Optics is a branch of physics which involves the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Practical applications of optics are found in a variety of technologies and everyday objects, including mirrors, lenses, telescopes, microscopes, etc. A curved or spherical mirror is a mirror with a curved reflective surface, which may be either convex (bulging outward) or concave (bulging inward). Most curved mirrors have surfaces that are shaped like part of a sphere. The behaviour of light reflected by a curved mirror is subject to the same laws as that of plane mirrors, which are known as the laws of reflection. The First law: the incident ray, the reflected ray, and the normal all lie on the same plane. The Second law: the angle between the incident ray and the normal is equal to the angle between the reflected ray and the normal.
A concave mirror has a reflecting surface that bulges inward (away from the incident light). Concave mirrors reflect light inward to focal point. They are used to focus light (see Figure 1a) . Concave mirrors show different image types depending on the distance between the object and the mirror. These mirrors are called "converging" because they tend to collect light that falls on them, refocusing parallel incoming rays toward a focus. Concave mirrors are used in some telescopes. The image on a concave mirror is virtual, upright and larger than the object if it is placed between focal point and mirror. If the object is placed beyond the focal point, its image is always real and inverted. But the image size depends on the position of the object (see Figure 2a).
Figure 1. Light rays traveling parallel to the axis of a) a concave mirror are reflected so that they all pass approximately through a common focal point f, b) a convex mirror are reflected so that they appear to come from a focal point f, located behind the mirror
A convex mirror is a curved mirror in which the reflective surface bulges toward the light source. Convex mirrors reflect light outwards and always form a virtual image, since the focus (f) and the centre of curvature (r) are both imaginary points "inside" the mirror, which cannot be reached (see Figure 1b). A collimated beam of light diverges after reflection from a convex mirror, since the normal to the surface differs with each spot on the mirror. The image on a convex mirror is always virtual, upright and smaller than the object. The ray diagram in Figure 2b represents how an image is formed by a spherical convex mirror. The figure uses two rays but remember that there are an infinite number of rays. Indeed, it is adequate to draw only two rays to locate the image of a point on an object. Therefore, we have taken just two rays coming from the top of object (candles in our case) and traced them as shown in Figure 2. When these two rays are extended backward (or forward in case of a concave mirror), their intersection locates the image point position. In a similar way, the two rays coming from each point of the object can be traced and the corresponding image point can be obtained, accordingly. In this way we can see the whole image of the object.
|Figure 2. Effect on image of object's position relative to mirror focal point f|
Optics Inspired Optimization (OIO) is an optics inspired population based evolutionary algorithm in which it is assumed that a number of artificial light points (points in Rn+1 whose mapping in Rn are potential solutions to the problem) are sitting in front of an artificial wavy mirror reflecting their images. OIO treats the surface of the function to be optimized as the reflecting mirror composed of peaks and valleys. Each peak is treated as a convex reflective surface and each valley is treated as a concave reflective surface. In this way, the artificial ray glittered from the artificial light point is reflected back artificially by the function surface, given that the reflecting surface is a part of a peak or a part of a valley, and the artificial image point (a new point in Rn+1 which is mapped in Rn as a new solution in the search domain) is formed upright (toward the light point position in the search space) or inverted (outward the light point position in the search space).
Figure 3 demonstrates the idea behind OIO to generate new solutions in the one dimensional search space. In this figure it is assumed that an artificial light point (object) in the joint search and objective space is in front of the function surface (mirror) in a particular distance from vertex (values on the X-axis form the search/solution space and values on the f(X)-axis form the objective space. The set of all points in the X-f(X)-coordinate system forms the joint search and objective space). The artificial image is formed in the joint search and objective space and its position and height is determined through mirror and magnification equations. Finally, mapping the artificial image position into the search space results the position of the new solution in the search space. Depending on the type of the reflecting part of the function surface (convex or concave) and depending on the position of the artificial light point (object) in the joint search and objective space, there are four different situations under which new solutions are generated (see Figure 3). The idea behind OIO is thus simple. Given an individual solution O in the population, a different solution F (vertex point) is picked randomly from the population. If F is worse than O, in terms of function/objective value, then it is assumed that the surface is convex and the new solution is generated upright somewhere toward O, on the line connecting O and F (see Figure 3a). If F is better than O then it is assumed that the surface is concave and the new solution is generated upright toward (see Figure 3b) or inverted outward (see Figure 3c and 3d) O, on the line connecting O and F in the search space.
As can be seen from the conceptual model of Figure 3, the logic of OIO to generate new solutions is able to serve exploration and exploitation during the search for an optimum. Exploration can be controlled relatively via allowing a larger jump in the solution space (see Figure 3b and 3c) while exploitation can be carried out by allowing a relatively smaller jump over the base solutions (see Figure 3a and 3d).
Figure 3. The idea behind generation of the new solutions in OIO
Figure 4 puts forward the entire flowchart of OIO for minimizing an unconstrained numerical function.
Figure 4. Flowchart of OIO algorithm
To find more details on the whole mechanizm of OIO and its mthematics please see the following paper.
|To download an introductory paper on OIO|
|To download the Powerpoint slides of OIO|
|To download the Matlab source code of OIO for unconstrained optimization|
|To download a schematic mechanizm of OIO|
Researches on Optics Inspired Optimization
Lalwani P, Banka H, C Kumar (2016). CRWO: Clustering and routing in wireless sensor networks using optics inspired optimization. Peer to Peer Networking and Applications, DOI 10.1007/s12083-016-0531-7.
Badrloo S, Husseinzadeh Kashan (2015). A new method for the quadratic assignment problem based on Optics Inspired Optimization.International Conference on Modern Research in Management and Industrial Engineering, Iran (in Persian).
Badrloo S, Husseinzadeh Kashan (2015). A new method for the travelling salesman problem based on Optics Inspired Optimization.International Conference on Modern Research in Management and Industrial Engineering, Iran (in Persian),
Moghadasi M (2015). Design of Image Processing methods using League Championship Algorithm and Optics Inspired Optimization. Ms.C Thesis, Azad University, Science and Research branch, Iran (in Persian).
Badrloo S (2015). A new method for solving combinatorial optimization problems with permutation based solution structure using optics inspired optimization. Ms.C Thesis, Azad University, Science and Research branch, Iran (in Persian).
Husseinzadeh Kashan A (2015). A New Metaheuristic for Optimization: Optics Inspired Optimization (OIO). Computers & Operations Research, 55, 99-125
Husseinzadeh Kashan A (2015). An Effective Algorithm for Constrained Optimization Based on Optics Inspired Optimization (OIO). Computers Aided Design, 63, 52-72.
Husseinzadeh Kashan A (2012). A New Metaheuristic for Optimization: Optics Inspired Optimization (OIO). Technical paper, Department of Industrial Engineering, Faculty of Engineering, Tarbiat Modares University, Tehran, Iran.