Contrast transformation cost function

The intensity channel’s contrast is augmented using local area enhancement models, including local mean and standard deviation. The subsequent function represents the contrast enhancement transfer function. (1)

In the aforementioned equation, T_f denotes the transformation function that enhances the intensity of the original image intensity value Ψ, which comprises M rows and N columns. While traditional techniques like histogram equalization yield superior results, they are computationally intensive. In contrast, we employed a modified statistical method, as outlined in [24], which necessitates fewer computations than the original proposed in [25]. The subsequent function illustrates enhancements in pixel-level intensity. (2)

In the aforementioned equation, κ(i, j) represents the original pixel intensity level, l(i, j) denotes the enhanced version, M signifies the global mean, and (i, j) indicates the central pixel upon which the operation is executed. ϑ(i, j) and ξ(i, j) represent the estimated local mean and standard deviation, respectively, computed from the neighboring pixels of k × k region. The non-zero value of ε enables the utilization of zero standard deviation while employing γ, allowing for the selection of a fraction of the mean value for subtraction. The decision variables, ε and γ, remain constant throughout the processing of an individual image. The contrast enhancement transformation function relies on the automatic estimation of decision variables. This function is employed to modify the intensity of the intensity channel at the pixel level. (3) here, R × S represents the dimension of the image. Mainly the objective function is based on the entropy H(Ψ*) and intensity E(Ψ*) functions of enhanced intensity channel. Sobel edge filter is used for detection of edges and edge intensity [32] using the following equations. (4) (5) (6)

Differential evolution

Differential Evolution (DE), introduced in [26], is an evolutionary method that utilizes global search parameters to identify optimal solutions. DE functions in two stages: 1) Initialization; 2) Evolution. During initialization, the population is produced randomly. Throughout the evolutionary process, the created population undergoes mutation, crossover, and selection procedures. The stages are reiterated iteratively to fulfill the selection criteria [27]. The fundamental flowchart of differential evolution is illustrated in Fig 4.

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Fig 4. Differential-Evolution process flowchart.

https://doi.org/10.1371/journal.pone.0315902.g004

Initialization: Initialization involves the generation of uniformly distributed population. Let , where Ξ^G is the search space and is a D−dimensional population vector with generation G and of size p i.e. . Following expression represents the uniform generation of population: (7) where {Φ_l, Φ_u} represents the lower and upper bounds of search space respectively.

Mutation: In mutation step, a mutation vector is generated for every target using the following expression: (8) where α ∈ {0, 1} is the scaling factor and {r1, r2, r3} ∈ {1, 2, …, p} are randomly selected but mutually different.

Crossover: After mutation, a new vector is generated, in crossover step, known as trial vector, . Crossover is performed among mutant vector and target vector using crossover probability C_p ∈ {0, 1}. (9) where i ∈ {1, 2, …, D}.

Selection: During the selection process, a comparison is performed between trial vector and target vector with respect to fitness criteria. The operation is performed with the following expression: (10)

In this paper we adopted the binary differential evolution, proposed in [28].

BAT optimization

BAT optimization is a metaheuristic algorithm [29]. The term “metaheuristic” combines two Greek words, meaning “to investigate a solution to a higher level”. In other words, metaheuristic techniques are computational models that try to enhance the candidate solutions to a near-optimal level through an iterative process in comparison to a given quality level. The key advantage of meta-heuristics is that they can explore very large spaces of contender solutions by making a few assumptions about the optimization problem. However, the limitation of metaheuristics is their inability to guarantee an optimal solution. Most metaheuristics are nature-inspired and population-based. Following are the key characteristics of the metaheuristic algorithms:

• Metaheuristics are approaches that lead to exploring the search space to extract the near-optimal solution.
• Meta-heuristic-based techniques range broadly from simple local search models to complex learning procedures.
• Meta-heuristic algorithms are non-deterministic approximations.
• Metaheuristics are not problem-specific.

BAT optimization is a metaheuristic algorithm inspired by the hunting behavior of bats. The model is based on the inherent traits of micro-bats, specifically their echolocation and biosonar capabilities. Before discussing the details of the BAT algorithm, first provide some details related to echolocation process of micro-bats.

Extraction of feature points

Feature points in an image are points that show significant presence, such as bright spots in a dark area or dark spots in bright areas. The ORB algorithm uses FAST for feature point extraction. In FAST, the image pixel is considered the corner point if it is significantly different from its neighborhood pixels. We divide the complete feature point extraction process into the following sub-processes.

1. At first, a pixel P is selected from the image with the brightness of I_p. By considering a brightness threshold T, the gray value of sixteen neighbor pixels in a circle is compared with the pixel P as shown in the Fig 5.
   If the difference of consecutive N points on the circle is greater than ε_th or less than ε_tl, the pixel P is considered as the feature point. (16) where I_x is the gray value of the point on the circle. Mostly, three-quarters of the total points on the circle is defined as the N. For example if at least 12 points around the circle exceeds the threshold, the center point is considered as corner otherwise it is rejected for feature point.
2. Feature Point Screening: In the original FAST algorithm, a feature point is selected by comparing the brightness value between the pixels; therefore, a large number of feature points are detected with no information related to direction. The modification of the original FAST algorithm in ORB is based on the Harris response value. The following equations calculate the Harris response for all features detected by the original FAST. (17) (18)
   In the above equations, R is the Harris response, M is a 2x2 matrix, w(x, y) is the image window function, while k is a constant ranging between [0.04 0.06]. I_xI_y are the feature point variations in the horizontal and vertical directions, respectively. However, the whole process is time consuming when applied to all points. Therefore, we can make a prediction based on the outcomes of four candidate points surrounding the central pixel, separated at 90-degree intervals. We reject a feature point without evaluating its neighboring points unless three out of four points exhibit sufficient gray level difference to verify all the points.
3. Creation of Image Scale Pyramid: The original FAST algorithm, used in the ORB algorithm, does not detect direction related information of feature points. However, the algorithm is modified for rotation invariance and scale invariance through using the grey level intensity centroid technique and by establishing image Gaussian pyramid respectively. At start, the moment of image block i.e. the orientation of the corner in a small image block can be defined as: (19) where α, β are the coordinates of the image pixel in the neighborhood of the feature point and λ_α,β represents the grey level intensity of the correspondent pixel. The centroid of image block is calculated by the following relation, (20) where υ₀₀ is the 0^th moment and represents the image block mass, while the centroid of the image block is represented by the 1^st moment of the image block i.e. (υ₁₀, υ₀₁). Finally, the orientation of the corner feature point is calculated with the following equation along with the correction if the feature point is dark with respect to its background. (21)
   The above relations make the FAST feature points rotation and scale invariance.

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Fig 5. Schematic representation of selection of neighboring pixels in FAST algorithm.

https://doi.org/10.1371/journal.pone.0315902.g005

Feature points descriptor

The feature descriptor’s goal is to characterize and distinguish the detected feature points. A feature descriptor is numerical information extracted from the image to distinguish one feature point from the others. In the ORB algorithm, an improved version of BRIEF [32] is used to calculate feature descriptors of the detected points. The BRIEF algorithm’s fundamental principle is to calculate the binary feature vector for all detected feature points using FAST. The BRIEF algorithm designates each feature point with a binary feature vector or feature descriptor, a string of 0 and 1, consisting of 128–512 bits. BRIEF’s core idea is based on the principle that it is possible to express the image neighborhood using a small amount of intensity contrast. (22) where λ_x is the gray level intensity of the pixel at position x around the feature point while λ_y is the gray level intensity at position y around the feature point. Moreover, Gaussian filtering is also applied at start to reduce the noise effect. Let’s random select a set of N pixels from a kxk neighborhood window with feature point at the center, normally N is taken as 128, 256 or 512. Lastly, an N-dimensional vector consisting of N binary strings is calculated as: (23)

In the BRIEF algorithm, only single pixel image neighborhood is adopted that is susceptible to noise. In order to solve this problem, a mxm pixel sub-window is used in the ORB with a neighborhood window of size kxk. In this case, the selection of sub-window obeys the Gaussian Distribution. The original BRIEF is undirected and is not invariant to rotation. A 2n matrix is defined for any n criterion feature set at position (x_i, y_i), (24)

A corrected Q matrix is constructed using the direction of neighborhood ϑ and the corresponding rotation matrix, Υ_ϑ, Q_ϑ = Υ_ϑQ. And the final Steered BRIEF descriptor is: (25)

Feature matching

Feature point matching entails the assessment of similarity between the feature points of two distinct images. After calculating the feature point descriptors, the subsequent step entails estimating the matching of feature points. Numerous feature matching algorithms exist, including Brute Force and Hamming distance. Hamming distance is utilized in BRIEF to quantify the number of differing characters between two strings of same length. Nonetheless, Brute Force is simpler since it calculates the distance between each feature point of image I_t and every feature descriptor of image I_t+ 1, and matching features based on the shortest distance. The FLANN (Fast Library for Approximate Nearest Neighbors) method is utilized alongside ORB in certain visual SLAM models [33].