A Mathematical Morphology Approach to the Star/Galaxy Characterization

A Mathematical Morphology Approach to the Star/Galaxy Characterization

Alcione Jandir Candéas ¹

Ulisses de Mendonça Braga Neto^{2 *}

Edson Costa de Barros Carvalho Filho¹

¹UFPE - Universidade Federal de Pernambuco

Departamento de Informática, Caixa Postal 7851

50732-970 Recife, PE, Brasil

{ ajc | ecdbcf}@di.ufpe.br

²The Johns Hopkins University

Department of Electrical and Computer Engineering

3400 N. Charles St. Baltimore MD 21218

[e-mail]

Abstract: The automated star/galaxy discrimination problem is a broadly studied issue of astronomical imaging. The success of a discrimination task depends on the features selected to characterize both classes of interest. In this work we propose an original approach to the characterization of these astronomical objects through the use of Mathematical Morphology, based on gray-level shape-size information. Our method consists of image preprocessing, segmentation and feature extraction steps, all of which employ Mathematical Morphology tools that were implemented using the MMach toolbox for the Khoros system. We briefly present a comparison between our segmentation results, based on the watershed method, and those of the SExtractor software package. The shape-size features are extracted through the use of the gray-level morphological pattern spectrum, which yields attractive analysis attributes that promise to be very suitable for future work in neural-network-based automatic classification.

Keywords: Star/Galaxy discrimination, gray-level morphology, granulometry, image preprocessing, segmentation, feature extraction

1 Introduction

The Image Processing and Analysis (IPA) field has been characterized by the increasing adoption of specific solutions to situations in which methods that are appropriate to some applications may be entirely inadequate to others. On the other hand, Mathematical Morphology (MM) emerges as a general theory that has provided a unified approach to deal with problems in Medicine, Geology, Geography, Remote Sensing, and many other fields. In spite of the diversity of purposes, all of these problems have a common feature: the need to extract shape information from images.

We employ Mathematical Morphology in the problem of star/galaxy discrimination, which is currently a broadly examined topic in the field of Astronomical Imaging. The shape information extracted with the use of MM tools proves to be suitable for the characterization of those astronomical, while promising to be effective for neural-network-based automatic classification.

This work also represents an attempt to call the attention of the general Image Processing and Analysis community back to Astronomical Imaging. The IPA field has grown strongly in the sixties due to the American space program, carried out by NASA. Since then, the technology developed has been applied in many other fields, leading to the development of new digital techniques for filtering, pattern recognition, formal grammar, neural networks, artificial intelligence, and others. By presenting an original Mathematical Morphology approach to the star/galaxy discrimination problem, we expect to renew the interest of the IPA community in astronomical applications, as well as to motivate and support multidisciplinary undertakings. We also point out that Astronomical Imaging is a new promising application for MM.

This paper is structured as follows: In Section 2 we outline some topics on astronomy, astronomical images and the problem of star/galaxy discrimination, while in Section 3 we present our method and the gray-level MM tools required by it. In Section 4, we show the results obtained and a comparison with the segmentation results of SExtractor (Source Extractor), a software that builds a catalog of objects from astronomical images. Finally, Section 5 gives some concluding remarks and directions for future work.

2 The Problem Context

2.1 Astronomy

The galactic structure and dynamics, the environmental effects on galactic arrangement and evolution, the diversity of galaxy morphologic types, and the large-scale distribution of dark and bright matter in the whole universe are some relevant current problems in Astrophysics, which is one of the various branches of Astronomy. These cosmological problems are usually studied in a statistical fashion [10] to treat deep surveys with large amount of stars and galaxies data collected over wide sky areas.

Most information we know with respect to the universe is due to the luminosity derived from space. With the advent of new ground-based spectroscopic measurement tools, in addition to the launching of space-based observatories such as the Hubble Space Telescope, deeper and more accurate galaxy and stellar catalogs are needed. Some of the most complete catalogs were visually compiled from photographic surveys. The current tendency is to use high-precision acquisition tools and fast scanning machines to compile galaxy catalogs over specific areas in the sky. The corresponding increase in image data due to the higher speed and accuracy of these instruments requires both new automated image detection and classification techniques.

2.2 Astronomical Images

The development of astronomical image acquisition methods and tools such as CCD (Charge Coupled Device) cameras and photographic plate scanners, besides the increase of available data in digital form, have stressed the relevance of computer interaction with a view to serving Astronomical purposes, which can customize and improve the results of observational programs. The digitized samples of the photographic plates have great utility in astronomical research.

A remarkable contribution to the star-galaxy surveys is the set of digitized images derived from the 936 pairs of photographic plates of the first epoch National Geographic-Palomar Observatory Sky Survey (POSS). These images were generated using the Automated Plate Scanner (APS) of the APS Project and Catalog at the University of Minnesota. A more detailed discussion about this project can be found in [8, 12].

The Digital Sky Survey (DSS) is a collection of image samples covering the entire sky and widely used for work in analysis and classification by the international astronomical community. It consists of a set of CD-ROMs produced by the Catalogs and Surveys Branch of the Space Telescope Science Institute (STScI). In fact, DSS celestial Northern hemisphere samples are based on POSS photographic data, while the Southern hemisphere ones come from SERC Southern Sky Survey, produced at Anglo-Australian Observatory, in Australia. All samples are available as FITS (Flexible Image Transport System) images, the standard data interchange and archival format employed worldwide by researchers in Astronomy.

2.3 Star/Galaxy Discrimination

Cataloguing astronomical objects automatically calls for a solution in IPA. However, due to the increase in the accuracy of the observational and registering tools in the last decades, not all of the currently detectable objects have been catalogued accordingly. This can be explained by the time-consuming, tiresome task of visual identification required by the cataloguing process. A single photographic plate can ordinarily produce thousands of detected objects. Therefore, it is necessary to develop fast image processing, recognition and classification by automated means.

Figure 1: Large scale galaxy-survey image sample

One of the most important topics in automatic cataloguing is the star/galaxy discrimination problem, which is currently greatly employed in the automatic identification of astronomical objects. This issue has been studied by many research groups and professionals, as can be seen in [11]. An interesting contribution toward this goal is the APS Project and Catalog, that has generated an Internet-accessible astronomical catalog of millions of stars and galaxies by using pattern recognition techniques for performing star-galaxy separation in large digital sky surveys. The implementation is based on photometric calibration and normalization of critical image parameter sets, using neural network image classifiers to automatically perform the expected discrimination.

However, all catalogues have limitations due to the adopted criteria. It is necessary to verify the catalog quality for completeness and correctness in order to produce reliable analysis results. This depends on many aspects, mainly the accuracy of the acquisition instrument correspondent to the image chosen, as well as the selected features and the reliability of the classifier employed. Once the image quality is guaranteed, adequate analysis tools are needed to extract relevant information from the data. The scope of the problem of how many and which parameters are necessary to an accurate characterization of a singular astronomical object is a very important topic to be considered. Indeed, this issue is currently being discussed by the astronomic research community [15].

3 The Proposed Method

Most works found in the literature are based on features extracted from photometric attributes. This means that these techniques employ pixel intensities, which are correlated to the object apparent brightness. In such works, a variety of methods have been employed in order to try to do achieve effective parameterization of the images combining the photometric attributes with some measure of object size such as the diameter, and the calculation of simple moments and gradients.

In this work we propose instead a morphological approach to the characterization of astronomical objects, which is closer to the human way of performing the visual analysis of astronomical images. This is accomplished by using Mathematical Morphology tools to extract gray-level shape information that can be useful for discriminating stars and galaxies in large scale galaxy-survey images, like those in DSS and POSS. These images typically contain thousands of astronomical objects of varying size and brightness, mostly stars and galaxies (Fig. 1 - this image sample is inverted for contrast enhancement purposes).

Both visual and automatic inspection of astronomic surveys have proved to be difficult tasks. A human viewer can perceive in the image that stars are well-defined objects, whereas galaxies present themselves as fuzzy blobs. Further, "large'' stars present a cross effect due to optical properties of the acquisition instrument. We propose to use these shape features in order to discriminate between the two classes of objects.

3.1 Gray-Level Mathematical Morphology

This section aims at presenting a brief review on the main gray-level MM notions and tools we utilize in this work. The interested reader can find a more thorough description in [5, 13, 16].

3.1.1 Preliminaries

Many common phenomena can be modeled as linear functions. In addition, linear methods often transform the problem to an easier one in the frequency domain. However, this represents a mathematical constraint that pressupposes the kind of processing one can perform, which can lead to undesirable side effects. Even proceeding with its own constraints, nonlinearity can be more appropriate to a particular application. For example, if shape information is key, one applies shape-based constraints.

In IPA the interest is in complex tasks such as discovering image size distributions or classifying objects. The primary advantage of nonlinear methods is their ability to selectively preserve structural information while accomplishing some task on the image.

Mathematical Morphology refers to a branch of nonlinear image processing and analysis which focuses on the geometric structure in an image. Its theoretical foundations lie both in Set Theory and the mathematical theory of order. Mathematical Morphology was born in the sixties, with the work of Georges Matheron and Jean Serra, in France. Since then, morphological image processing has grown in both theory and application worldwide.

The original theory was restricted to binary images, but as time passed, it became clear that gray-level images could be treated in the same algebraic framework as in the binary case. Grayscale Morphology was developed extending the principles of MM from 2 to 3 dimensions, based on the intuitive notion of continuous tone images visualized as 3-dimensional surfaces, whose topologies can be modified by probing them with geometric structures. If the image "background" tends to be" dark", the objects of the image should be visualized in the topographic model as "peaks", and as" valleys" for a brighter background. Those MM principles can be applied to sets in n-dimensional Euclidean spaces.

The gray-level MM operators deal with functions f : E ® [0,255] Î Z, where E stands for the usual planar digital grid, a matrix representing some portion of the cartesian grid which points are the image pixels positions. f is a gray-level function on the points of Euclidean 2-space, denoting the gray-level values at each point. This corresponds to the usual concept of gray-level images. The basic geometrical idea behind these operators is to probe the image with a function defined on a small subset of the image domain of definition, called a structuring element, in order to extract shape-size information through the way it "fits" the image.

3.1.2 Gray-scale Operators

In gray-scale morphology, minimum and maximum play central roles as counterparts of intersection and union in binary morphology. The basic dual gray-level MM operators are dilation and erosion, defined respectively by [1]:

(1)

(2)

where f stands for the image and g and B denote, respectively, the structuring element values and domain of definition (B Ì E). B ^t is necessary to make dilation dual to erosion. The addition and subtraction operators used in (1) and (2) are not the usual ones, but the Heijman's operators which consist of slightly altered versions necessary to comply with algebraic constraints (for more details, see [7]).

The composition of erosion and dilation yields the opening and closing operators, which are given respectively by:

(3)

(4)

Openings and closings are morphological filters [14], which have good noise-removal properties. From the 3-dimensional topographic model, we can see that performing gray-scale openings and closings by appropriately selecting the structuring element size and shape has the property of removing image details. The closing operator is the most often used in this work, since it acts on the gray-level "background" and we consider inverted images.

3.1.3 Morphological Tools

Many image transformations can be derived from the basic morphological operators. In the following, we present the tools we need in our method.

Top-Hat: A nice enhancement tool can be obtained from openings or closings, known as the top-hat transform. It has the property of enhancing "peaks" or" valleys" by applying respectively the opening or closing operator. The top-hat transform operators for both cases are defined by:

(5)

(6)

After smoothing the image with the opening (resp. closing) operation, subtracting the original image yields the expected enhanced result, as can be visualized in Fig.2.

Figure 2: Top-Hat transform: a,e) structuring element; b,f) original image; c,g) opening (resp. closing); d,h) result from subtraction

We consider the closing case, so that the resulting image should be inverted again. Due to the simplification properties of the closing operator, the top-hat defined above in (6) has the property of enhancing "valleys", that is, darker regions in the image, serving as a good background remover.

The choice of the structuring element size is very important. If it is too small, the background removal is affected by the presence of random noise. If it is too large, it cannot reproduce the small scale variations of the faint objects. Therefore, a good compromise has to be found.

Granulometry: By using a suitable family of closings (resp.openings) one gets a powerful tool for describing shape and size in image analysis, namely the granulometries [4]. By considering an image as a collection of "grains", whether or not an individual grain will pass through a sieve depends on its size and shape relative to the mesh of the sieve.

By increasing the mesh size, while keeping the basic mesh shape, more of the image will pass through in each step. The result tends to no grains remaining. Thus, the method is based on" sieving" an image followed by measure of the residue left on the sieve.

In practice, granulometries consist of a sequence of closings (resp.openings) by a set of appropriately selected increasingly larger structuring elements belonging to the same family, that is, keeping the basic shape which is said to be the generator of the granulometry. By measuring the volume under the image after each closing, we can build a size-distribution curve:

(7)

where l is the parameterization of the closing family, V(l) is the image background volume (i.e. the positive integral) at each iteration and L is the parameter associated to the largest structuring element, which is selected to be the one large enough to "wipe out" or completely close the object of interest. Note that L is itself a useful analysis attribute.

Due to the closing properties, V(l) corresponds to the background volume. To achieve an accurate value of the residue left, it is necessary to subtract the initial volume V(0) from the last volume found. The idea behind the residue varying at each iteration, has direct relation with the object shape and size.

The function above is monotone increasing in the interval [0,1], so it may be viewed as a cumulative probability distribution. Its associated probability density function is called the Pattern Spectrum of the image relative to the granulometry, which is given by the following discrete derivative:

(8)

Due to the properties of F(l) mentioned above, the function G (l) is non-negative and ideally should become null for l > LAMBDA.

The pattern spectrum is a powerful tool for characterizing shape and size in image analysis [9]. In a manner reminiscent of the Fourier spectrum, it shows the decomposition of a given object in terms of a fundamental shape "scaled" by the increasing values of the parameter l.

We can define a useful analysis attribute, the average size, which is the expected value of the pattern spectrum:

(9)

Another very useful attribute based on the pattern spectrum is the average roughness:

(10)

The average roughness is in fact the information-theoretical entropy of the pattern spectrum [9], quantifying the shape-size complexity - how the object shape differs from the shape expected. The larger it gets, the more complex with respect to the structuring elements used the image is.

Watershed: The watershed transformation is a powerful Mathematical Morphology tool for segmentation [17]. This methodology has a number of variants. It can be defined entirely in terms of geodesic MM operators [13], but the most intuitive approach is to think of an image as a topographic model, and suppose that water is oozing and rising at equal speed from every regional minimum, starting from the lowest one and then from each of the others as soon as the global water level reaches its altitude. Dams are built in the places where water from different minima would merge, separating the watersheds.

Figure 3: Watershed divide lines in a gray-level image viewed as a topographic model

The dams rising above the water surface constitute the watershed divide lines, which are composed of closed contours that involve each of the regional minima and correspond to the crest lines of the relief, achieving a good segmentation by the single line.

Last Erosion: The concept of last erosion> is related to binary images and derives from successive erosions of an image by a single structuring element. At each step, connected components of pixels can be reduced, separated or even disappear. The residues derived from each component constitute the last erosion of the image, which is often used as marker sets for further processing. Different approaches of this operator is provided by [6].

3.2 The Detailed Procedure

The main idea behind IPA is to deal with procedures with a view to improving quality or enhance relevant features. Processing an image can be approached in different steps, that can be individually considered by the interested researchers. These steps can be basically summarized as:

• Image Acquisition

• Preprocessing

• Segmentation

• Representation (Feature Extraction)

In this section we are going to outline the steps comprised in our method (see Fig. 4) for the MM approach to the star/galaxy characterization in large scale galaxy-survey images.

Figure 4: IPA steps organization in our method.

3.2.1 Image Acquisition

The FITS images used in this work were taken from DSS large-scale sky survey (see section 2.2). Taking into account the occurrence of galaxies, we have selected for this study the following images:

• Group 1701: (J2000)

  ¹ a

  ² [02 08 32.1] d [-55 39 6.5]

• ABELL 3667: (J2000) a [20 10 50.3] d [-56 40 23]

• ABELL 3698: (J2000) a [20 35 08.1] d [-25 15 34]

• ABELL 3705: (J2000) a [20 41 55.5] d [-35 16 06]

• ABELL 3775: (J2000) a [21 31 24.8] d [-43 18 38]

• O0140: (J2000) a [01 04 34.3] d [-23 50 08]

Each of these images are a 530x530 frame, except Group 1701, which is a 500x500 one³.

3.2.2 Preprocessing

Image Inversion: This is a good practice in astronomical imaging, whose purpose is to obtain object contrast enhancement for human viewers. In our case there is another reason: although most Mathematical Morphology operators come in dual sets with respect to gray-level complementation, the inversion is necessary to get the right topographic image model (the objects of interest become "valleys") required by the watershed approach in the segmentation step.

Top-hat transform: It is often necessary to enhance the image, diminishing environmental conditions effects (luminosity, atmospheric density, and so on) at the surface acquisition time - for instance optical effects like halos around large stars can affect the segmentation step, corrupting objects within those regions and modifying the "cross-like" shape of the stars we would like to emphasize in typical DSS images. This background removal operation is performed with a top-hat transform of the inverted image, using as structuring element a flat Euclidean disk (a circle digital approximation) of radius r=8. This value is necessary to guarantee the halo removal of the larger star in the samples utilized in our work. After the top-hat is applied, the image is normalized back to the full dynamic range and inverted once again.

Open-close filter by a flat structuring element: A composition of an opening followed by a closing is used to prevent from an oversegmentation by removing background noise that is enhanced by the top-hat step. The flat structuring element is a zero-valued function having just its domain of definition, that is, a set. In our case, the set is the 3 x 3 square.

Change of Homotopy: Residual background noise commonly leads to undesired regions without objects in the segmentated image. This recurrent difficulty in works on astronomical classification is often treated by visual inspection to reject those regions, in order to prepare the training base. For fully automated techniques, a good practice when using the watershed segmentation method consists of first regularizing the image changing its homotopy[16]. It is necessary to obtain appropriate markers, or connected sets of pixels, to identify each individual object desired to be extracted. This involves prior knowledge to the images under study. A background marker also is required. Markers that are reasonably insensitive to residual background noise in the enhanced image can be obtained by threshold. In our case, we select a high threshold value (210) to preserve faint objects. However, that led to situations where a single marker was found for overlapping objects. Applying the Last Erosion operation following the threshold allows to get satisfactory markers. Applying the straight watershed to the thresholded image yields divide lines that can be used as a background marker. Once the markers have been extracted, a morphological procedure allows to impose this set of markers to create regional minima at their locations.

3.2.3 Segmentation

Region segmentation by the watershedmethod: Initially we applied the direct watershed transformation to the preprocessed image (with no prior change of homotopy). Using the flooding interpretation of the watershed (see section 3.1), the output is a tessellation of the image where each closed contour defines a region which probably contains an object inside. The watershed is able to handle most of the object overlapping occurrences (this is called "deblending" in the astronomical literature). By using the change of homotopy from the top-hat image, it was able to handle the empty regions.

3.2.4 Feature Extraction

Computation of the pattern spectrum and associated values: Once the regions probably containing an object have been detected, the procedure enters its feature extraction phase. Firstly, each region in the image has to be properly isolated and centered in a local neighborhood image, to be computed its closing pattern spectrum and associated measures of average size and roughness. In order to minimize border effects on the computation, we pad the region with the mean value of the entire original image.

We consider three different families of 3-dimensional structuring elements, associated to the non-planar digital disks of increasingly radius according to the city-block, euclidean, and chessboard distances, which consist respectively in a pyramid, semi-sphere and cube digital approximation. The parameterization for the granulometry is thus the digital disk radius. In fact the average size attribute may be seen as the object average radius according to the respective metric [4].

As we discussed in section 3.1, the largest radius L should correspond to the closing that removes the object, when the sequence of closings should terminate. We determine L by means of a homogeneity criterion, namely, the gray-level standard deviation of each closed image based on the volume of the image at each iteration, and then select a range [0.00001,0.00006] beyond which the computation stops.

4 Experimental Results

In this section we present and comment some results we have obtained from the application of the proposed method. We compare our segmentation results with the ones from SExtractor, a software package for astronomical imaging.

4.1 SExtractor Overview

SExtractor is a software particularly oriented toward reduction of large scale galaxy-survey data [3]. One of its main purposes is the deblending of overlapping extended astronomical objects. The complete analysis of the image is done in many steps. This modularity enables one to implement his parameters, such as: frame buffer, magnitude threshold, connectivity, and others.

In short, the first step consists in a complex computation of a background map per object. In order to suppress possible local overestimations due to bright stars, a median filter is applied to the resulting background map. The whole process takes a considerable time.

Once the background map has drawn up, the program will search for connected sets of pixels above a given threshold, using a one-pass 8-connectivity algorithm. It still has an option to apply a convolution to the image, that can spend too much processing time. All the pixels above the threshold are stored in a pixel stack, which size is set by the user. When it has no more room to store incoming pixels, it "sacrifices" the largest object (for instance, a very bright star or a large galaxy in the image) extracted so far.

SExtractor deblending method is based on multithresholding by passing the connected set of pixels through a sort of filter to try to split it into eventual overlapping components. This technique gives a sort of 2-dimensional "model" of the light distribution within the objects, where it is probed by the algorithm to meet junction thresholds.

This method is unable to deblend components that are so close, as interacting pairs of optically faint galaxies. Other typical problematic cases include extended Sc galaxies⁴. A" cleaning" procedure is still done to filter spurious detections made often in the neighborhood of objects with shallow profiles (for example, elliptical galaxies), examining the objects detected before sent to an output catalog.

4.2 Implementation of the MM Approach

We implemented our method on the Khoros image processing and visualization system, version 1.0, running in a Sun IPX workstation under Unix and X11R5. Khoros is a very popular IPA open platform developed at New Mexico University and freely available through anonymous ftp. Khoros turned out to be a very convenient tool for algorithm development and fast prototyping due to its visual programming environment - Cantata, which allows basic operators composition in a manner MM operations are done. We used the MMach operators, a Khoros toolbox that implements Mathematical Morphology techniques [2], also available through anonymous ftp at São Paulo University.

We would like to point out that there is a bug in the Khoros 1.0 conversion routine from FITS to VIFF, the format used by Khoros, so that we could not input the original images directly. The solution adopted was to use the XV 3.10 package to convert the images to formats readable by Khoros, such as PBM and (uncompressed) TIFF.

(a)
(b)
(c)
(d)

Figure 5: Sequence of operations for background smoothing and object region segmentation for the ABELL 3698 survey: (a) original image; (b) inverted image; (c) result of closing operation by a flat euclidean disk (radius = 8); (d) normalization after subtracting the original inverted image;

(e)
(f)
(g)
(h)

Figure 6: (e) inverted result of the top-hat transform for halos removal; (f) background moise smoothing by an open-close filtering; (g) labelled regions by the watershed implementation; (h) watershed divide lines superimposed to the original inverted image (ABELL 3698)

(a)
(b)
(c)
(d)

Figure 7: Sequence of operations for background smoothing and object region segmentation for the ABELL 3775 survey: (a) original image; (b) inverted image; (c) result of closing operation by a flat euclidean disk (radius = 8); (d) normalization after subtracting the original inverted image;

(e)
(f)
(g)
(h)

Figure 8: (e) inverted result of the top-hat transform for halos removal; (f) background moise smoothing by an open-close filtering; (g) labelled regions by the watershed implementation; (h) watershed divide lines superimposed to the original inverted image (ABELL 3775) We would like to point out that there is a bug in the Khoros 1.0 conversion routine from FITS to VIFF, the format used by Khoros, so that we could not input the original images directly. The solution adopted was to use the XV 3.10 package to convert the images to formats readable by Khoros, such as PBM and (uncompressed) TIFF.

4.2.1 Preliminary Results

We see in Figs. 5-6 and Figs. 7-8, the sequence of operations used to remove background noise and to obtain the segmented regions applying the direct watershed to the preprocessed image as described in section 3.2, using two of the selected images (ABELL 3698 and ABELL 3775 respectively), without change of homotopy.

We can see that the top-hat operation has successfully removed the halos and enhanced the objects by smoothing the background. The segmentation obtained with the direct watershed was near-optimal, where most detected regions correspond to one and only one object. We can see some overlapping objects were satisfactorily separated, but a few problems remain - for example, close or interacting object pairs as optical" partners" that were not separated.

There are some regions which do not correspond to any object, due to residual background noise, as can be examined in Fig. 8-h (This is a common problem for the other deblending techniques found in the literature). In addition, the large Sc galaxy in the bottom of the image in Fig. 6-h was taken apart by the segmentation, due to the non-homogeneity of the spiral arms (in fact, a galaxy is a large set of astronomical objects). As described in section 4.1, SExtractor deals with the latter case, although it does not consider the empty-region one. This oversegmentation is handled by many methods through visual inspection and post-processing techniques.

Another advantage derived from the watershed segmentation is that we have the option of including or not the basins that touch the image's border (Figs. 5-h and 8-h depict the "no border" case). This is done automatically in the MMach implementation of the watershed, while other methods do not offer this benefit, requiring visual inspection by human experts in order to accomplish it. In many applications it is strongly recommended to remove particles which are close to the image border. When dealing with astronomical images, they may introduce problems when performing the analysis step.

In table 1 we can see the number of objects detected by our method for the six images considered, with separate columns corresponding to the "no border" option being turned off and on respectively, as well as the results obtained by SExtractor with a satisfactory parameterization, both before and after oversegmentation post-processing (There is no post-processing in our method at this time).

The pattern spectrum extraction requires that each segmented region be considered separately. This is done by labelling the regions (this is already the output of the watershed MMach implementation, as can be seen in Figs. 6-g and 8-g), thresholding and subsequent masking the preprocessed image. The mean value of the original image used to pad the region proved to be a good estimate of the background gray-level value.

Image Proposed Method SExtractor Border No border Original Post-proc A3667 1879 1767 1785 1700 A3698 1890 1785 1831 1722 A3705 1728 1638 1684 1620 A3775 1890 1785 1308 1257 G1701 1672 1572 1543 1457 O0140 1834 1731 1484 1413

Table 1: Comparison of preliminary segmentation results

In Fig. 9, we can see four of the padded regions in image A3698, along with the respective pattern spectrum and analysis attributes for each metric utilized. We use the symbols and R for the average and largest size, instead of the previously defined and L, to stress the fact that the parameterization of the granulometry corresponds to the radius according to the selected metric.

The regions shown correspond to two stars and two galaxies of different apparent sizes. The most significant results to a suitable star/galaxy characterization task were the ones obtained using the city-block metric. The family can be seen as an increasing sequence of scaled square-based pyramids, which best approximates the symmetrical topology model of stellar objects, also satisfying the requirements for a valid granulometry, since the family is generated by iterated Minkowski sums of the digital disk of unitary radius.

For the two "largest" objects, as given by the average radius and largest radius (according to the respective metric induced by the particular family of structuring elements used), the pattern spectrum in Figure 9 clearly differentiates between them, since the "large" star produces a bimodal curve, due to the cross effect discussed earlier, whereas the galaxy is associated with a unimodal curve due to its uniform granulometry.

The first "peak" in the pattern spectrum of the" large" star corresponds to the cross effect removal, featuring the figure complexity relative to the geometric shape probed. This is also reflected in the average roughness attribute, which has a higher value for the star.

As for the "smaller" objects, we can see that the pattern spectrum for the star rises faster, as a result of the sharp gray-level stellar profile, characterized by the gray-level abrupt change from the object to the background. The galaxy, as a non-stellar object, has a smoother profile in the city-block metric, as indicated by its pattern spectrum. However, in this specific case, the average roughness attribute was inconclusive.

Figure 9:
Pattern spectrum characterization of four representative objects using different metrics. From top to bottom: city-block, euclidean and chessboard

4.2.2 Optimizing the Watershed Segmentation

Although the preliminary results seemed satisfactory, a great number of selected empty regions due to residual background noise will certainly cause an overhead in the classification step. Even if the classifier could clearly identify when there is not any object inside a region, based on appropriate selected features as the pattern spectrum and radius thanks to the stopping criterion established, this is still a disadvantage if we realize that the classifier will work with an excessive number of images.

An adjustment process, called change of homotopy, is used in order to optimize the watershed segmentation, requiring the selection of a marker image for further processing that best identifies individual objects, requiring a previous knowledge about the expected result. In our case, we initially tried to find a marker which keeps most of the faint astronomical objects. An immediate solution consists of marking manually the objects, but an automatic extraction of the markers is preferred.

The markers obtained by applying the change of homotopy to the top-hat transform for the ABELL3698 survey in Fig. 6-e provided a satisfactory marking of the objects and background, thereby yielding a segmentation result in accordance with visual identification, as illustrated by Fig. 10. We can perceive that overlapping objects in the preliminary segmentation are now separated. The Sc galaxy, however, is still taken appart by the method.

In table 2 we have the amount of objects visually identified from 5 of the images considered, as well as the number of objects yielded by our optimized segmentation method.

Image Change of Homotopy Visual Identification A3667 651 592 A3698 725 688 A3775 421 379 G1701 159 184 O0140 199 209

Table 2: Comparison of optimized segmentation results

5 Concluding Remarks

In this work we have introduced an original approach to the problem of star/galaxy characterization, based on Gray-level Mathematical Morphology operations. The proposed method is based on high-level gray shape-size information, which is closer to the human visual characterization of astronomical objects, whereas other approaches use pixel-based geometrical and statistical attributes. The purpose of our work was to find a compromise between refinement in both detection and measurements of astronomical objects using a set of robust MM tools, renewing the interest of the IPA community for astronomical imaging.

The basic intuitive principle of MM consisting in extract shape information based on morphological transformations by a carefully selected structuring element depending on the structural information expected makes MM attractive to be applied in many issues involved with IPA.

The watershed is a simple technique that has proved to be a very good tool for region segmentation. Other methods, like SExtractor, use heuristical approaches to this end, while our preprocessing step followed by the direct application of watershed yielded comparable results.

We considered the use of marker-based homotopy change of the image prior to the application of the watershed, as long as the marker could be obtained automatically, in order to treat the empty-region problem, cutting down on probable overhead to a future classification step. The minima imposition used in watershed-based segmentation, assuming that the markers are relevant image features, proved to be a powerful technique. We have yet to consider a post-processing step for dealing with the broken-galaxy, which will further improve the segmentation step.

We have shown that the pattern spectrum is a good shape-size descriptor. We think we can get even better results by looking for a more suitable family of structuring elements for the granulometry. We have shown that the pattern spectrum is a good shape-size descriptor. We think we can get even better results by looking for a more suitable family of structuring elements for the granulometry. Although the city-block family used in this work was able to give satisfactory results, it is very simple, which does not take into account detailed structural object information. For instance, we know that a distinctive feature of galaxies is the fuzzyness and slow gray-level variation, and this could be better explored with a family of properly scaled gaussian functions as structuring elements.

Figure 10:
(a) markers obtained to the objects and the background; (b) ABELL3698 image segmented by changing the homotopy of the top-hat transform

Gray-scale granulometry is an emerging issue, and it can not be found an exhaustive literature as for the binary case. There is not a stop criterion already defined for the granulometries. It was necessary to create our own homogeneity criterion to stop the sequence of closings based on a standard deviation from the residue left after each "sieving", which achieved satisfactory results.

Most objects can be satisfactorily treated by our method, except the very "small" ones, which can be detected but are hard to characterize by the pattern spectrum. The" small" objects which are not too faint can be handled by other methods in the literature with the use of photometric attributes. The faint ones are treated subjectively, by requiring the assistance of a human viewer to try to identify these objects visually or reject them for the classification step. But we would like to point out that even then, all the methods found in the literature to the star/galaxy characterization are not completely automated in the classification step for all cases.

We intend to further use our method to build a fully automatic neural-network-based classifier using the shape-size features extracted from the morphological operations. For this purpose, 2 classes could be originally established: stellar and non-stellar objects, while the latter one can still be subdivided in: galaxy or unidentified object. Further works could extend the classification to take into account the galaxy morphological types according to the Hubble sequence. We think this can be accomplished if the set of markers in the preprocessing step is used to modify the homotopy of the gradient function in such a way that its watersheds coincide with the contours of the objects shape.

Acknowledgements The authors would like to thank Dr. Laerte Sodré and his group at IAG-USP (Instituto Astronômico e Geofísico da Universidade de São Paulo) for kindly providing the images used in this work and their results obtained with the SExtractor software, and also for the remarks on astronomical imaging technical literature. We also thank Drs. Gerald Banon, Ivo Busko and Steve Odewahn for the incentive received. We are grateful to the referees for their comments.

References

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[2] J. Barrera, G.J.F. Banon, and R.A. Lotufo. A Mathematical Morphology toolbox for the KHOROS system. Technical Report RT-MAC-9403, Instituto de Matemática e Estatística - Universidade Estadual de São Paulo, São Paulo, SP - Brasil, Jan 1994.

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*

The second author was supported at the time this work was performed by CNPq DCR grant 300804/95-4

1

estimated center coordinates to year 2000

2

right ascension (a,in hours, minutes and seconds) and declination (d, in degrees, minutes and seconds) - celestial coordinations of the equatorial system

3

500x500 pixels on DSS means 15'x15' arcsec on the sky

4

according to the Hubble sequence, a Sc galaxy classification consists in having a not so relevant center relative to an open spiral structure, with well-defined arms

Publication Dates

• Publication in this collection
  13 Oct 1998
• Date of issue
  Apr 1997