You ask what is the use of classification, arrangement, systematization. I answer you; order and simplification are the first steps toward the mastery of a subject the actual enemy is the unknown

Allan Sandage
Astronomer

Morphology and Evolution in Galaxies and Clusters of Galaxies: Minkowski Functionals as Shape Descriptors


Morphology of heavenly objects such as galaxies, clusters, super-clusters and voids of galaxies can provide important clues for understanding the past and present physical processes which play significant role in their formation. Theoretical models of the structure formation in the universe are currently based on the hierarchical clustering scenario, the most essential component of which is the idea of merging. During the evolution of the universe, small systems merge due to gravitational attraction resulting in formation of larger clumps. Merging of two or more galaxies is a violent process that significantly disturbs the shape. Hence the morphological study of galaxies and/or clusters of galaxies at present and high red-shifts may reveal important information about the rate of merging at different red-shifts and thus put a stringent constraints on the models of the structure formation.

The full morphological description of structures requires both topological and geometrical characteristics, and in general, is a formidable task. In practice, one would like to have as much information as possible expressed in terms of few meaningful and robust parameters as possible. This is, in principle, very difficult if not impossible to achieve. In 1903, H. Minkowski showed (Minkowski H., Math. Ann, 57, 447) that one need to construct three measures (in 2D; four in 3D) for full morphological description of an object. The measures are the area, perimeter, and genus, or equivalently, the Euler characteristic (EC) of the object (in 2D; the volume, (surface) area, integrated mean curvature, and integrated Gaussian curvature or genus or Euler characteristics in 3D). In 1957, H. Hadwiger (Hadwiger H., 1957, Vorlesungen uber Inhalt, Oberflache und Isoperimetrie, Springer Verlag) showed that these measures posses properties such as motion invariance, additivity and continuity (see below for detail). He showed that the measures having these properties form a complete set. In other words, there exists only three (in 2D; four in 3D) linearly independent measures and any (new) measure that follow the properties above can be expressed be expressed as the linear sum of the existing three (in 2D; four in 3D) .

In each dimension, there exists a hierarchy of the MFs. For example, in 2D there are functionals of rank 0, rank 1 and rank 2; these are known, respectively, as the scalar, vector, tensor functionals. The analyses of Minkowski (1903) and Hadwiger (1957) were limited to scalar functionals. The generalization to vector functionals was given by Hadwiger & Schnieder (Hadwiger H. & Schnieder R., 1971, Elemente der Mathematik, 26, 49), Schnieder (Schnieder R., 1972, Abh. Math. Sem., Univ. Hamburg, 37, 112 & 202), and Hadwiger & Meier (Hadwiger H. & Meier C., 1974, Math. Nachr., 56, 361). They noted that both in 2D and 3D, the scalar and vector functional are "isomorphic". It means that in these dimensions there is a one-to-one corresponds between the two functionals. In other words, the number of scalar functionals is equal to those of the vector functionals . The next order tensor functionals were provided later by McMullen (McMullen P., Rend. Circ. Mat. Palermo (2), 1997, 50, 259-271). For a moderate discussion on vector and tensor functionals including their properties in 2D and 3D, one can consult Beisbart (Beisbart C., 2001, PhD Thesis, Ludwig-Maximilians University, Munich, Germany).

The MFs have been used for detection and studies of possible non-Gaussianity in CMB maps and shapes of the images of simulated clusters of galaxies. The most important aspect is that the MFs provide a non-parametric description of objects (in 2D and 3D) implying that no specific or prior assumptions are need to identify their shapes . The technique based on the MFs is numerically very efficient and therefore is applicable to large data sets (e. g., SDSS image catalogue).

We have developed a set of structural measures derived from 2D MFs, and applied them to simulated elliptic galaxies and to real galaxies as imaged in the 2MASS survey. To our knowledge, this is the first time this set of measures from the MFs have been applied to both simulated and real (galaxy) data. We have also developed and used structural measures for morphological analysis of simulated and real galaxy-clusters. We believe that the use of MFs based shape descriptors could be viable along with the use of conventional structural parameters used in the galactic morphological analysis.

The set of MFs in 2D

The MFs form a complete set of measures that can be used as a statistical tool for quantifying geometry and topology of a "convex structure". By definition, a convex structure is a pattern where a line drawn in between any two points on that structure remains inside that structure. Examples of 2D convex structures include circle, ellipse, square, parallelogram, and triangle. In 3D, the configurations of convex bodies are spheroid, ellipsoid, cube, tetrahedron, and parallelepiped. The convex bodies form a very special class of objects and almost all objects in our everyday life are non-convex (concave) in nature.



A schematic diagram showing a convex (left panel) and a non-convex (right panel) structure in 2D


To see the full list of MFs in 2D click here . Note that the scalar and vector functionals are well understood measures but apart from first three tensor functionals shown in the table , the physical meaning of the remaining four is still unclear. Therefore so far we deal with only three tensors in 2D instead of all seven. For full treatment of the MFs click here .

Properties of MFs

In D-dimension, there exists a hierarchy of the MFs. For example, in 2D there are functionals of rank 0, rank 1 and rank 2; these are known as, respectively, scalar, vector, tensor functionals. The scalar functionals are coordinate independent but the vectors and tensors depend on coordinate.
In D-dimension, the scalar functionals obey a set of covariance properties such as motion invariance (includes translation and rotation), additivity, and continuity (see below).
In D-dimension, the number of scalar and vector functionals is D+1 where as the number of tensor functionals is 3D+1.
Tensor functionals are symmetric.
The set of the MFs in D-dimension completely characterize the geometry (i.e., shape) and topology of a "convex structure".
The MFs are additive measures. This property implys that the functionals can be used to measure the local as well as global properties of a system. It allows one to calculate the functionals for a larger system by summing up the local contributions.
The parameters constructed from the MFs are generic and robust.

To have a better feeling of the properties of scalar functionals lets have a look at the figure shown below. In this figure we show motion invariance, additivity, and continuity (from top to bottom respectively) where M represents any measure such as area, perimeter or genus, K denotes the convex structure (an isophotal contour of a galaxy image for example), g is the movement operator and depends on position and rotational angle, and U denotes the union of two bodies.



It is crucial to note that images of galaxies or clusters of galaxies are non-convex in nature. Then how will it be possible to apply the MFs measures for these objects ? The answer to this question lies in the "additivity" property of the scalar MFs . If we look at the definition of additivity in the figure above (middle part of the figure), we notice that the object on the left side of the expression is constructed from three convex bodies on the right side. The object on the left is non-convex but we can estimate the area, perimeter, and genus of the object without any difficulty. If we take this as the working definition, then we can, in fact, measure the geometric shape and topology of any object.

Parameters derived from MFs

We treat an image as a set of contour lines corresponding to a set of surface brightness levels. A contour is constructed by linear interpolation at a given level. For every contour, the first step of the functional analysis provides the following parameters.

Three scalars: A_S, P_S, and Euler Characteristics (EC) . Note that EC is also represented by the symbol, \chi. For galaxy morphology EC is always 1 and so it is not an interesting parameter. It becomes an important parameter for galaxy-cluster analysis since these systems are multi-component structures and EC is equal to or greater than 1 .

Three vectors or centroids: A_i, P_i, and \chi_i.

Three symmetric tensors A_{ij}, P_{ij}, and \chi_{ij} (here i, j = 1, 2) giving a total of nine components of (three from each tensor). For more click here .

All the parameters above are in Cartesian coordinate .

In the next step, the eigenvalues (\lambda_1 and \lambda_2; \lambda_1 > \lambda_2) of the tensors are found taking centroids as the origins of corresponding tensors.
After calculating the eigenvalues, we proceed to construct the axes and orientations of the ``auxiliary ellipse'' (AE). To construct the area tensor AE, for example, we take the eigenvalues of A_{ij} and ask what possible ellipse may have exactly the same tensor. When we find that particular ellipse, we label it as the area tensor AE. The orientation of the semi-major axis of the AE with respect to the positive x-axis is taken as its orientation.
The AEs corresponding to the perimeter and EC tensors are constructed in a similar manner. To discern morphologically different objects, therefore, we use ellipticities (\epsilon_i) and orientations (\Phi_i) of the AEs rather than the eigenvalues of the tensors. We define ellipticity of the AEs as \epsilon_i = 1 - b_i/a_i, where i corresponds to one of the three tensors, and a, b are the semi-major and semi-minor axes of the AEs.

The use of AEs effectively relates a contour to an ellipse: the similarity of three AEs is a strong evidence that the shape of the contour is elliptical. For example, in case of a perfect elliptic contour, the AEs will be the same. In particular, the areas of all three AEs will be equal to the area of the contour, i. e., A_A = A_P = A_{\chi} = A_S, and the perimeters of the ellipses will be equal to the perimeter of the contour, i. e., P_A = P_P = P_{\chi} = P_S. In addition, the orientations of all three ellipses will coincide with the orientation of the contour. Therefore, if plotted, all three AEs will be on top of each other, overlapping with the contour. For that contour, all three vector centroids will also coincide with each other and with the center of the contour. Note that the latter alone does not guarantee that the contour itself is elliptical in nature since for any centrally symmetric contour the centroids would coincide. However, for a non-elliptical contour all three AEs will be different in sizes and orientations.

Advantages of having the hierarchy of MFs

Note that being independent of coordinate the scalar functionals do not have the ability to predict the directional dependence of any structure such as its position with respect to any background geometry (i. e., an origin) and its orientation. Let us have a look at the following figure:



At the top panels we see two elliptic structures. These structures are equal in area, perimeter, and EC but different in orientation. They even have the same centroids. If we analyze these objects using only scalar and vector functionals we will conclude that the objects have identical shape. We will not be able to tell whether the objects are oriented in any preferred direction or not. This information can easily be extracted by the higher order, tensor functionals. The power of the MFs lies in the fact that in situation where lower order functionals fail to provide information crucial for any structural analysis we can search for this information using higher order functionals. We encounter this type of situation in galaxy morphology where we need the measurement of both shape and orientation.
Note that the structures at the top are single component objects. What can we learn about multi-component objects ? To find that let us look at this figure once again but this time we consider the bottom panels. Each objects in these panels have three components and hence multi-structured. For this illustration the objects are drawn such that the scalar functional will give identical result: equal area, equal perimeter, equal number of components (EC = 3 this time) for both objects. But The vector functionals will tell the structural difference immediately because of different spatial locations of the components. In other words, we can say that the centroids for the object at the left panel will be different than those at the right panel. Using this information we can find the degree of asymmetry for each object. Subsequent analysis with tensor functional will of course reveal more information such as degree of elongation of the overall structure. Note that the objects at the bottom panels represent a very special case where the components are similar. It happens rarely in reality, morphology of clusters of galaxies is an example, and even the lowest order functionals will also be able to probe structural differences.

Minkowski Measures vs Conventional Non-parametric Approach: A Comparison

The non-parametric approach for shape analysis, such as moments of inertia technique, has been known to the astronomical community for some time (Carter D., 1978, MNRAS, 182, 797; Carter D. & Metcalf N., 1980, MNRAS, 191, 325). It should be noted here that the morphological analyses based only on the moments of inertia would provide incomplete and sometimes misleading results.
As an example, let us assume that one has a galaxy image which has been kept in a black box and analyzed using simply the inertia tensor without having a priori knowledge of the shape of the image. The analysis based only on the moments of inertia will provide a resultant ellipticity of the object regardless of its actual shape. Using this result one can always infer an elliptical shape for the unseen object. If one raises the question of the likeliness of the elliptical shape of the object, the analysis based on the inertia tensor alone will not be able to give a satisfactory answer. One needs to invoke additional measure(s) in order to justify the result. It is at this point where the measures derived from the set of MFs appear to be effective. Subsequent analyses of the image using moments of the perimeter and genus tensor enables one to pin down the type of the galaxy and thus ensures the objectivity of the analysis.
Let us have a look at the following figures.



The leftmost panel at the top shows the isophotal contour of a simulated spiral galaxy. Let us assume that we do not know its shape and use moments technique to find it. This method will provide the shape of this object in terms of ellipticity, \epsilon. Let us take that ellipticity (\epsilon) and construct an ellipse which has exactly the same ellipticity as what we get after moment analysis. This ellipse is shown by red solid line in the middle panel at the top superimposed on the galaxy contour.
Now instead of a spiral galaxy contour if we have that of an elliptical galaxy (leftmost panel at the bottom) with the same flattening (\epsilon) as what we obtained for the spiral, then the moments technique will give an estimate which, obviously, will be \epsilon.
Problem starts right at this point.

If we take ellipticity as a reliable structural measure then with these two estimates without any prior knowledge of the contours we can not tell whether the contours are elliptical or non-elliptical in nature. To resolve the problem, apart from inertia tensor, we must invoke some other measures to pin down the real shape of the contour. Interestingly tensor MFs can be those desired measures. Measurements provided by all three tensors will be markedly different for the non-elliptic contour (rightmost panel at the top) whereas for a perfect elliptic contour they will be identical to one another (rightmost panel at the bottom).In the figure above, the solid, dashed , and dashed-dot lines in red , green , and blue represent the Minkowski measures derived, respectively, from the area, perimeter and genus tensor.

The ellipticities obtained from different AEs provide information (regarding shapes) similar to the conventional shape measure based on inertia tensor. The main difference is that the conventional method finds the eigenvalue of the inertia tensor for an annular region enclosing mass density or surface brightness. The method based on MFs, however, finds the eigenvalues of contour(s) where the region enclosed by the contour(s) is assumed to be homogeneous and to have constant surface density. Note the important fact that for a contour enclosing a region of of homogeneous density, the area tensor MF (A_{ij}) is equivalent to inertia tensor.

Functional aspect of MFs

A functional is a function of a function. It is a real number. A functional takes all the values of a function over its domain as the input and assigns an aggregate value as the output.
For 2D data analysis one can choose some specific contours of interest and thereby create "objects" encompassed by those contours. The shapes of objects thus formed therefore depend entirely on the forms of the contours which in turn depend solely on the 2D coordinates x and y. Therefore contours can be as the functions of coordinates (domain) and any other function depending on contours could well be attributed as the functionals of the contours.
Morphological analysis of any 2D (convex) object based on scalar Minkowski measures such as area, perimeter or genus quantifies the shape of the object which depend entirely on the choice of the contours. Since these scalar measures are functions of contours it will not be surprising that one can consider them as functionals.
In the following figures we show two contours C_1 and C_2. Both of these contours have same area (A), length of the perimeter (P), and genus (\chi = 1), yet the contours are different since they are functions of different set of coordinates (x,y).






Application of Minkowski Functionals: Galaxies

Detecting Spurious Contamination Induced by Foreground Stars on Galaxy Image: Efficiency of Minkowski Functionals as Image Filtering Tool


Minkowski shape measures are sensitive to detect image contamination by a foreground star. Here we briefly highlight this matter. We show four different Near-Infra Red (NIR) galaxies imaged in J, H, Ks bands where each image has a foreground star embedded in the galaxy image body. For each image the spacing between adjacent contours is the same in all three bands. Notice that the appearance of the star changes with wavelengths.



The J band images of NGC 4278, NGC 3193, NGC 3379, and NGC 5077

Unsmoothed contours of J band images of NGC 4278, NGC 3193, NGC 3379, and NGC 5077




The H band images of NGC 4278, NGC 3193, NGC 3379, and NGC 5077

Unsmoothed contours of H band images of NGC 4278, NGC 3193, NGC 3379, and NGC 5077




The Ks band images of NGC 4278, NGC 3193, NGC 3379, and NGC 5077

Unsmoothed contours of Ks band images of NGC 4278, NGC 3193, NGC 3379, and NGC 5077






Measures from Minkowski Functional for contours of NGC 4278, NGC 3193, NGC 3379, and NGC 5077 for all three bands

Ellipticity (left panel) and relative differences in the areas enclosed by different Auxiliary Ellipses (AEs; right panel) for elliptical galaxies where a foreground star is embedded in the galaxy images. The parameters are plotted as a function of area inside galaxy contours at different radii from the galaxy center, i. e., as a function of scalar area (As).

The figure shows information from all three Near-Infra Red (NIR) bands: red, green, and blue colors represent, respectively, J, H, and Ks band. On the left panel, the dotted line represents ellipticity from the scalar functional; the solid, dashed, and dashed-dotted line represent the ellipticities of the area, perimeter, and genus AEs, respectively. On the right panel, the relative differences in areas enclosed by the three AEs with respect to scalar area As are shown. The line style is similar to left panel. The horizontal dashed line corresponds to the Ks band 3sigma_n isophote estimate provided by the 2MASS survey.

A sharp kink in the ellipticity profiles is the signature of the embedded foreground star (right panel). A similar feature can also be seen from the plot showing the ratios of the sizes of AEs (left panel). From these figures shown above, we see that a foreground star embedded in an image causes galaxy isophotes to deviate from their original shapes. It usually adds small, lobe-like features to the otherwise smoothed and spherical contours. The functionals easily pick up this type of signal present on the contour and translate it to the shape parameters. This demonstration highlights the fact that MFs based measures can be used for automatic detection of features attached to the image body.

A Technique to Reduce the Effect of Noise in Galaxy Image Analysis: Contour Smoothing Approach

Background noise distorts the original profile of a galaxy and therefore its isophotal contours will always be deviated from their true shapes. In galaxy morphology finding the true shape of a galaxy profile is extremely important to understand its evolution, dynamical state and environmental effects.
The effects of noise can be reduced in several ways. One approach is to smoothing the image map with some filter, the other could be incorporating systematic effects into estimates of the parameters by introducing corrections. We introduce a simple linear technique that smoothes not the whole map but only its contours chosen some specific levels above background. The method of smoothing is based on replacing the set of contour points by a new set each point of which is placed exactly in the middle of two adjacent points in the original set. This set of new points construct a new contour what we call a contour one time smoothed from the previous one. The Minkowski parameters are then computed for this smoothed contour. The procedure is applied iteratively many times depending on the length of the contour and the level of the noise.
To implement this method finding a lower limit to begin smoothing and an upper limit up to which one should stop is very crucial as without any knowledge of the true shape one can in fact smooth a contour so much that for using this particular approach the resulting contour would ultimately appear as circular and eventually with further smoothing as a point. To get a control over this situation one can convolve simulated galaxy profiles with noises of known properties, find effective limits for smoothing and then calibrate it for real galaxy images.
We follow this approach to select smoothing limits where we clip the galaxy image at a certain level above the background noise (\sigma_n) and find the total number of contour points, N. Next we set the first smoothing limit equal to the one-tenth (N/10) of the number of contour points. Detail analysis shows that this is a reasonable choice to start the smoothing. The subsequent higher number of smoothing is just the integer multiple of the first.
Note that the required smoothing and the accuracy with which one can determine the original shape depends crucially on the steepness of the profile. Higher the gradient of the profile more accurate one would be to find the true shape and less amount smoothing will be required.



The J-H-Ks composite image of NGC 5044



Smoothed Ks band isophotal contour of NGC 5044


The J-H-Ks composite image of the 2MASS E0 galaxy NGC 5044 (figure on the right) and a representative example of the smoothed Ks isophotal contour of the galaxy shown by the heavy solid line (figure on the left). Three different points (open) circle, square and star represent three vector MFs (a measure of asymmetry) and the three closed contours shown by the dotted, dashed, and dashed-dot lines represent the shapes of the three AEs constructed, respectively, from the area, perimeter, and genus tensors. Note that except this figure for all other figures we use solid, dashed, and dashed-dot lines represent the measures derived from the area, perimeter, and genus tensors.
The galaxy contours at different rms noise levels (5, 15, 25\sigma_n, from left to right) are shown after different amount of smoothing is applied to the contour points (from top to bottom). The wiggling due the background noise is reduced by the contour smoothing as shown in the bottom panel. It also removes any apparent asymmetry present and makes the tensor ellipses to converge with one another.

Application of Minkowski Functionals: Clusters of Galaxies



A qualitative demonstration of two dimensional structures with different intrinsic properties. Clumpiness, asymmetry, centroid shift and twist are shown in panels 1 to 4, respectively. Different colors represent isophotal regions with different intensity where the intensity increases inwards.To analyze properties of these objects we can derive various parameters from the MFs.


This part is still under construction. please click here .


Acknowledgment: The J-H-Ks composite image of NGC 5044 and J, H, and Ks band images of NGC 4278, NGC 3193, NGC 3379, and NGC 5077 have been acquired from the 2MASS archive.