====== GeoDMA 0.2 Features ======

GeoDMA has metrics integrating Polygons, Cells, and Images. Through image segmentation, GeoDMA creates Polygons. 

We provide a list of 3 feature types, including: 

  * Segmentation-based **spectral** features 
  * Segmentation-based **spatial** features 
  * Landscape-based features 

===== Segmentation-based spectral features =====

All spectral metrics are calculated inside a polygon, when $X = p$, or inside a cell, when $X = C$ The spectral channel is defined by $B$. 

Some of the following equations describe features based on the Gray-Level Cooccurrence Matrix - GLCM. The term $p_{ij}$ is the normalized frequency in which two neighboring cells separated by a fixed shift occur on the image, one with gray tone $i$ and the other with gray tone $j$. The constant $D$ is the dimension of the GLCM, which has the same gray value range of the original image. 

| Name | Description | Formula | Range | Units |
| rX_amplitude_B | Defines the amplitude of the pixels inside the object. The amplitude means the maximum pixel value minus the minimum pixel value. | $px_{max} - px_{min}$ | $\geq 0$ | $px$ |
| rX_dissimilarity_B | Measures how different the elements of the GLCM are from each other and it is high when the local region has a high contrast. | $i - j$ | $\geq 0$ | - |
| rX_entropy_B | Measures the disorder in an image. When the image is not uniform, many GLCM elements have small values, resulting in large entropy. | $-\sum_{i=1}^{D - 1} \sum_{j=1}^{D - 1} p_{ij} . \log{p_{ij}}$ | $\geq 0$ | - |
| rX_homogeneity_B | Assumes higher values for smaller differences in the GLCM. | $\sum_{i=1}^{D - 1} \sum_{j=1}^{D - 1} \frac{p_{ij}}{1 + (i - j)^2}$ | $\geq 0$ | - |
| rX_mean_B | Returns the average value for all $N$ pixels inside the object. | $ \frac{\sum_{i=1}^N px_i}{N}$ | $\geq 0$ | $px$ | 
| rX_mode_B | Returns the most occurring value (mode) for all $N$ pixels inside the object. When the object is multimodal, the first value is assumed. | | $\geq 0$ | $px$ |
| rX_std_B | Returns the standard deviation of all $N$ pixels ($\mu$ is the mean value). | $\sqrt{\frac{1}{N-1}\sum_{i=1}^N \left(px_i - \mu \right)^2}$ | $\geq 0$ | $px$ |


===== Segmentation-based spatial features =====

| Name | Description | Formula | Range | Units |
| p_angle | Represents the main angle of an object. It is obtained by computing the minimum circumscribing ellipse, and the angle of the biggest radius of the ellipse suits to the object's angle. | | $\left[0, \pi\right]$ | $rad$ |
| p_area | Returns the area of the object. When measured in pixels is equal to $N$. | | $\geq 0$ | $px^2$ |
| p_box_area | Returns the bounding box area of an object, measured in pixels. | | $\geq 0$ | $px^2$ | 
| p_circle | Relates the areas of the object and the smallest circumscribing circle around the object. In the equation, $R$ is the maximum distance between the centroid and all vertices. | $1 - \frac{N}{\pi R^2}$ | $[0, 1)$ | $px^2$ |
| p_elliptic_fit | Finds the minimum circumscribing ellipse to the object and returns the ratio between the object's area and the ellipse area. | | $\left[0, 1\right]$ | - |
| p_fractal_dimension | Returns the fractal dimension of an object. | $2 \frac{\log {\frac{\textit{perimeter}}{4}} }{\log N}$ | $[1, 2]$ | - |
| p_gyration_radius | This feature equals the average distance between each pixel position in one object and the object's centroid. The more similar to a circle is the object, the more likely the centroid will be inside it, and therefore this feature will be closer to 0. | $\textit{pos}_i - \textit{pos}_C|}{N}$ | $\geq 0$ | $px$ |
| p_length | It is the height of the object's bounding box. | | $\geq 0$ | $px$ |
| p_perimeter | It is the amount of pixels in the object's border. | | $\geq 0$ | $px$ |
| p_perimeter_area_ratio | Calculates the ratio between the perimeter and the area of an object. | $\frac{\textit{perimeter}}{N}$ | $\geq 0$ | $px^{-1}$ |
| p_rectangular_fit | This feature fits a minimum rectangle outside the object and calculates the ratio between its area and the area of this rectangle. The closer to $1$ is this feature, the most similar to a rectangle. | | $\left[0, 1\right]$ | - |
| p_width | It is the width of the object's bounding box. | | $\geq 0$ | $px$ |


=====  Landscape-based features =====

When the unit is hectares, the value is divided by $10^4$. Please note that most of the following features are based on [[http://www.umass.edu/landeco/research/fragstats|Fragstats software]].

| Name | Description | Formula | Range | Units |
| c_ca | Class Area means the sum of areas of a cell. | $\sum_{j=1}^n a_j$ | $\geq 0$ | $ha$ |
| c_percentland | $\%Land$ equals the sum of the areas ($m^2$) of all patches of the corresponding patch type, divided by total landscape area ($m^2$). $\%Land$ equals the percentage the landscape comprised of the corresponding patch type. | $\frac{\sum_{j=1}^n a_j}{A} \times 100$ | $\left[0, 100\right]$ | $\%$ |
| c_pd | Patch Density equals the number of patches of the corresponding patch type divided by total landscape area. | $\frac{n}{A}$ | $\geq 0$ | Patches |
| c_mps | Mean Patch Size equals the sum of the areas ($m^2$) of all patches of the corresponding patch type, divided by the number of patches of the same type. | $\frac{\sum_{j=1}^n a_j}{n} 10^{-4}$ | $\geq 0$| $ha$ |
| c_pssd | Patch Size Std is the root mean squared error (deviation from the mean) in patch size. This is the population standard deviation, not the sample standard deviation. | $\sqrt{\frac{\sum_{j=1}^n \left(a_j - MPS \right)^2}{n}} 10^{-4}$ | $\geq 0$| $ha$ |
| c_lsi | Landscape Shape Index equals the sum of the landscape boundary and all edge segments ($m$) within the boundary. This sum involves the corresponding patch type (including borders), divided by the square root of the total landscape area ($m^2$). | $\frac{\sum_{j=1}^n e_j}{2\sqrt{\pi \times A}}$| $\geq 1$| - |
| c_msi | Mean Shape Index equals the sum of the patch perimeter ($m$) divided by the square root of patch area ($m^2$) for each patch of the corresponding patch type. | $\frac{\sum_{j=1}^n \frac{j}{2\sqrt{\pi \times a_j}}}{n}$| $\geq 1$| - |
| c_awmsi | Area-Weighted MSI equals the sum, across all patches of the corresponding patch type, of each patch perimeter ($m$) divided by the square root of patch area ($m^2$). | $\sum_{j=1}^n \left[ \frac{j}{2 \sqrt{\pi \times a_j}} \times \frac{a_j}{\sum_{j=1}^n a_j} \right]$| $\geq 1$| - |
| c_mpfd | MPFD stands for the Mean Patch Fractal Dimension. | | | |
| c_awmpfd| AWMPFD stands for Area-weighted Mean Patch Fractal Dimension. | | | |
| c_ed | Edge Density equals the sum of the lengths ($m$) of all edge segments involving the corresponding patch type, divided by the total landscape area ($m^2$). | $\frac{\sum_{j=1}^m e_j}{A} 10^{-4}$| $\geq 0$| $m/ha$ |
| c_mpar | Mean Perimeter Area Ratio equals the sum of ratios between perimeters and areas, divided by the number of patches of the same type. | $\frac{ \sum_{j=1}^n \frac{j}{a_j}}{n}$| $\geq 0$| $m^{-1}$ |
| c_pscov | Patch Size Coefficient of Variation calculates the ratio between the features _c_pssd_ and _c_mps_. | $\frac{PSSD}{MPS} \times 100$| $\geq 0$| - |
| c_np | NP equals the number of patches inside a particular landsacape. | $n$ | $\geq 0$ | - |
| c_te | TE equals the total size of the edge. | $\sum_{j=0}^n e_j$| $\geq 0$ | $ha$ |
| c_iji | IJI stands for Interspersion and Juxtaposition Index.  The observed interspersion over the maximum possible interspersion for the given number of patch types. | $\frac{-\sum_{j=1}^n e_j \ln(e_j)}{\ln(n - 1)}$ | $[0, 100]$ | $\%$ |