Divisão de Observação da Terra e Geoinformática

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====== GeoDMA 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 $B$, and the total number of bands is defined by $NB$. 

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 |
| AMPL_BAND   | Amplitude defines the amplitude of the pixels inside the object. The amplitude means the maximum pixel value minus the minimum pixel value. | $amp = px_{max} - px_{min}$ | $\geq 0$ | $px$ |
| BRATIO_BAND  | Band Ratio describes the contribution of the given band to the region. | $bandRatio = \frac{\mu_{B_i}}{\sum_{j = 1}^{NB} \mu_{B_j}}$ | $\geq 0$ | $px$ |
| COUNT_BAND       | Count defines the total number of pixels inside the object, including pixels with dummy values.                                         | $N_{tot} = cout(X)$         | $\geq 0$ | $N$  |
| CONTSE_BAND | Contrast SE returns a measure of the intensity contrast between a pixel and its southeast neighbor over the object. Contrast is 0 for a constant object. It is also known as //Sum of Squares Variance// | $Contrast = \sum_{i=0}^{D-1}\sum_{j=0}^{D-1}p_{i,j} . |i - j|^2$         | $[0, (size(GLCM, 1)-1)^2]$ | - |
| DISSE_BAND | Dissimilarity SE measures how different the elements of the GLCM are from each other and it is high when the local region has a high contrast. | $dissimilarity = \sum_{i=0}^{D-1}\sum_{j=0}^{D-1}p_{i,j}|i-j|$ | $\geq 0$ | - |
| ENERGSE_BAND    | Energy SE returns the squared root of Angular Second Moment, computed by the sum of the squared elements in GLCM. Energy is 1 for a constant image.  | $energy = \sum_{i=0}^{D-1} \sum{j=0}^{D-1} p_{i,j}^2}$ | $[0,1]$ | - |
| ENTRSE_BAND   | Entropy SE measures the disorder in an image. When the image is not uniform, many GLCM elements have small values, resulting in large entropy. | $entropy = -\sum_{i=1}^{D - 1} \sum_{j=1}^{D - 1} p_{ij} . \log{p_{ij}}$ | $\geq 0$ | - |
| HOMOGSE_BAND | Homogeneity SE assumes higher values for smaller differences in the GLCM. Also called //Inverse Difference Moment//. Homogeneity is 1 for a diagonal GLCM. | $homog = \sum_{i=1}^{D - 1} \sum_{j=1}^{D - 1} \frac{p_{ij}}{1 + (i - j)^2}$ | $\geq 0$ | - |
| KURT_BAND     | Kurtosis returns the kurtosis value for all the valid pixels (not dummy) inside the object.                                                | $K = \frac{n(n+1)(n-1)}{(n-2)(n-3)} . \frac{\sum_{i=1}^{N}(px_i-\mu)^4}{(\sum_{i=1}^N (px_i - \mu)^2)^2$ | $\geq 1$ | -  |
| MAX_VAL_BAND  | Maximum Value computes the maximum gray level value (not dummy) inside the object.                                                | $maxVal = max(X)$ | $X$ | $px$  |
| MEAN_BAND         | Mean computes the average value for all $N$ pixels inside the object. | $\mu = \frac{\sum_{i=1}^N px_i}{N}$ | $\geq 0$ | $px$ |
| MEDIAN_BAND       | Median computes the median for all $N$ pixels inside the object. | $median = p_{\frac{(n+1)}{2}}$ | $X$ | $px$ | 
| MIN_VAL_BAND  | Minimum Value computes the minimum gray level value (not dummy) inside the object.                                                | $minVal = min(X)$ | $X$ | $px$  |
| MODE_BAND         | Mode returns the most occurring gray level value (mode) for all $N$ (not dummy) pixels inside the object. When the object is multimodal, the first value is assumed. | | $X$ | $px$ |
| N_MOD_BAND    | Num Modes returns the number of modes for the object.         | | $\geq 1$ | $px$ |
| SKEW_BAND     | Returns the Skewness value for all the valid pixels (not dummy) inside the object.                                                | $S = \frac{n\sqrt{n-1}}{n-2} . \frac{\sum_{i=1}^{N}(px_i-\mu)^3}{(\sum_{i=1}^N (px_i - \mu)^2)^\frac{3}{2}$ | - | -  |
| STDDEV_BAND | Returns the Standard Deviation of all $N$ (not dummy) pixels ($\mu$ is the mean value) inside the object. | $\sigma = \sqrt{\frac{1}{N-1}\sum_{i=1}^N \left(px_i - \mu \right)^2}$ | $\geq 0$ | $px$ |
| SUM_BAND | Returns the Sum of all $N$ (not dummy) pixels inside the object.                                           | $sum = \sum_{i = 1}^{N}px_i$ | $\geq 0$ | $px$ |
| VLDCNT_BAND | Valid Count defines the total number of pixels inside the object with not dummy values.                                           | $N = count(X)$ | $\geq 0$ | $N$ |
| VARCOEF_BAND | Returns the Coefficient of Variation of the values for all the valid pixels (not dummy) inside the object               | $c_v = \frac{100\sigma}{\mu}$ | $[0,100]$ | % |
| VAR_BAND | Returns the Variance of all $N$ (not dummy) pixels ($\mu$ is the mean value) inside the object. | $s^2 = \frac{\sum_{i=1}^N \left(px_i - \mu \right)^2}{N-1}$ | $\geq 0$ | $px$ |
===== Segmentation-based spatial features ===== 


| Name | Description | Formula | Range | Units |
| POL_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. | | $\left[-\pi, \pi]$ | $rad$ |
| POL_AREA | Returns the area of the object, measured in the unit of measure of the current Spatial Reference System.                                   | | $\geq 0$ | $[m,degrees,...]^2$ |
| PBOX_AREA | Returns the bounding box area of an object, measured in the unit of measure of the current Spatial Reference System. | | $\geq 0$ | $[m,degrees,...]^2$ | 
| PBOX_PERIM | Returns the perimeter of the bounding box of an object, measured in the unit of measure of the current Spatial Reference System. | | $\geq 0$ | $[m,degrees,...]$ | 
| PCIRCLE| 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. | $1 - \frac{\textit{area}}{\pi R^2}$ | $[0, 1)$ | $[m,degrees,...]^2$ |
| PELLIP_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_FRACDIM | Returns the fractal dimension of an object. | $2 \frac{\log {\frac{\textit{perimeter}}{4}} }{\log area}$ | $[1, 2]$ | - |
| PGYRATIUS | This feature is equals the average distance between each vertex of the polygon and it'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. | $\frac{\sum{\sqrt{(x_i - x_C)^2 + (x_i - x_C)^2}}}{N}$ | $\geq 0$ | $[m, degrees, ...]$ |
| POLRADIUS | Returns the polygon radius. It corresponds to the maximum distance between the polygon centroid its vertexes. | $R$ | $>0$ | - |
| PBOX_LEN | It is the height of the object's bounding box, measured in the unit of measure of the current Spatial Reference System. | | $\geq 0$ | $[m, degrees, ...]$ |
| P_PERIM | Returns the perimeter of the object, measured in the unit of measure of the current Spatial Reference System. | | $\geq 0$ | $[m,degrees,...]$ |
| P_PERARAT | Calculates the ratio between the perimeter and the area of an object. | $\frac{\textit{perimeter}}{\textit{area}}$ | $\geq 0$ | $px^{-1}$ |
| PRECTFIT | 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. | $\frac{\textit{area}}{\textit{minBoxArea}}$ | $\left[0, 1\right]$ | - |
| PBOX_WIDTH | Returns the width of the object's bounding box, measured in the unit of measure of the current Spatial Reference System. | | $\geq 0$ | $[m, degrees, ...]$ |
| P_COMPAC | Returns the compacity of the object. | $\frac{\textit{perimAreaRatio}}{\sqrt{\textit{area}}}}$ | $\geq 0$ | - |
| PDENSITY | This feature corresponds to the ratio between the polygon area and the polygon radius. | $\frac{\textit{area}}{\textit{radius}}$ | $\geq 0$ | - |
| PSHAPEIDX | This feature corresponds to the ratio between the polygon perimeter and the squared root of the polygon area. | $\frac{\textit{perimeter}}{4 * \sqrt{\textit{area}}}$ | $\geq 0$ | - |


=====  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 given class inside a cell. | $CA = \sum_{j=1}^n a_j$ | $\geq 0$ | $ha$ |
| c_PERCENTLAND | $\%Land$ equals the sum of the areas ($m^2$) of all patches of the corresponding patch type (class), divided by total landscape area ($m^2$). $\%Land$ is equals to the percentage the landscape comprised of the corresponding patch type (class). | $PLAND = \frac{\sum_{j=1}^n a_j}{A} \times 100$ | $\left[0, 100\right]$ | $\%$ |
| c_PD | PD stands for Patch Density, which is equals the number of patches in the landscape, divided by total landscape area ($m^2$), multiplied by 10,000 and 100 (to convert to 100 hectares). Note, PD does not include background patches or patches in the landscape border, if present. However, total landscape area ($A$) includes any internal background present. | $PD = \frac{n}{A} \times 10000 \times 100$ | $\geq 0$ | Number/100ha |
| c_MPS | MPS stands for Mean Patch Size, which is equals to 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. | $MPS = \frac{\sum_{j=1}^n a_j}{n} 10^{-4}$ | $\geq 0$| $ha$ |
| c_PSSD | PSSD stands for Patch Size Standard Deviation, which is the root mean squared error (deviation from the mean) in patch size. This is the population standard deviation, not the sample standard deviation. | $PSSD = \sqrt{\frac{\sum_{j=1}^n \left(a_j - MPS \right)^2}{n}} 10^{-4}$ | $\geq 0$| $ha$ |
| c_LSI | LSI stands for Landscape Shape Index, which is 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 two times the square root of the total landscape area ($m^2$) multiplied by pi ($\pi$). | $LSI = \frac{\sum_{j=1}^n e_j}{2\sqrt{\pi \times A}}$ | $\geq 1$| - |
| c_MSI | MSI stands for Mean Shape Index, which is equals the sum of the patch perimeter ($m$) by divided two times the square root of patch area ($m^2$) multiplied by pi ($\pi$) for each patch of the corresponding patch type, divided by the the number of patches of the same patch type (class). | $MSI = \frac{\sum_{j=1}^n \frac{p_j}{2 \times \sqrt{\pi \times a_j}}}{n}$| $\geq 1$| - |
| c_AWMSI | AWMSI stands for Area-Weighted MSI, which is 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 two times the square root of the total landscape area ($m^2$) multiplied by pi ($\pi$). This first term is multiplied by the area of the corresponding patch, divided by the sum of the areas of all patches of the same patch type (class).| $AWMSI = \sum_{j=1}^n \left[ \frac{p_j}{2 \sqrt{\pi \times a_j}} \times \frac{a_j}{\sum_{j=1}^n a_j} \right]$| $\geq 1$| - |
| c_MPFD | MPFD stands for the Mean Patch Fractal Dimension. | $MPFD = \frac{\sum_{j=1}^{n} \frac{2 \times \ln{p_j}}{\ln{a_j}}}{n}$ | | |
| c_AWMPFD| AWMPFD stands for Area-weighted Mean Patch Fractal Dimension. | $AWMPFD = \sum_{j = 1}^{n} [\frac{2 \times \ln{p_j}}{\ln{a_j}} \times \frac{a_j}{\sum_{j = 1}^{n} a_{ij}}]$ | | |
| c_ED | ED stands for Edge Density, which is equals the sum of the lengths ($m$) of all edge segments involving the corresponding patch type, divided by the total landscape area ($m^2$). | $ED = \frac{\sum_{j=1}^m e_j}{A} 10^{4}$| $\geq 0$| $m/ha$ |
| c_MPAR | MPAR stands for Mean Perimeter Area Ratio, which is equals the sum of ratios between perimeters and areas, divided by the number of patches of the same type. | $MPAR = \frac{ \sum_{j=1}^n \frac{p_j}{a_j}}{n}$| $\geq 0$| $m^{-1}$ |
| c_PSCOV | PSCOV stands for Patch Size Coefficient of Variation, which calculates the ratio between the features c_PSSD and c_MPS. | $PSCOV = \frac{PSSD}{MPS} \times 100$| $\geq 0$| - |
| c_NP | NP stands for Number of Patches, which is equals to the number of patches of a corresponding patch type (class) inside a particular landsacape. | $NP = n$ | $\geq 0$ | - |
| c_TE | TE stands for Total Edges, which is equals the total size of the edges of the all patches of the given patch type (class). | $TE = \sum_{j=0}^n e_j$ | $\geq 0$ | $m$ |
| c_IJI | IJI stands for Interspersion and Juxtaposition Index.  The observed interspersion over the maximum possible interspersion for the given number of patch types. It only exists for $n > 3$. | $IJI = \frac{-\sum_{j=1}^n (\frac{e_j}{\sum_{k=1}^n e_k}) \times \ln{(\frac{e_j}{\sum_{k=1}^n e_k})}}{\ln(m - 1)} \times 100$ | $[0, 100]$ | $\%$ |
| c_TABO | TABO stands for the Total Area of the Biggest Object that intersects the landscape. |  |  | $ha$ |
| c_BIA | BIA stands for the Biggest Intersection Area. |  |  | $ha$ |
| c_TAOBIA | TAOBIA stands for the Total Area of the Object with Biggest Intersection Area. |  |  | $ha$ |
| PR | PR stands for Patch Richness, which is equals the number of different patch types (classes) present within the landscape boundary. | $PR = m$ | $\geq0$ |  |
| PRD | PRD stands for Patch Richness Density, which is equals the number of different patch types present within the landscape boundary divided by total landscape area ($m^2$), multiplied by $10,000$ and $100$ (to convert to $100$ hectares). Note, total landscape area ($A$) includes any internal background present. | $PRD = \frac{m}{A} \times 10000 \times 100$ | $\geq0$ | $Number/100 ha$ |
| SHDI | SHDI stands for Shannon's Diversity Index, which is equals to minus the sum, across all patch types, of the proportional abundance of each patch type multiplied by that proportion. Note, $P_i$, which is the proportion of the landscape occupied by patch type (class) $i$, is based on total landscape area ($A$) excluding any internal background present. | $SHDI = -\sum_{i = 0}^{m} P_i \times \ln{P_i}$ | $\geq0$ | |
| SIDI | SHDI stands for Simpson's Diversity Index, which is equals to 1 minus the sum, across all patch types, of the proportional abundance of each patch type squared. Note, $P_i$, which is the proportion of the landscape occupied by patch type (class) $i$, is based on total landscape area ($A$) excluding any internal background present. | $SIDI = 1 - \sum_{i = 0}^{m} P_i^2$ | $0 \leq SIDI < 1$ | |
| SHEI | SHEI stands for Shannon's Evenness Index, which is equals to minus the sum, across all patch types, of the proportional abundance of each patch type multiplied by that proportion, divided by the logarithm of the number of patch types. In other words, the observed Shannon's Diversity Index (SHDI) divided by the maximum Shannon's Diversity Index for that number of patch types. Note, $P_i$, which is the proportion of the landscape occupied by patch type (class) $i$, is based on total landscape area ($A$) excluding any internal background present. | $SHEI = \frac{-\sum_{i = 0}^{m} P_i \times \ln{P_i}}{\ln(m)}$ | $0 \leq SHEI \leq 1$ | |
| SIEI | SIEI stands for Simpson's Evenness Index, which is equals to 1 minus the sum, across all patch types, of the proportional abundance of each patch type squared, divided by 1 minus 1 divided by the number of patch types. In other words, the observed Simpson's Diversity Index (SIDI) divided by the maximum Simpson's Diversity Index for that number of patch types. Note, $P_i$, which is the proportion of the landscape occupied by patch type (class) $i$, is based on total landscape area ($A$) excluding any internal background present. | $SIEI = \frac{1 - \sum_{i = 0}^{m} P_i^2}{1 - (\frac{1}{m})}$ | $0 \leq SIEI \leq 1$ | |