54 | 5 |

SF | 656 | 19 | 3 |

S01 | 645 | 11 | 2 |

S02 | 975 | 12 | 3 |

SIM | 15375 | 45 | 4 |

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## Share and Cite

**MDPI and ACS Style**

De Borba, A.A.; Muhuri, A.; Marengoni, M.; Frery, A.C.
Feature Selection for Edge Detection in PolSAR Images. *Remote Sens.* **2023**, *15*, 2479.
https://doi.org/10.3390/rs15092479

**AMA Style**

De Borba AA, Muhuri A, Marengoni M, Frery AC.
Feature Selection for Edge Detection in PolSAR Images. *Remote Sensing*. 2023; 15(9):2479.
https://doi.org/10.3390/rs15092479

**Chicago/Turabian Style**

De Borba, Anderson A., Arnab Muhuri, Mauricio Marengoni, and Alejandro C. Frery.
2023. "Feature Selection for Edge Detection in PolSAR Images" *Remote Sensing* 15, no. 9: 2479.
https://doi.org/10.3390/rs15092479