Application of optimization heuristics for complex astronomical object model identification
Abstract
Detection and localization of astronomical objects are two of the most fundamental topics in astronomical science where localization uses detection results. Object localization is based on modeling of point spread function and estimation of its parameters. Commonly used models as Gauss or Moffat in objects localization provide good approximation of analyzed objects but cannot be sufficient in the case of exact applications such as object energy estimation. Thus the use of sophisticated models is upon the place. One of the key roles plays also the way of the objective function estimation. The least square method is often used, but it expects data with normal distribution, thus there is a question of a maximum likelihood method application. Another important factor of presented problem is choice of the right optimization method. Classical methods for objective function minimization usually require a good initial estimate for all parameters and differentiation of the objective function with respect to model parameters. The results indicated that stochastic methods such as simulated annealing or harmony search achieved better results than the classical optimization methods.
Keywords
Astronomical image Object identification Interference Turbulence Focusing Optimization heuristics1 Introduction
Astronomy is a natural science that deals with the study of objects such as planets, moons, stars, etc. Identification of mentioned objects is one of the most fundamental topics in astronomical science and the branch dealing with object identification is called astronomical photometry. Identification can be understood as a sequence of two steps. The first step is object detection to find the regions where object occurs. The next step is its localization (modeling), i.e., we want to get exact information about its parameters. Those information can be further used in energy estimation or objects’ interactions where we want to know how close the objects are we able to distinguish using their deconvolution.
The main aim of this article is to introduce different approach to modeling of result PSF and compare it with commonly used methods. There are introduced physical phenomena of interference on the thin lens, atmospheric turbulence and focusing, which influence resulting PSF. Except for these simple phenomena description, there is also described method of their combination in the frequency domain using convolution theorem. Optimization of described PSF models is based on authors’ previous work dealing with detection (Mojzis et al. 2012) and identification (Mojzis et al. 2012) of analyzed objects based on analysis of dark and light frames (Howell 2006) using hypotheses testing. This topic is mentioned in Sect. 2 to provide a general overview how the authors reached the objective function derived in Sect. 3.2.
Analyzed images can be considered as functions with many local minima and maxima. Thus the second aim of this work is an application and comparison of classical optimization methods and optimization heuristics. The authors want to show that the second approach is more suitable with respect to the finding of the best solution and repeatability of obtained result.
2 Object detection via multiple hypothesis testing
Astronomical systems represent special kind of imaging systems that use CCD image sensors (Buil 1991). Processing of images acquired by them assumes good knowledge of their properties. Analysis of these systems can give the answer on many questions about application of algorithms for further processing of acquired data. CCD sensor works as a photon counter, thus it can be supposed that the output of astronomical imaging systems is a Poisson random variable.
Analysis of these systems is based on the dark frames processing. The important question about the used CCD sensor is if it has the same properties in each image cell (pixel). The answer on this question is given by a statistical test described in the following section.
2.1 Noise model
2.2 Tests for the Poisson distribution
2.3 Multiple hypothesis testing
2.4 False discovery rate detection
When the Sidak’s correction is applied to the \(p\) values given by the Eq. (21), then the values that occur under \(p_{{\mathrm {crit.}}}\) curve are statistically significant. These values thus represent areas of the light image where objects may occur.
3 Object modeling
3.1 Basic PSF models
3.2 Objective function definition and its optimization
Optimization of models introduced in Sect. 3.1 is based on minimization of objective function. Its inference can be performed using Least Square Method (LSM), but as mentioned in Mojzis et al. (2012), data acquired by astronomical CCD camera are Poisson distributed. Thus we can use different approach based on Maximum Likelihood Estimate (MLE), see Mojzis et al. (2012).
In statistics, MLE is a method of estimating the parameters of a statistical model, Eq. (23). When it is applied to a data set and given statistical model, MLE provides estimates for the model’s parameters. For a fixed set of data and certain statistical model, it produces a distribution that gives to the measured data the greatest probability, i.e., estimated parameters maximize the likelihood function (Pawitan 2001; Severini 2001).
Let us now consider astronomical image defined by Eq. (1), noise model, Eq. (2) and average dark, Eq. (3).
3.3 Optimization methods
For the purpose of the objective function optimization, there was applied fmincon function. This function is based on derivatives evaluation, which may be problem in the case of discontinuous functions and also good initial estimates of optimized parameters are necessary. Another problem can occur with finding of local minima except global minima of evaluated function. This is the reason, why there were optimization heuristics applied and compared too.
3.3.1 Controlled Random Search
Control random search (CRS) (Price 1977) is based on random search (RS) principle, but it combines the random search and modeseeking routines into a single continuous process. CRS algorithms are populationsetbased algorithms specially developed for treating global optimization problems. Like genetic and differential evolution algorithms, a CRS aims to maximize (or minimize) a certain objective function between members of an evolving population of trial solutions.
Random Search (Rastrigin 1963) is a family of numerical optimization methods that does not require the gradient of the problem to be optimized, and RS can hence be used on functions that are not continuous or differentiable. Such optimization methods are also known as directsearch, derivativefree, or blackbox methods.
The name Random Search is attributed to Rastrigin (1963) who made an early presentation of RS along with basic mathematical analysis. RS works by iteratively moving to better positions in the search space, which are sampled from a hypersphere surrounding the current position.
3.3.2 Cuckoo Search
Cuckoo Search (CS) (Yang and Deb 2009; Gandomi et al. 2013) is inspired by the behavior of cuckoo species by laying their eggs in the nests of other birds. Some of the birds can be in the direct conflict with the cuckoo and can either throw these alien eggs away or simply abandon its nest and build a new nest somewhere else.
Each egg in a nest represents a solution, and a cuckoo egg represents a new solution. The aim is to use the new and potentially better solutions (cuckoos) to replace a notsogood solution in the nests. In the simplest form, each nest has one egg. The algorithm can be extended to more complicated cases in which each nest has multiple eggs representing a set of solutions.
3.3.3 Harmony Search
Harmony Search (HS) (Geem et al. 2011; Geem 2009) is inspired by musician improvisation process. It imitates the natural phenomenon of musicians’ behavior when they cooperate the pitches of their instruments together to achieve a fantastic harmony as measured by esthetic standards. This musicians’ prolonged and intense process led them to the perfect state. It is a very successful metaheuristic algorithm that can explore the search space of a given data in parallel optimization environment, where each solution (harmony) vector is generated by intelligently exploring and exploiting a search space.
3.3.4 Simulated Annealing
Simulated Annealing (SA) (Kirkpatrick et al. 1983; Laarhoven and Aarts 1987) is a generic probabilistic metaheuristic for the global optimization. It is inspired by annealing in metallurgy, a technique involving heating and controlled cooling of a material to increase the size of its crystals and reduce their defects.
The state of some physical systems, and the function \(F(x)\) to be minimized is analogous to the internal energy of the system in that state. The goal is to bring the system, from an arbitrary initial state, to a state with the minimum possible energy.
At each step, the SA heuristic considers some neighboring state \(s'\) of the current state \(s\), and probabilistically decides between moving the system to state \(s'\) or staying in state \(s\). These probabilities ultimately lead the system to move to states of lower energy. Typically this step is repeated until the system reaches a state that is good enough for the application, or until a given computation budget has been exhausted.
3.3.5 Artificial Bee Colony algorithm
Artificial Bee Colony (ABC) (Karaboga 2005; Karaboga and Bastruk 2008) is an optimization algorithm based on the intelligent foraging behavior of honey bee swarm.
In the ABC model, the colony consists of three groups of bees: employed bees, onlookers and scouts. It is assumed that there is only one artificial employed bee for each food source. In other words, the number of employed bees in the colony is equal to the number of food sources around the hive. Employed bees go to their food source and come back to hive and dance on this area. The employed bee whose food source has been abandoned becomes a scout and starts to search for finding a new food source. Onlookers watch the dances of employed bees and choose food sources depending on dances.
3.3.6 Backtracking Search algorithm
Backtracking Search Algorithm (BSA) (Knuth 1968; McGregor 1982) is a general algorithm for finding a solution to some computational problems, notably constraint satisfaction problems, that incrementally builds candidates to the solutions, and abandons each partial candidate \(c\), called backtracks, as soon as it determines that \(c\) cannot possibly be completed to a valid solution.
BSA enumerates a set of partial candidates that, in principle, could be completed in various ways to give all the possible solutions to the given problem. The completion is done incrementally, by a sequence of candidate extension steps. Conceptually, the partial candidates are represented as the nodes of a tree structure, the potential search tree. Each partial candidate is the parent of the candidates that differ from it by a single extension step. The leaves of the tree are the partial candidates that cannot be extended further.
BSA traverses this search tree recursively, from the root down, in depthfirst order. At each node \(c\), the algorithm checks whether c can be completed to a valid solution. If it cannot, the whole subtree rooted at c is skipped. Otherwise, the algorithm checks whether \(c\) itself is a valid solution, and if so reports it to the user and recursively enumerates all subtrees of \(c\). The two tests and the children of each node are defined by usergiven procedures. Therefore, the actual search tree that is traversed by the algorithm is only a part of the potential tree. The total cost of the algorithm is the number of nodes of the actual tree times the cost of obtaining and processing each node. This fact should be considered when choosing the potential search tree and implementing the pruning test.
3.3.7 Differential Search algorithm
Differential Search algorithm (DSA) (Civicioglu 2012) is an effective evolutionary algorithm for solving realvalued numerical optimization problems. DSA was inspired by migration of superorganisms utilizing the concept of stablemotion.
In DSA, the search space is simulated as the food areas and each point in the search space corresponds to an artificialsuperorganism migration. The goal of this migration is to find the global optimal solution of the problem. During this process, the artificialsuperorganism checks which randomly selected positions can be retained temporarily. If such a tested position is suitable to be retained for some time, the artificialsuperorganism uses this migration model to settle at the discovered position and then continues its migration from this position on.
3.3.8 Particle Swarm Optimization algorithm
Particle Swarm Optimization (PSO) (Poli 2008; Clerc 2006) is a computational method that optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality. PSO optimizes a problem by having a population of candidate solutions, here dubbed particles, and moving these particles around in the search space according to simple mathematical formulae over the particle’s position and velocity. Each particle’s movement is influenced by its local best known position but, is also guided toward the best known positions in the search space, which are updated as better positions are found by other particles. This is expected to move the swarm toward the best solutions.
Basic variant of the PSO algorithm works by having a population (swarm) of candidate solutions (particles). These particles are moved around in the search space according to a few simple formulae. The movements of the particles are guided by their own best known position in the search space as well as the entire swarm’s best known position. When improved positions are being discovered, these will then come to guide the movements of the swarm. The process is repeated and by doing so it is hoped, but not guaranteed, that a satisfactory solution will eventually be discovered.
4 Results
Localization and modeling methods introduced in the previous sections were tested on real astronomical data. For system analysis presentation were used data acquired by MAIA (Meteor Automatic Imager and Analyzer System) (Koten et al. 2011). Detection and modeling methods were then applied to chosen parts of astronomical image presented in Fig. 1. This image was acquired on 8 August 2004 by BOOTES 2 astronomical imaging system, where SBIG ST9 (LPT) astronomical camera was used. This camera was equipped with Meade optics (focal length 30 cm, lens speed 10). These data are not adjusted by the dark frame subtraction (Buil 1991).
Table 1 contains dark frames’ analysis results. The dark frames were acquired by the MAIA system for different values of CCD sensor temperatures. From mentioned table, it is possible to say that the imaging system has Poisson distribution with \(\lambda _1 =\cdots =\lambda _n\) for all CCD sensor temperatures. Thus the CCD sensor has same properties in each pixel at given sensor temperatures.
MAIA dark frames’ analysis for different sensor temperatures
Temperature (K)  Image size  Number of frames  Number of accepted \(H_0\)  Number of rejected \(H_0\)  FDR 

268.15  \(765\times 510\)  100  374,567  15,583  0 
273.15  \(765\times 510\)  100  387,923  2,227  0 
278.15  \(765\times 510\)  100  383,503  6,647  0 
283.15  \(765\times 510\)  100  383,662  6,488  0 
288.15  \(765\times 510\)  100  383,966  6,184  0 
293.15  \(765\times 510\)  100  383,379  6,771  0 
Figure 3 presents analyzed light and dark images and Fig. 4 shows results of object detection algorithm using FDR evaluation.

small object—maximum intensity in the analyzed area is less than 21,845,

medium object—maximum intensity in the analyzed area is higher than 21,845 and less than 43,690,

large object—maximum intensity in the analyzed area exceeds 43,690 and the top is given by the system resolution properties, thus 65,535.
Parameters of basic PSFs
PSF model  Parameters 

Gauss  \(\sigma \) 
Moffat  \(\sigma \), \(\beta \) 
Interference  \(\Omega \) 
Focusing  \(\rho \) 
Turbulence  \(\psi \), \(\sigma \) 
Lower and upper bounds of model parameters
Model  Bound  \(p_1\)  \(p_2\)  \(p_3\)  \(p_4\)  \(p_5\)  \(p_6\) 

Gauss  LB  1  1  1  1  
UB  \(2\max (\mathrm {x}(k,l)\mathrm {d}(k,l))\)  \(M\)  \(N\)  \(\max (M,N)\)  
Moffat  LB  1  1  1  1  0.01  
UB  \(2\max (\mathrm {x}(k,l)\mathrm {d}(k,l))\)  \(M\)  \(N\)  \(\max (M,N)\)  50  
Interturb  LB  1  1  1  0.01  0.0001  0.0001 
UB  \(2\max (\mathrm {x}(k,l)\mathrm {d}(k,l))\)  \(M\)  \(N\)  10  10  10  
Interfoc  LB  1  1  1  0.01  0.01  
UB  \(2\max (\mathrm {x}(k,l)\mathrm {d}(k,l))\)  \(M\)  \(N\)  10  10 
Optimization results of Gauss model—small object
Method  Function values  Optimum parameters  

\(\phi _{\mathrm {min}}\)  \(\phi _{\mathrm {max}}\)  \(\phi _{\mathrm {average}}\)  \(\phi _{\mathrm {std}}\)  \(p_1\)  \(p_2\)  \(p_3\)  \(p_4\)  
fmincon  6.15\(\times 10^3\)  1.32\(\times 10^5\)  1.12\(\times 10^4\)  2.50\(\times 10^4\)  1.76\(\times 10^4\)  11.42  14.39  1.38 
CRS  \(7.90\times 10^3\)  \(3.23\times 10^4\)  \(1.78\times 10^4\)  \(5.77\times 10^3\)  1.84\(\times 10^4\)  11.49  14.21  1.29 
CS  \(6.15\times 10^3\)  \(1.51\times 10^4\)  \(7.30\times 10^3\)  \(1.76\times 10^3\)  1.73\(\times 10^4\)  11.41  14.39  1.39 
HS  \(6.15\times 10^3\)  \(6.17\times 10^3\)  \(6.15\times 10^3\)  \(4.77\times 10^2\)  1.75\(\times 10^4\)  11.41  14.39  1.38 
SA  \(6.18\times 10^3\)  \(2.14\times 10^4\)  \(9.90\times 10^3\)  \(3.01\times 10^3\)  1.79\(\times 10^4\)  11.47  14.46  1.39 
ABC  \(6.18\times 10^3\)  \(6.48\times 10^3\)  \(6.46\times 10^3\)  \(5.01\times 10^0\)  1.76\(\times 10^4\)  11.47  14.46  1.40 
BSA  \(6.46\times 10^3\)  \(3.30\times 10^4\)  \(1.63\times 10^4\)  \(6.25\times 10^3\)  1.90\(\times 10^4\)  11.51  14.42  1.39 
DSA  \(6.18\times 10^3\)  \(1.58\times 10^4\)  \(7.67\times 10^3\)  \(1.85\times 10^3\)  1.74\(\times 10^4\)  11.47  14.46  1.41 
PSO  \(6.21\times 10^3\)  \(2.24\times 10^4\)  \(1.04\times 10^4\)  \(3.16\times 10^3\)  1.80\(\times 10^4\)  11.53  14.53  1.40 
Optimization results of Moffat model—small object
Method  Function values  Optimum parameters  

\(\phi _{\mathrm {min}}\)  \(\phi _{\mathrm {max}}\)  \(\phi _{\mathrm {average}}\)  \(\phi _{\mathrm {std}}\)  \(p_1\)  \(p_2\)  \(p_3\)  \(p_4\)  \(p_5\)  
fmincon  \(6.11\times 10^3\)  \(9.36\times 10^4\)  \(9.61\times 10^3\)  \(1.73\times 10^4\)  \(1.82\times 10^4\)  11.42  14.39  4.36  11.07 
CRS  \(6.49\times 10^3\)  \(1.99\times 10^4\)  1.19\(\times 10^4\)  3.11\(\times 10^3\)  \(1.97\times 10^4\)  11.42  14.40  8.40  43.74 
CS  \(6.13\times 10^3\)  \(1.37\times 10^4\)  6.89\(\times 10^3\)  1.36\(\times 10^3\)  \(1.78\times 10^4\)  11.41  14.39  7.09  27.23 
HS  \(6.11\times 10^3\)  \(6.13\times 10^3\)  6.12\(\times 10^3\)  4.84\(\times 10^2\)  \(1.83\times 10^4\)  11.42  14.39  4.23  10.50 
SA  \(6.14\times 10^3\)  \(1.55\times 10^4\)  6.95\(\times 10^3\)  1.40\(\times 10^3\)  \(1.83\times 10^4\)  11.47  14.46  4.66  12.41 
ABC  \(6.14\times 10^3\)  \(6.44\times 10^3\)  6.43\(\times 10^3\)  5.08\(\times 10^0\)  \(1.84\times 10^4\)  11.47  14.46  4.27  10.60 
BSA  \(6.53\times 10^3\)  \(2.13\times 10^4\)  1.21\(\times 10^4\)  3.31\(\times 10^3\)  \(1.64\times 10^4\)  11.53  14.40  9.27  41.67 
DSA  \(6.16\times 10^3\)  \(1.44\times 10^4\)  7.23\(\times 10^3\)  1.43\(\times 10^3\)  \(1.79\times 10^4\)  11.47  14.46  7.16  27.50 
PSO  \(6.18\times 10^3\)  \(1.62\times 10^4\)  7.30\(\times 10^3\)  1.47\(\times 10^3\)  \(1.84\times 10^4\)  11.53  14.54  4.71  12.54 
Optimization results of INTERTURB model—small object
Method  Function values  Optimum parameters  

\(\phi _{\mathrm {min}}\)  \(\phi _{\mathrm {max}}\)  \(\phi _{\mathrm {average}}\)  \(\phi _{\mathrm {std}}\)  \(p_1\)  \(p_2\)  \(p_3\)  \(p_4\)  \(p_5\)  \(p_6\)  
fmincon  \(9.71\times 10^3\)  \(1.13\times 10^5\)  \(2.42\times 10^4\)  2.20\(\times 10^4\)  \(1.67\times 10^4\)  14.40  11.37  2.48  \(3.47\times 10^{4}\)  \(9.96\times 10^{1}\) 
CRS  \(1.10\times 10^4\)  \(1.79\times 10^4\)  \(1.31\times 10^4\)  \(1.50\times 10^3\)  \(2.29\times 10^4\)  14.52  11.33  9.80  \(9.47\times 10^{1}\)  \(5.03\times 10^{3}\) 
CS  \(9.79\times 10^3\)  \(1.97\times 10^4\)  \(1.11\times 10^4\)  \(1.65\times 10^3\)  \(1.72\times 10^4\)  14.45  11.36  2.50  \(1.11\times 10^{1}\)  \(6.83\times 10^{2}\) 
HS  \(9.69\times 10^3\)  \(3.16\times 10^4\)  \(1.08\times 10^4\)  \(4.71\times 10^3\)  \(1.66\times 10^4\)  14.40  11.36  2.45  \(1.17\times 10^{4}\)  \(6.51\times 10^{4}\) 
SA  \(9.77\times 10^3\)  \(3.41\times 10^4\)  \(1.29\times 10^4\)  \(4.85\times 10^3\)  \(1.71\times 10^4\)  14.49  11.43  2.51  \(7.27\times 10^{1}\)  \(1.19\times 10^{3}\) 
ABC  \(9.74\times 10^3\)  \(3.32\times 10^4\)  \(1.14\times 10^4\)  \(4.95\times 10^3\)  \(1.67\times 10^4\)  14.47  11.42  2.47  \(1.18\times 10^{4}\)  \(6.58\times 10^{4}\) 
BSA  \(1.11\times 10^4\)  \(1.92\times 10^4\)  \(1.40\times 10^4\)  \(1.81\times 10^3\)  \(2.56\times 10^4\)  14.58  11.51  9.94  \(1.84\times 10^{1}\)  \(4.05\times 10^{2}\) 
DSA  \(9.87\times 10^3\)  \(1.53\times 10^4\)  \(1.15\times 10^4\)  \(9.12\times 10^2\)  \(1.52\times 10^4\)  14.48  11.41  2.32  \(7.32\times 10^{2}\)  \(7.35\times 10^{3}\) 
PSO  \(9.79\times 10^3\)  \(3.49\times 10^4\)  \(1.19\times 10^4\)  \(5.19\times 10^3\)  \(1.67\times 10^4\)  14.55  11.48  2.50  \(1.19\times 10^{4}\)  \(6.64\times 10^{4}\) 
Optimization results of INTERFOC model  small object
Method  Function values  Optimum parameters  

\(\phi _{\mathrm {min}}\)  \(\phi _{\mathrm {max}}\)  \(\phi _{\mathrm {average}}\)  \(\phi _{\mathrm {std}}\)  \(p_1\)  \(p_2\)  \(p_3\)  \(p_4\)  \(p_5\)  
fmincon  \(2.01\times 10^4\)  \(1.31\times 10^5\)  \(6.42\times 10^4\)  \(3.11\times 10^4\)  \(8.31\times 10^3\)  14.46  11.15  1.00  0.10 
CRS  \(7.50\times 10^3\)  \(2.66\times 10^4\)  \(1.79\times 10^4\)  \(4.66\times 10^3\)  \(2.41\times 10^4\)  14.51  11.35  4.03  1.30 
CS  \(5.99\times 10^3\)  \(1.56\times 10^4\)  \(9.22\times 10^3\)  \(2.72\times 10^3\)  \(2.15\times 10^4\)  14.45  11.41  1.96  1.04 
HS  \(5.84\times 10^3\)  \(2.88\times 10^4\)  \(1.13\times 10^4\)  \(7.92\times 10^3\)  \(2.05\times 10^4\)  14.46  11.42  1.98  1.05 
SA  \(5.87\times 10^3\)  \(3.61\times 10^4\)  \(1.31\times 10^4\)  \(9.41\times 10^3\)  \(2.04\times 10^4\)  14.53  11.47  2.02  1.10 
ABC  \(5.87\times 10^3\)  \(3.02\times 10^4\)  \(1.19\times 10^4\)  \(8.32\times 10^3\)  \(2.06\times 10^4\)  14.53  11.47  2.00  1.06 
BSA  \(8.65\times 10^3\)  \(3.03\times 10^4\)  \(1.84\times 10^4\)  \(5.17\times 10^3\)  \(3.04\times 10^4\)  14.44  11.43  4.93  1.24 
DSA  \(6.02\times 10^3\)  \(1.64\times 10^4\)  \(9.68times10^3\)  \(2.86\times 10^3\)  \(2.16\times 10^4\)  14.53  11.46  1.98  1.05 
PSO  \(5.90\times 10^3\)  \(3.79\times 10^4\)  \(1.37\times 10^4\)  \(9.88\times 10^3\)  \(2.05\times 10^4\)  14.60  11.53  2.04  1.12 
Optimization results of Gauss model—medium object
Method  Function values  Optimum parameters  

\(\phi _{\mathrm {min}}\)  \(\phi _{\mathrm {max}}\)  \(\phi _{\mathrm {average}}\)  \(\phi _{\mathrm {std}}\)  \(p_1\)  \(p_2\)  \(p_3\)  \(p_4\)  
fmincon  \(1.10\times 10^4\)  \(3.80\times 10^5\)  \(7.74\times 10^4\)  \(1.43\times 10^5\)  \(2.75\times 10^4\)  26.94  33.44  1.69 
CRS  \(1.32\times 10^4\)  \(7.07\times 10^4\)  3.79\(\times 10^4\)  1.18\(\times 10^4\)  3.01\(\times 10^4\)  27.00  33.27  1.64 
CS  \(1.10\times 10^4\)  \(3.01\times 10^4\)  1.45\(\times 10^4\)  4.59\(\times 10^3\)  2.77\(\times 10^4\)  26.95  33.44  1.68 
HS  \(1.10\times 10^4\)  \(1.10\times 10^4\)  1.10\(\times 10^4\)  6.92\(\times 10^3\)  2.75\(\times 10^4\)  26.94  33.44  1.69 
SA  \(1.16\times 10^4\)  \(4.64\times 10^4\)  2.52\(\times 10^4\)  9.38\(\times 10^3\)  2.70\(\times 10^4\)  27.08  33.60  1.72 
ABC  \(1.11\times 10^4\)  \(1.16\times 10^4\)  1.15\(\times 10^4\)  7.27\(\times 10^3\)  2.76\(\times 10^4\)  27.08  33.60  1.70 
BSA  \(1.89\times 10^4\)  \(6.63\times 10^4\)  4.15\(\times 10^4\)  1.22\(\times 10^4\)  3.20\(\times 10^4\)  26.85  33.59  1.77 
DSA  \(1.11\times 10^4\)  \(3.16\times 10^4\)  1.53\(\times 10^4\)  4.82\(\times 10^3\)  2.78\(\times 10^4\)  27.08  33.60  1.70 
PSO  \(1.16\times 10^4\)  \(4.87\times 10^4\)  2.64\(\times 10^4\)  9.84\(\times 10^3\)  2.71\(\times 10^4\)  27.21  33.77  1.74 
Optimization results of Moffat model—medium object
Method  Function values  Optimum parameters  

\(\phi _{\mathrm {min}}\)  \(\phi _{\mathrm {max}}\)  \(\phi _{\mathrm {average}}\)  \(\phi _{\mathrm {std}}\)  \(p_1\)  \(p_2\)  \(p_3\)  \(p_4\)  \(p_5\)  
fmincon  \(1.11\times 10^4\)  \(2.48\times 10^5\)  1.58\(\times 10^4\)  \(3.35\times 10^4\)  \(2.78\times 10^4\)  26.94  33.44  10.00  36.34 
CRS  \(1.33\times 10^4\)  \(4.94\times 10^4\)  \(2.86\times 10^4\)  \(8.64\times 10^3\)  2.72\(\times 10^4\)  26.81  33.30  8.32  23.61 
CS  \(1.12\times 10^4\)  \(1.89\times 10^4\)  \(1.28\times 10^4\)  \(1.64\times 10^3\)  2.79\(\times 10^4\)  26.94  33.44  6.24  14.78 
HS  \(1.11\times 10^4\)  \(1.11\times 10^4\)  \(1.11\times 10^4\)  \(5.90\times 10^3\)  2.79\(\times 10^4\)  26.94  33.44  10.00  36.39 
SA  \(1.11\times 10^4\)  \(3.30\times 10^4\)  \(1.39\times 10^4\)  \(3.76\times 10^3\)  2.79\(\times 10^4\)  27.08  33.60  10.10  36.76 
ABC  \(1.11\times 10^4\)  \(1.16\times 10^4\)  \(1.16\times 10^4\)  \(6.20\times 10^3\)  2.80\(\times 10^4\)  27.08  33.60  10.10  36.75 
BSA  \(1.22\times 10^4\)  \(5.36\times 10^4\)  \(2.90\times 10^4\)  \(1.04\times 10^4\)  2.78\(\times 10^4\)  27.05  33.65  7.82  24.34 
DSA  \(1.12\times 10^4\)  \(1.98\times 10^4\)  \(1.35\times 10^4\)  \(1.72\times 10^3\)  2.80\(\times 10^4\)  27.08  33.61  6.30  14.93 
PSO  \(1.12\times 10^4\)  \(3.47\times 10^4\)  \(1.46\times 10^4\)  \(3.94\times 10^3\)  2.80\(\times 10^4\)  27.21  33.77  10.20  37.13 
Optimization results of INTERTURB model—medium object
Method  Function values  Optimum parameters  

\(\phi _{\mathrm {min}}\)  \(\phi _{\mathrm {max}}\)  \(\phi _{\mathrm {average}}\)  \(\phi _{\mathrm {std}}\)  \(p_1\)  \(p_2\)  \(p_3\)  \(p_4\)  \(p_5\)  \(p_6\)  
fmincon  \(1.32\times 10^4\)  \(2.99\times 10^5\)  \(5.91\times 10^4\)  \(6.83\times 10^4\)  \(2.51\times 10^4\)  33.45  27.00  1.98  1.79\(\times 10^{2}\)  1.00 
CRS  \(2.02\times 10^4\)  \(3.53\times 10^4\)  \(2.82\times 10^4\)  \(2.88\times 10^3\)  \(2.84\times 10^4\)  33.31  26.82  2.40  \(7.55\times 10^{1}\)  \(8.35\times 10^{2}\) 
CS  \(1.33\times 10^4\)  \(2.43\times 10^4\)  \(1.81\times 10^4\)  \(4.08\times 10^3\)  \(2.54\times 10^4\)  33.41  27.00  1.99  \(2.38\times 10^{2}\)  \(5.41\times 10^{1}\) 
HS  \(1.32\times 10^4\)  \(1.44\times 10^4\)  \(1.33\times 10^4\)  \(1.72\times 10^2\)  \(2.45\times 10^4\)  33.45  27.00  1.95  \(2.16\times 10^{6}\)  \(6.16\times 10^{1}\) 
SA  \(1.32\times 10^4\)  \(1.44\times 10^4\)  \(1.33\times 10^4\)  \(1.72\times 10^2\)  \(2.45\times 10^4\)  33.63  27.13  1.98  \(6.82\times 10^{3}\)  \(4.35\times 10^{1}\) 
ABC  \(1.33\times 10^4\)  \(1.51\times 10^4\)  \(1.39\times 10^4\)  \(1.81\times 10^2\)  \(2.47\times 10^4\)  33.61  27.13  1.97  \(2.18\times 10^{6}\)  \(6.22\times 10^{1}\) 
BSA  \(1.67\times 10^4\)  \(4.00\times 10^4\)  \(2.91\times 10^4\)  \(3.84\times 10^3\)  \(2.37\times 10^4\)  33.78  26.96  1.93  \(2.10\times 10^{1}\)  \(3.37\times 10^{1}\) 
DSA  \(1.34\times 10^4\)  \(2.56\times 10^4\)  \(2.13\times 10^4\)  \(4.25\times 10^3\)  \(2.49\times 10^4\)  33.59  27.13  2.00  \(2.27\times 10^{2}\)  \(7.21\times 10^{1}\) 
PSO  \(1.33\times 10^4\)  \(1.59\times 10^4\)  \(1.46\times 10^4\)  \(1.90\times 10^2\)  \(2.48\times 10^4\)  33.78  27.27  1.99  \(2.20\times 10^{6}\)  \(6.28\times 10^{1}\) 
Optimization results of INTERFOC model—medium object
Method  Function values  Optimum parameters  

\(\phi _{\mathrm {min}}\)  \(\phi _{\mathrm {max}}\)  \(\phi _{\mathrm {average}}\)  \(\phi _{\mathrm {std}}\)  \(p_1\)  \(p_2\)  \(p_3\)  \(p_4\)  \(p_5\)  
fmincon  \(2.75\times 10^4\)  \(3.75\times 10^5\)  \(2.53\times 10^5\)  \(1.22\times 10^5\)  \(1.56\times 10^4\)  33.36  27.00  1.00  0.10 
CRS  \(2.38\times 10^4\)  \(7.26\times 10^4\)  \(4.66\times 10^4\)  \(1.31\times 10^4\)  \(2.30\times 10^4\)  33.10  27.21  1.74  2.26 
CS  \(9.40\times 10^3\)  \(3.52\times 10^4\)  \(1.40\times 10^4\)  \(4.34\times 10^3\)  \(2.65\times 10^4\)  33.41  26.99  1.64  1.81 
HS  \(9.23\times 10^3\)  \(6.84\times 10^4\)  \(1.57\times 10^4\)  \(1.59\times 10^4\)  \(2.68\times 10^4\)  33.44  26.98  1.69  1.83 
SA  \(9.60\times 10^3\)  \(1.04\times 10^5\)  \(4.34\times 10^4\)  \(3.00\times 10^4\)  \(2.90\times 10^4\)  33.61  27.10  1.85  1.87 
ABC  \(9.27\times 10^3\)  \(7.19\times 10^4\)  \(1.64\times 10^4\)  \(1.67\times 10^4\)  \(2.70\times 10^4\)  33.61  27.11  1.71  1.85 
BSA  \(1.53\times 10^4\)  \(7.53\times 10^4\)  \(4.54\times 10^4\)  \(1.60\times 10^4\)  \(2.93\times 10^4\)  33.71  27.24  2.63  1.78 
DSA  \(9.44\times 10^3\)  \(3.69\times 10^4\)  \(1.46\times 10^4\)  \(4.56\times 10^3\)  \(2.67\times 10^4\)  33.58  27.12  1.66  1.83 
PSO  \(9.64\times 10^3\)  \(1.09\times 10^5\)  \(4.55\times 10^4\)  \(3.15\times 10^4\)  \(2.92\times 10^4\)  33.78  27.23  1.87  1.89 
Optimization results of Gauss model—large object
Method  Function values  Optimum parameters  

\(\phi _{\mathrm {min}}\)  \(\phi _{\mathrm {max}}\)  \(\phi _{\mathrm {average}}\)  \(\phi _{\mathrm {std}}\)  \(p_1\)  \(p_2\)  \(p_3\)  \(p_4\)  
fmincon  \(5.53\times 10^4\)  \(9.55\times 10^5\)  \(1.63\times 10^5\)  \(2.95\times 10^5\)  \(6.95\times 10^4\)  31.71  23.52  1.97 
CRS  \(6.15\times 10^4\)  \(1.55\times 10^5\)  \(1.05\times 10^5\)  1.94\(\times 10^4\)  \(6.58\times 10^4\)  31.82  23.39  1.99 
CS  \(5.76\times 10^4\)  \(1.18\times 10^5\)  \(8.35\times 10^4\)  1.71\(\times 10^4\)  \(6.61\times 10^4\)  31.74  23.51  2.00 
HS  \(5.63\times 10^4\)  \(5.63\times 10^4\)  \(5.63\times 10^4\)  8.81\(\times 10^4\)  \(6.79\times 10^4\)  31.71  23.52  1.98 
SA  \(5.72\times 10^4\)  \(1.31\times 10^5\)  \(8.58\times 10^4\)  2.13\(\times 10^4\)  \(6.72\times 10^4\)  31.87  23.63  2.01 
ABC  \(5.66\times 10^4\)  \(5.91\times 10^4\)  \(5.91\times 10^4\)  9.25\(\times 10^4\)  \(6.83\times 10^4\)  31.87  23.64  2.00 
BSA  \(6.44\times 10^4\)  \(1.79\times 10^5\)  \(1.07\times 10^5\)  2.42\(\times 10^4\)  \(6.04\times 10^4\)  31.98  23.58  2.07 
DSA  \(5.79\times 10^4\)  \(1.24\times 10^5\)  \(8.76\times 10^4\)  1.80\(\times 10^4\)  \(6.64\times 10^4\)  31.90  23.63  2.02 
PSO  \(5.75\times 10^4\)  \(1.37\times 10^5\)  \(9.01\times 10^4\)  2.24\(\times 10^4\)  \(6.76\times 10^4\)  32.03  23.75  2.03 
Optimization results of Moffat model—large object
Method  Function values  Optimum parameters  

\(\phi _{\mathrm {min}}\)  \(\phi _{\mathrm {max}}\)  \(\phi _{\mathrm {average}}\)  \(\phi _{\mathrm {std}}\)  \(p_1\)  \(p_2\)  \(p_3\)  \(p_4\)  \(p_5\)  
fmincon  \(5.83\times 10^4\)  \(2.90\times 10^5\)  \(7.69\times 10^4\)  \(6.35\times 10^4\)  \(6.79\times 10^4\)  31.71  23.51  10.00  26.49 
CRS  \(7.31\times 10^4\)  \(1.56\times 10^5\)  \(1.06\times 10^5\)  \(2.11\times 10^4\)  \(6.51\times 10^4\)  31.75  23.25  9.22  20.07 
CS  \(5.82\times 10^4\)  \(1.01\times 10^5\)  \(6.79\times 10^4\)  \(9.78\times 10^3\)  \(6.82\times 10^4\)  31.71  23.51  9.95  26.44 
HS  \(5.73\times 10^4\)  \(5.83\times 10^4\)  \(5.75\times 10^4\)  \(1.79\times 10^2\)  \(6.95\times 10^4\)  31.71  23.51  10.00  26.79 
SA  \(5.80\times 10^4\)  \(9.10\times 10^4\)  \(6.76\times 10^4\)  \(7.16\times 10^3\)  \(6.98\times 10^4\)  31.87  23.63  10.10  27.83 
ABC  \(5.76\times 10^4\)  \(6.12\times 10^4\)  \(6.04\times 10^4\)  \(1.88\times 10^2\)  \(6.98\times 10^4\)  31.87  23.63  10.10  27.06 
BSA  \(7.18\times 10^4\)  \(1.52\times 10^5\)  \(1.08\times 10^5\)  \(1.53\times 10^4\)  \(5.93\times 10^4\)  31.70  23.68  8.85  20.47 
DSA  \(5.85\times 10^4\)  \(1.06\times 10^5\)  \(7.13\times 10^4\)  \(1.03\times 10^4\)  \(6.85\times 10^4\)  31.86  23.62  10.05  26.71 
PSO  \(5.83\times 10^4\)  \(9.55\times 10^4\)  \(7.10\times 10^4\)  \(7.52\times 10^3\)  \(7.01\times 10^4\)  32.03  23.74  10.20  28.11 
Optimization results of INTERTURB model—large object
Method  Function values  Optimum parameters  

\(\phi _{\mathrm {min}}\)  \(\phi _{\mathrm {max}}\)  \(\phi _{\mathrm {average}}\)  \(\phi _{\mathrm {std}}\)  \(p_1\)  \(p_2\)  \(p_3\)  \(p_4\)  \(p_5\)  \(p_6\)  
fmincon  9.97\(\times 10^4\)  3.23\(\times 10^5\)  1.64\(\times 10^5\)  7.24\(\times 10^4\)  5.08\(\times 10^4\)  23.46  31.86  1.53  2.05\(\times 10^{5}\)  4.96\(\times 10^{1}\) 
CRS  1.06\(\times 10^5\)  1.55\(\times 10^5\)  1.33\(\times 10^5\)  1.18\(\times 10^4\)  5.04\(\times 10^4\)  23.54  31.62  1.63  1.33\(\times 10^{1}\)  5.99\(\times 10^{1}\) 
CS  9.98\(\times 10^4\)  1.05\(\times 10^5\)  1.01\(\times 10^5\)  1.26\(\times 10^3\)  5.19\(\times 10^4\)  23.47  31.85  1.55  2.60\(\times 10^{2}\)  2.32\(\times 10^{1}\) 
HS  9.97\(\times 10^4\)  1.03\(\times 10^5\)  9.98\(\times 10^4\)  4.51\(\times 10^2\)  5.08\(\times 10^4\)  23.46  31.86  1.53  3.89\(\times 10^{6}\)  6.64\(\times 10^{2}\) 
SA  1.00\(\times 10^5\)  1.10\(\times 10^5\)  1.06\(\times 10^5\)  1.33\(\times 10^3\)  5.21\(\times 10^4\)  23.59  32.01  1.56  2.62\(\times 10^{2}\)  2.35\(\times 10^{1}\) 
ABC  1.00\(\times 10^5\)  1.08\(\times 10^5\)  1.05\(\times 10^5\)  4.73\(\times 10^2\)  5.11\(\times 10^4\)  23.58  32.02  1.55  3.93\(\times 10^{6}\)  6.71\(\times 10^{2}\) 
BSA  1.09\(\times 10^5\)  1.62\(\times 10^5\)  1.44\(\times 10^5\)  1.08\(\times 10^4\)  5.99\(\times 10^4\)  23.45  32.18  1.66  1.01\(\times 10^{1}\)  9.57\(\times 10^{1}\) 
DSA  1.00\(\times 10^5\)  1.30\(\times 10^5\)  1.10\(\times 10^5\)  5.04\(\times 10^3\)  5.32\(\times 10^4\)  23.59  32.02  1.58  1.44\(\times 10^{2}\)  2.83\(\times 10^{1}\) 
PSO  1.01\(\times 10^5\)  1.14\(\times 10^5\)  1.10\(\times 10^5\)  4.97\(\times 10^2\)  5.13\(\times 10^4\)  23.70  32.18  1.57  3.97\(\times 10^{6}\)  6.77\(\times 10^{2}\) 
Optimization results of INTERFOC model—large object
Method  Function values  Optimum parameters  

\(\phi _{\mathrm {min}}\)  \(\phi _{\mathrm {max}}\)  \(\phi _{\mathrm {average}}\)  \(\phi _{\mathrm {std}}\)  \(p_1\)  \(p_2\)  \(p_3\)  \(p_4\)  \(p_5\)  
fmincon  7.47\(\times 10^4\)  9.55\(\times 10^5\)  6.14\(\times 10^5\)  2.90\(\times 10^5\)  4.45\(\times 10^4\)  23.49  31.78  1.00  0.10 
CRS  \(5.50\times 10^4\)  \(1.84\times 10^5\)  \(1.05\times 10^5\)  \(2.98\times 10^4\)  \(3.68\times 10^4\)  23.51  31.82  2.59  2.55 
CS  \(4.10\times 10^4\)  \(1.07\times 10^5\)  \(5.73\times 10^4\)  \(1.30\times 10^4\)  \(5.29\times 10^4\)  23.55  31.71  2.49  2.27 
HS  \(3.87\times 10^4\)  \(1.69\times 10^5\)  \(4.67\times 10^4\)  \(2.69\times 10^4\)  \(6.07\times 10^4\)  23.55  31.71  2.55  2.19 
SA  \(3.91\times 10^4\)  \(1.92\times 10^5\)  \(5.85\times 10^4\)  \(3.30\times 10^4\)  \(5.97\times 10^4\)  23.68  31.87  2.53  2.22 
ABC  \(3.89\times 10^4\)  \(1.77\times 10^5\)  \(4.90\times 10^4\)  \(2.83\times 10^4\)  \(6.10\times 10^4\)  23.67  31.87  2.58  2.21 
BSA  \(5.00\times 10^4\)  \(1.66\times 10^5\)  \(1.05\times 10^5\)  \(2.99\times 10^4\)  \(5.37\times 10^4\)  23.55  31.89  6.36  2.31 
DSA  \(4.12\times 10^4\)  \(1.12\times 10^5\)  \(6.01\times 10^4\)  \(1.37\times 10^4\)  \(5.32\times 10^4\)  23.66  31.87  2.52  2.29 
PSO  \(3.93\times 10^4\)  \(2.02\times 10^5\)  \(6.14\times 10^4\)  \(3.46\times 10^4\)  \(6.00\times 10^4\)  23.80  32.03  2.55  2.24 
For each optimization method, there were performed 50 cycles of the objective function \(\phi \) minimization. All optimization methods were set to 50,000 evaluations of the objective function in each cycle. Lower and upper bounds of optimized models were estimated empirically.
In Tables 4, 5, 6, 7, there are summarized results of small object models optimization, namely Gauss model in Table 4, Moffat in Table 5 and combined model of interference and either turbulence or focusing in Tables 6 and 7, respectively.
From Tables 4, 5, 6, 7, it is obvious that the best value of objective function \(\phi \) was reached by the application of interference and focusing combination, where this result was achieved by harmonic search optimization method. When the particular optimization methods are compared, then there are not as big differences, when the results for Gauss and Moffat models are compared. All the methods gave almost similar minimum values of \(\phi \), but when we look at standard deviation of \(\phi \), then the most stable method is HS algorithm. Other genetic algorithms have again comparable \(\phi \) standard deviation, but the fmincon was the worst. In the case of Gauss and Moffat model, the fmincon function gives comparable results to genetic algorithms in the solution of the given problem. From Table 7 is visible that the fmincon function is not suitable for more complicated models such as used combination of simple functions. The best value of the objective function was again reached by HS followed by the SA, but also CS gave good result with lower standard deviation then the two previously mentioned algorithms.
Tables 8, 9, 10, 11 show results of medium object localization. It is again obvious that the best results with respect to \(\phi _{\mathrm {min}}\) were reached by HS algorithm followed by the SA, CS and MCS methods. The Standard deviations of \(\phi \) were again lowest in case of HS algorithm for Gauss and Moffat models. For the third used model, there the lowest value of \(\phi _{\mathrm {std}}\) was achieved with CS algorithm with respect to the value of \(\phi _{\mathrm {min}}\) but HS gives also satisfactory results.
Last four tables, Tables 12, 13, 14, 15 summarize results of localization for large astronomical object. In the case of the object used in this article whereof parameters were estimated, there the best results with respect to \(\phi _{\mathrm {min}}\) and \(\phi _{\mathrm {std}}\) were reached by the HS methods for all three models.
When minimum values of \(\phi \) are compared then it can be said that the fmincon function gives comparable results as optimization heuristics for all three types of objects when the Gauss and Moffat models are applied. On the other hand, as it was previously mentioned, it does not give satisfactory results for the third used model.
With regard to the standard deviations then it can be said that the fmincon function is not suitable from the point of view of the solution repeatability. When the results from Tables 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 and 15 are compared then it can be written that the best method is HS algorithm. It always gives the lowest estimate of \(\phi \) but from stability point of view gives best results CS when \(\phi _{\mathrm {std}}\) values are compared.
5 Conclusion
In this paper, there were described algorithms used for analysis of astronomical images, especially for detection of objects and their modeling. There was also explained how authors derived objective function from their hypothesis using Poisson distributed data.
There was mentioned method of objects detection based on the presumption that the analyzed data are Poisson distributed. This presumption was verified on dark images acquired by MAIA astronomical system, Table 1, and led to the inference of objects detection algorithm. Presented algorithm was derived using relationship between Poisson and NB distribution, which cumulative distribution function allows us comparison of information present in the light and dark images. In combination with FDR evaluation using Sidak’s correction was presented new object detection algorithm, Fig. 4.
The hypothesis of Poisson distributed data and consideration of image model allowed us to derive objective function, which can be used with different object models that has been optimized. There were described two commonly used object models, i.e., Gauss and Moffat and there was also introduced more complicated model, which suppose combination of interference and either interference or focusing phenomena. For the purpose of the objective function optimization, there were used different approaches. The first one was application of the Matlab fmincon function which is based on classic optimization algorithms. The second approach was application of optimization heuristics like cuckoo search, harmony search, etc.
In this article, there were analyzed three different cuts of used astronomical image, where each cut contains one astronomical object represented by PSF with different maximum intensity. These PSFs were modeled using previously mentioned approaches and algorithms. From the results, Table 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 and 15 it is obvious that the combined model of interference and focusing is better for fitting of astronomical objects than the two simpler models and the second combined model, i.e., interference plus turbulence. When the optimization methods are compared, then it can be said that the fmincon function is suitable for the first two simple models, but it is not acceptable for the third used model. It is also not acceptable from the point of view of the \(\phi _{\mathrm {std}}\) values, where the fmincon functions have higher values than the optimization heuristics for all used object models. When the \(\phi _{\mathrm {min}}\) and \(\phi _{\mathrm {std}}\) values are compared then as the best heuristic can be marked HS algorithm, but from results, stability point of view gives the best result CS algorithm.
Results and approaches presented in this article are supposed to find their use in our future work, which will be focused on further research of astronomical images where we would like analyze if these images can be distributed by the different law than the Poisson one. Results will also find its use in further research focused on pair, triplet and quadruplet interactions of astronomical objects and their deconvolution.
Notes
Acknowledgments
This work has been supported by the financial support from specific university research MSMT No 20/2013, research project MSM 6046137306 of the Ministry of Education, Youth and Sports of the Czech Republic and by the grant No. GA1425251S Nonlinear imaging systems with spatially variant point spread function of the Czech Science Foundation.
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