Basic intensity features
Basic features characerize the intensity and its variation of an object.
intensity
Basic intensity features are simple statistics on the original gray value distribution of an object. In contrast to texture features basic intensity features cannot describe substructures and repeated subpatterns of the object.
 intensity_average

average original gray value
 intensity_standarddeviation

standard deviation of the original gray values
 intensity_weightedaverage

average original gray values weighted by the distance between the pixel and center of gravity: $$\frac{1}{N}\sum x  \overline{x} f(x)$$
 intensity_weighteddistance

average distance to the center of gravity: $$\frac{1}{N}\sum x  \overline{x}$$
 intensity_weightedinverseaverage

average original gray values weighted by the inverse distance between the pixel and center of gravity: $$\frac{1}{N}\sum\frac{f(x)}{x  \overline{x} + 1}$$
 intensitynormalized_average

average normalized gray value
 intensitynormalized_standarddeviation

standard deviation of the normalized gray values
 intensitynormalized_weightedaverage

average normalized gray values weighted by the distance between the pixel and center of gravity: $$\frac{1}{N}\sum x  \overline{x} f(x)$$
 intensitynormalized_weighteddistance

average distance to the center of gravity: $$\frac{1}{N}\sum x  \overline{x}$$
 intensitynormalized_weightedinverseaverage

average normalized gray values weighted by the inverse distance between the pixel and center of gravity: $$\frac{1}{N}\sum\frac{f(x)}{x  \overline{x} + 1}$$
 circularity_standard

rough estimation of the 'roundness' of the object: $$\frac{P}{2 \sqrt{\pi N}}$$It should be one for a perfect circle, becoming larger with decreasing similarity to a circle
 circularity_robustaverage

robust way of calculating circularity, as it does not rely on the perimeter, but on the average distance $\overline{\Delta}$: $$\frac{1 + \sqrt{\pi}\overline{\Delta} }{\sqrt{N}}  1$$ The measure tends towards 0 for a perfect circle
 circularity_robustmax

robust way of calculating circularity, as it does not rely on the perimeter, but on the maximal distance $\Delta_{max}$: $$\frac{1 + \sqrt{\pi}\Delta_{max} }{\sqrt{N}}  1$$ The measure tends towards 0 for a perfect circle
 area_

number of object pixels $N$
 expansion_max

maximal distance $\Delta_{\max}$ between center of gravity and border pixels
 expansion_min

minimal distance $\Delta_{\min}$ between center of gravity and border pixels
 expansion_ratio

ratio of minimal to maximal distance between center of gravity and border pixels (which takes 1 as amaximal value for a perfectly round object and 0 for an object with the center of gravity lying on its border (due to a strong concavity)
 perimeter_

object perimeter $P$
 convexhull_averageclumpdisplacement

difference between the center of gravity of the object and the centers of the connected components in $D$, weighted by their area
 convexhull_arearatio

ratio of the area of the object to the area of its convex hull. This feature equals 1 for convex sets, if not, it is smaller
 convexhull_numberofconnectedcomponents

number of connected components of $D$
 convexhull_areamax0

area of the biggest connected components of $D$.
 convexhull_areamax1

area of the second biggest connected components of $D$.
 convexhull_areamax2

area of the thrid biggest connected components of $D$.
 convexhull_areamean

mean area of the connected components of $D$.
 convexhull_rugosity

ratio of the perimeter of the object to the perimeter of its convex hull. For convex objects, this feature equals 1, if the object is not convex, it is smaller than 1
 convexhull_numberoflargeconnectedcomponents

number of large connected components (larger than a certain threshold) of $D$
 convexhull_areavariance

area variance of the connected components of $D$
 distancemapdynamics_numberofmaxima

number of maxima in the distance map
 distancemapdynamics_radius

Parameter:
highest := 0, 1, 2, 3radius of the first four highest dynamics of the distance map
 granulometry_area1

Parameter:
operation := close, openNormalized change of area after applying subsequent moprhological operations $\Omega_{s_i}$ with $s_i = 1$: $$\frac{\mathrm{Area}(\Omega_1 f  \Omega_0 f)}{\mathrm{Area}(f)}$$
 granulometry_area2

Parameter:
operation := close, openNormalized change of area after applying subsequent moprhological operations $\Omega_{s_i}$ with $s_i = 2$: $$\frac{\mathrm{Area}(\Omega_2 f  \Omega_1 f)}{\mathrm{Area}(f)}$$
 granulometry_area3

Parameter:
operation := close, openNormalized change of area after applying subsequent moprhological operations $\Omega_{s_i}$ with $s_i = 3$: $$\frac{\mathrm{Area}(\Omega_3 f  \Omega_2 f)}{\mathrm{Area}(f)}$$
 granulometry_area5

Parameter:
operation := close, openNormalized change of area after applying subsequent moprhological operations $\Omega_{s_i}$ with $s_i = 5$: $$\frac{\mathrm{Area}(\Omega_5 f  \Omega_3 f)}{\mathrm{Area}(f)}$$
 granulometry_area7

Parameter:
operation := close, openNormalized change of area after applying subsequent moprhological operations $\Omega_{s_i}$ with $s_i = 7$: $$\frac{\mathrm{Area}(\Omega_7 f  \Omega_5 f)}{\mathrm{Area}(f)}$$
 granulometry_volume1

Parameter:
operation := close, openNormalized change of volume after applying subsequent moprhological operations $\Omega_{s_i}$ with $s_i = 1$: $$\frac{\mathrm{Vol}(\Omega_1 f  \Omega_0 f)}{\mathrm{Vol}(f)}$$
 granulometry_volume2

Parameter:
operation := close, openNormalized change of volume after applying subsequent moprhological operations $\Omega_{s_i}$ with $s_i = 2$: $$\frac{\mathrm{Vol}(\Omega_2 f  \Omega_1 f)}{\mathrm{Vol}(f)}$$
 granulometry_volume3

Parameter:
operation := close, openNormalized change of volume after applying subsequent moprhological operations $\Omega_{s_i}$ with $s_i = 3$: $$\frac{\mathrm{Vol}(\Omega_3 f  \Omega_2 f)}{\mathrm{Vol}(f)}$$
 granulometry_volume5

Parameter:
operation := close, openNormalized change of volume after applying subsequent moprhological operations $\Omega_{s_i}$ with $s_i = 5$: $$\frac{\mathrm{Vol}(\Omega_5 f  \Omega_3 f)}{\mathrm{Vol}(f)}$$
 granulometry_volume7

Parameter:
operation := close, openNormalized change of volume after applying subsequent moprhological operations $\Omega_{s_i}$ with $s_i = 7$: $$\frac{\mathrm{Vol}(\Omega_7 f  \Omega_5 f)}{\mathrm{Vol}(f)}$$
 haralick_ASM

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Angular Second Moment (Energy) $$\sum P_{i,j}^2$$
 haralick_CON

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Contrast (Inertia) $$\sum P_{i,j}(ij)^2$$
 haralick_COR

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Correlation $$ \sum \frac{(i\mu)(j\mu)}{\sigma^2} P(i,j)$$
 haralick_COV

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Coefficient of variation $$\frac{\sigma^2}{\mu}$$
 haralick_DAV

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Difference average $$\sum_k k \sum_{ij=k} P(i,j)$$
 haralick_ENT

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Entropy $$ \sum P_{i,j} \log{P_{i,j}}$$
 haralick_IDM

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Inverse Difference Moment (Homogeneity) $$\sum \frac{P_{i,j}}{1 + (ij)^2}$$
 haralick_PRO

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Prominence $$\sum (i+j2\mu)^4 P(i,j)$$
 haralick_SAV

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Sum average: the average of partial sums, subject to the condition $i+j = k$: $$\sum_k k \sum_{i+j=k} P(i,j)$$
 haralick_SET

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Sum entropy $$\sum_k \sum_{ij=k} P(i,j) \log{\sum_{ij=k} P(i,j)}$$
 haralick_SHA

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Shade $$\sum (i+j2\mu)^3 P(i,j)$$
 haralick_SVA

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Sum variance $$\sum_k (k  SAV)^2 \sum_{ij=k} P(i,j)$$
 haralick_VAR

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Variance $$\sum (i\mu)^2 P(i,j)$$
 haralick_average

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Average $\mu$
 haralick_variance

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Variance $\sigma^2$
 haralicknormalized_ASM

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Angular Second Moment (Energy) $$\sum P_{i,j}^2$$
 haralicknormalized_CON

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Contrast (Inertia) $$\sum P_{i,j}(ij)^2$$
 haralicknormalized_COR

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Correlation $$ \sum \frac{(i\mu)(j\mu)}{\sigma^2} P(i,j)$$
 haralicknormalized_COV

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Coefficient of variation $$\frac{\sigma^2}{\mu}$$
 haralicknormalized_DAV

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Difference average $$\sum_k k \sum_{ij=k} P(i,j)$$
 haralicknormalized_ENT

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Entropy $$ \sum P_{i,j} \log{P_{i,j}}$$
 haralicknormalized_IDM

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Inverse Difference Moment (Homogeneity) $$\sum \frac{P_{i,j}}{1 + (ij)^2}$$
 haralicknormalized_PRO

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Prominence $$\sum (i+j2\mu)^4 P(i,j)$$
 haralicknormalized_SAV

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Sum average: the average of partial sums, subject to the condition $i+j = k$: $$\sum_k k \sum_{i+j=k} P(i,j)$$
 haralicknormalized_SET

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Sum entropy $$\sum_k \sum_{ij=k} P(i,j) \log{\sum_{ij=k} P(i,j)}$$
 haralicknormalized_SHA

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Shade $$\sum (i+j2\mu)^3 P(i,j)$$
 haralicknormalized_SVA

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Sum variance $$\sum_k (k  SAV)^2 \sum_{ij=k} P(i,j)$$
 haralicknormalized_VAR

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Variance $$\sum (i\mu)^2 P(i,j)$$
 haralicknormalized_average

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Average $\mu$
 haralicknormalized_variance

Parameter:
angularstep := 45
distance := 1, 2, 4, 8Variance $\sigma^2$
 algebraicinvariant_

Parameter:
I := 1, 2, 3, 4, 5, 6, 7The $I$th Humoment (algebraic invarient)
 eccentricity_

like circularity, but under the hypothesis of an ellipse
 ellipse_ratioaxis

ratio of minor axis to major axis
 ellipse_majoraxis

length of major axis of the ellipse having the same second order moments as the object
 ellipse_minoraxis

length of minor axis of the ellipse having the same second order moments as the object
 gyration_radius

radius of a circle centered in the origin with the same second order moments as the object
 gyration_ratio

ratio of the gyration radius to the maximal distance $\Delta_{max}$
 principle_gyration_ratio

ratio of the two principle_gyrations
 principle_gyration_x

distance between the principal axis and a line with the same second order moment as the object
 principle_gyration_y

distance between the axis perpendicular to the principal one and a line with the same second order moment as the object
 skewness_x

projection skewness (normalized third order moment) in the direction of the principal axis
 skewness_y

projection skewness (normalized third order moment) in the direction perpendicular to the principal axis
 statisticalgeometry_CAREA_average

Parameter:
on := foreground, backgroundAverage averageclumparea (average area of the connected components)
 statisticalgeometry_CAREA_max

Parameter:
on := foreground, backgroundMaximal averageclumparea (average area of the connected components)
 statisticalgeometry_CAREA_sample_average

Parameter:
on := foreground, backgroundSample average averageclumparea (average area of the connected components)
 statisticalgeometry_CAREA_sample_standarddeviation

Parameter:
on := foreground, backgroundSample standard deviation of averageclumparea (average area of the connected components)
 statisticalgeometry_DISP_average

Parameter:
on := foreground, backgroundAverage averageclumpdisplacement (estimation of the average distance between center of gravity of the connected component and the center of gravity of the object)
 statisticalgeometry_DISP_max

Parameter:
on := foreground, backgroundMaximal averageclumpdisplacement (estimation of the average distance between center of gravity of the connected component and the center of gravity of the object)
 statisticalgeometry_DISP_sample_average

Parameter:
on := foreground, backgroundSample average averageclumpdisplacement (estimation of the average distance between center of gravity of the connected component and the center of gravity of the object)
 statisticalgeometry_DISP_sample_standarddeviation

Parameter:
on := foreground, backgroundSample standard deviation of averageclumpdisplacement (estimation of the average distance between center of gravity of the connected component and the center of gravity of the object)
 statisticalgeometry_INTERIA_average

Parameter:
on := foreground, backgroundAverage averageclumpinteria (like average clump displacement, but weighted by the area of each connected component)
 statisticalgeometry_INTERIA_max

Parameter:
on := foreground, backgroundMaximal averageclumpinteria (like average clump displacement, but weighted by the area of each connected component)
 statisticalgeometry_INTERIA_sample_average

Parameter:
on := foreground, backgroundSample average averageclumpinteria (like average clump displacement, but weighted by the area of each connected component)
 statisticalgeometry_INTERIA_sample_standarddeviation

Parameter:
on := foreground, backgroundSample standard deviation averageclumpinteria (like average clump displacement, but weighted by the area of each connected component)
 statisticalgeometry_IRGL_average

Parameter:
on := foreground, backgroundAverage irregularity
 statisticalgeometry_IRGL_max

Parameter:
on := foreground, backgroundMaximal irregularity
 statisticalgeometry_IRGL_sample_average

Parameter:
on := foreground, backgroundSample average irregularity
 statisticalgeometry_IRGL_sample_standarddeviation

Parameter:
on := foreground, backgroundSample standard deviation of irregularity
 statisticalgeometry_NCA_average

Parameter:
on := foreground, backgroundAverage normalized number of connected components (number of connected components divided by the area of the object)
 statisticalgeometry_NCA_max

Parameter:
on := foreground, backgroundMaximal normalized number of connected components (number of connected components divided by the area of the object)
 statisticalgeometry_NCA_sample_average

Parameter:
on := foreground, backgroundSample average normalized number of connected components (number of connected components divided by the area of the object)
 statisticalgeometry_NCA_sample_standarddeviation

Parameter:
on := foreground, backgroundSample standard deviation of normalized number of connected components (number of connected components divided by the area of the object)
 statisticalgeometry_TAREA_average

Parameter:
on := foreground, backgroundAverage total clump area
 statisticalgeometry_TAREA_max

Parameter:
on := foreground, backgroundMaximal total clump area
 statisticalgeometry_TAREA_sample_average

Parameter:
on := foreground, backgroundSample average total clump area
 statisticalgeometry_TAREA_sample_standarddeviation

Parameter:
on := foreground, backgroundSample standard deviation of total clump area
intensitynormalized
Basic intensitynormalized features are simple statistics on the normalized gray value distribution of an object. In contrast to texture features basic intensity features cannot describe substructures and repeated subpatterns of the object.
Basic shape features
Basic features are characterizing the shape of an object.
circularity
Circularity features estimate of the roundness of the object
geometry
Geometry features describe basic geometric properties of an object.
Convex hull features
The convex hull of a binary image is the smallest convex set containing the image. Several object features can be derived from the convex hull and from the set difference.
convexhull
The convex hull $\mathrm{Conv}(X)$ of a set $X$ (binary image) is the smallest convex set containing $X$. Important features can be derived from the convex hull and from the set difference $D = \mathrm{Conv}(X) \setminus X$. To study the maximal areas of the connected components of $D$ is useful in order to find binuclear cell nuclei for instance. A perfect binuclear cell should have two concavities of more or less the same area. A trinuclear cell will have three concavities of similar size, if the three nuclei are positioned on the corners of a triangle. If three nuclei are positioned in a row, the number of concavities is either higher or the concavities have no symmetric size. The size distribution of concavities is certainly a powerful feature. But we cannot conclude directly the number of possibly involved cell nuclei. The reason is that a 'bended' row of several nuclei results on one side in one large concavity and on the other side in several small concavities. In order to deal with this problem, it would be good, to be able to control the degree or size of concavity we are interested in. This can be done by the use of granulometries.
Distance map features
Distance map features are object features derived from the euclidean distance transform (distance map) of the image
distancemapdynamics
Let $D_X$ be the distance map of set $X$ (binary image), i.e. $D_X(x)$ is the distance of pixel $x \in X$ to the nearest pixel $y \notin X$. If the object corresponds to an ellipse, one can expect one prominent maximum in the distance function. If it could be decomposed into two overlapping ellipses, where each of these ellipses are well recognizable, on would expect two prominent maxima. Actually, if the basic shapes are sufficiently prominent, the number of prominent maxima should be the same as the number of the basic shapes.
Granulometry features
Granulomety features characterize the size distribution of objects (or structures) in an image. For this, the image is successively simplified by morphological operators.
granulometry
Granulometry allows one to study the size distribution of objects (orstructures) in an image. For this, the image is successively simplified by operators which remove all (bright or dark) structures up to a certain size. A record is kept of how much is removed from the image with each filtering step, leading to a distribution of measurements $m_i$, which can be seen as size dependent texture or shape descriptors. Each antiextensive (extensive), increasing and absorptive operator $\psi$ with $\psi_0 f = f$ can be used for the definition of a granulometry [3]. These properties make sure that the family of operators behaves like a set of sieves with increasing size. The sieves remove larger grains of the substance to be sieved. In practice, morphological openings ($\gamma$: antiextensive) and closings ($\phi$: extensive) are mostly used for this purpose. Let $\gamma_{sB}$ be a morphological opening with the structuring element $B$ of size $s$. Then the family $\Gamma = \{ \gamma_{sB}\}$ defines a series of operators which can be successively applied to the image $f$. For each step, we record a measure (e.g. the sum of gray level values) of what has been removed from the image. Hence, we obtain a list of measures $m_i = m(s_i)$ with $s_i \in \{s_1, s_2, s_3, \ldots \}$. We have used the volume and the area, where the image volume $\mathrm{Vol}(\cdot)$ means the sum of gray levels and the area $\mathrm{Area}(\cdot)$means the number of non zero pixels.$$\begin{eqnarray} v^{\phi}_i &=& \frac{Vol(\phi_{s_{i+1}} f  \phi_{s_i} f)}{Vol(f)} \nonumber \\ a^{\phi}_i &=& \frac{Area(\phi_{s_{i+1}} f  \phi_{s_i} f)}{Area(f)} \nonumber \\ v^{\gamma}_i &=& \frac{Vol(\gamma_{s_i} f  \gamma_{s_{i+1}} f)}{Vol(f)} \nonumber \\ a^{\gamma}_i &=& \frac{Area(\gamma_{s_i} f  \gamma_{s_{i+1}} f)}{Area(f)} \end{eqnarray}$$ The features $v^{\phi}_i$ and $v^{\gamma}_i$ describe the texture in terms of size of dark and bright substructures. If one of these values is particularly high, there were many substructures of this size in the object.There may be a problem with the last structuring element: if the object is too small, the last opening will eventually remove the whole object. In this case, we would record an abnormally high value. The features $a^{\gamma}_i$ and $a^{\phi}_i$ describe the irregularity of the shape. Whereas $a^{\phi}_i$ characterize the concavities, $a^{\gamma}_i$ describes prominent spikes. These features are inherently invariant to rotation, translation, and, due to the division by $\mathrm{Vol}(f)$ and $\mathrm{Area}(f)$, to scaling.
Haralick features
The Haralick features that aim at characterizing the texture of objects by means of joint distribution of pixel value combinations.
haralick
The Haralick features computed on the original gray values. Haralick fetaures characterize the texture of objects by means of jointdistribution of pixel value combinations.In histograms, one tries to analyze the frequency of certain gray level values in animage. The inconvenience of this representation is that the spatial distribution of thesevalues is completely lost. One method to address this is to record combinationsof pixel values at a certain distance. This can be done by the cooccurrencematrix depending on distance $d$ and angle $\Phi$, which takes as element $c^{d,\phi}_{i,j}$ the number of pixel pairs $x, y$ with distance $d$ at angle $\phi$, fulfilling $f(x) = i$ and $f(y) =j$ In order to obtain rotational invariance, the mean of the cooccurrence matrices is calculated for four different angles $\phi_i = 0^0, 45^0, 90^0, 145^0$ and for different distances $d =1, 2, 4, 8$. Let $$P_{i,j}=\frac{c_{i,j}}{N}$$ be the cooccurrence probability for values $i$ and $j$ (for a given distance $d$ in all four angles) and let $\mu = \sum_j j \sum_i P(i,j)$ the mean gray level value and $\sigma^2 = \sum_j (j\mu)^2 \sum_i P(i,j)$ its variance. Then, the following features can be calculated from the averaged cooccurrence matrix.
For a detailed discussion on Haralick features, see [1]
haralicknormalized
The Haralick features computed on the normalized gray values. Haralick fetaures characterize the texture of objects by means of jointdistribution of pixel value combinations.In histograms, one tries to analyze the frequency of certain gray level values in animage. The inconvenience of this representation is that the spatial distribution of thesevalues is completely lost. One method to address this is to record combinationsof pixel values at a certain distance. This can be done by the cooccurrencematrix depending on distance $d$ and angle $\Phi$, which takes as element $c^{d,\phi}_{i,j}$ the number of pixel pairs $x, y$ with distance $d$ at angle $\phi$, fulfilling $f(x) = i$ and $f(y) =j$ In order to obtain rotational invariance, the mean of the cooccurrence matrices is calculated for four different angles $\phi_i = 0^0, 45^0, 90^0, 145^0$ and for different distances $d =1, 2, 4, 8$. Let $$P_{i,j}=\frac{c_{i,j}}{N}$$ be the cooccurrence probability for values $i$ and $j$ (for a given distance $d$ in all four angles) and let $\mu = \sum_j j \sum_i P(i,j)$ the mean gray level value and $\sigma^2 = \sum_j (j\mu)^2 \sum_i P(i,j)$ its variance. Then, the following features can be calculated from the averaged cooccurrence matrix.
For a detailed discussion on Haralick features, see [1]
Moments
Moments and derived features have been are defined to characterize distributions of pixel values (like histograms).
moments
Moments and derived features have been initially defined to characterize distributions of values (like histograms), but they can also be used as shape or texture descriptors. Discrete moments $m_{pq}$ are defined as: $$ \label{equ:moments} m_{pq} = \sum x^p y^q f(x,y)$$ with $f(x,y) \in \{0,1\}$ for shape descriptors and $f(x,y)$ the gray level value for gray level descriptors. In pattern recognition tasks where objects are to be characterized independently from their position and orientation, translation and rotation invariance are required. This can be achieved by several means. One possibility is to use moment invariants [6] as features, i.e. to define polynomial combinations of moments which are invariant with respect to an affine transformation. A second possibility is to find the principal axis of the pattern and to calculate the moments with respect to this axis and the axis perpendicular to it. This corresponds to a rotation of the object in such a way that its principal axis coincides with the xaxis. These moments are called standard moments [3]. Furthermore, we can see from equation \ref{equ:moments}, that the moments can be written as $m_{pq} = \langle x^p y^q, f(x,y) \rangle$ with $\langle\cdot, \cdot\rangle$ the scalar product. This means that $m_{pq}$ is nothing else than the projection of $f$ on the monomial $x^p y^q$. As the $x^p y^q$ are not orthogonal, the decomposition is suboptimal in terms of redundancy. Therefore, the monomials can be replaced by orthogonal polynomials, like Zernike or Legendre polynomials. A comparison of the resulting methods can be found in [7]. In [3], the authors claim that standard moments give similar performance in pattern recognition tasks as Zernike moments. We have not followed the approach of Zernike or Legendre moments.
Statistical geometric features
Statistical geometric features collect statistics of shape and texture measurements applied to the binarized image at several thresholds levels
statisticalgeometry
Statistical geometric features (or Levelset features) are shape features for different levelsets (i.e. results of different thresholds) and, as such, texture features. For each of the following features, a distribution of values is calculated according to the set of thresholds. For each of these distributions, the maximal feature value, the average feature value, the sample mean and the sample standard deviation are calculated as statistics (max_value, avg_value, sample_mean, sample_sd). Furthermore, all features are calculated on the foreground and on the background after thresholding.