GA Based Rational Cubic B-Spline Representation for Still Image Interpolation

In this paper, an image interpolation scheme is designed for 2D natural images. A local support rational cubic spline with control parameters, as interpolatory function, is being optimized using Genetic Algorithm (GA). GA is applied to determine the appropriate values of control parameter used in the description of rational cubic spline. Three state-of-the-art Image Quality Assessment (IQA) models with traditional one are involved for comparison with existing image interpolation schemes and perceptual quality check of resulting images. The results show that the proposed scheme is better than the existing ones in comparison.


Introduction
Interpolation plays a significant role in the field of digital image processing; applications like image sample data rate conversion, geometrical transformations, visualization etc. Still image interpolation is one of the three categories of digital image interpolation which are classified according to the types of digital images to be interpolated. Others are named as multi frame image interpolation and image sequence/video interpolation. The techniques used for interpolating still images deal only with single image frames. In order to interpolate each digital image, these techniques used only the spatial information within each image frame. Whereas the techniques specified for multi frame image interpolation use several image frames to interpolate digital image frames of a scene. However in case of image sequence/video interpolation both spatial and temporal data informations are used to get improved image interpolation results. Further literature on still image, multi frame image and image sequence/video interpolation is respectively available in Jensen and Anastassiou (1995), Kim and Su (1993) and Chang et al. (2001). Since 1970s, several piecewise polynomials have been considered to investigate the problems related to interpolation in the field of image processing by numerous writers, among whom are Hou and Andrews (1978), Parker et al. (1983) and Keys (1981). Piecewise cubic convolution along with high order B-splines are the most popular image interpolation functions. As splines did not have any concern to realistic stochastic models of digital images, so they are considered as a perfect fit for image processing.
In this paper, a local support rational cubic spline with control parameter is presented to investigate problems related to still image interpolation. The rational spline is optimized using Genetic Algorithm (GA). GA is applied to determine the appropriate values of control parameters of rational cubic spline. To examine smooth and improved quality results of interpolated images three structure similarity based Image Quality Assessment (IQA) metrics with traditional one are exercised here.
Rest of the paper is arranged as follows. The process of construction of local support rational spline and its extended interpolatory functions is presented in Section 2. Optimal interpolation using GA is addressed in Section 3 and Section 4. Section 5 includes all the experimental results and assessments. Finally concluding remarks are made in Section 6.

Rational Cubic B-Spline
In this section the construction method of local support based rational spline presented by Gregory and Sarfraz (1990) is reviewed and extended, in some way, to its one and two dimensional interpolatory functions. Let be the given set data points defined over the interval where be the partition of A piecewise rational cubic function is defined over each sub interval as: (1) where and be the tension parameters defined in each sub interval . The rational cubic function (1) has the following interpolating properties: and (2) where denotes the derivatives with respect to and are the first derivatives at the knots , . For each sub interval , the rational cubic function (1) can be written in the form (3) where the function are the rational basis functions and found to be non-negative for with ∑ and defined here as: , , , , with co-efficient of interpolation which are defined as: In order to construct local support basis for rational cubic B-spline representation, let us first review the method presented by Gregory and Sarfraz (1990). Let the additional knots and be introduced on both sides the interval with as control parameters defined on this extended partition. A rational cubic spline function , can be introduced, such that On the remaining two intervals will have the rational cubic form defined as: where the function are defined same as above. The requirement that the function is continuous up to second order, in particular at , and , may satisfy the following properties: where . The graphical view of the rational cubic function is shown in Figure 1. Now, to determine the local support basis for rational cubic spline, take the difference of as: where the function are themselves determined by getting the difference of the functions as: By the definition of the function in equation (8), it can be noticed that The graphical view of the rational cubic function is shown in Figure 2.
Therefore, an obvious representation of the rational cubic spline basis on an subinterval can be computed from equations (4) to equation (7) as: The graphical view of the rational basis function , is shown in Figure 3. Furthermore, both the 1-dimensional rational cubic B-spline and its extended 2dimensional interpolatory functions will be formulated here using the local support basis (10). Let be the required rational spline interpolatory function defined as: where are coefficient of interpolation those can be determined from some set of discrete data at certain given spatial points. Substituting the value of from equation (10) in equation (11)   (12) where So, the vector form of equation (12) can be presented as: (13) where , and with Next, to extend the above one dimensional case to its two dimensions, equation (11) elaborates in the following expression is defined same as above. In the same manner ̃ ( ̃) is the rational cubic B-spline basis corresponds to the set of knots ̃ with tension parameter ̃ Finally, for , ̃ [ ̃ ̃ the function ( ̃) is vector form for 2-dimension rational spline is presented as: and the matrix is a corresponding extension of , defined as: ] and is same as defined in equation (13) with corresponding extended matrix ̃ .

Problem Optimization using Genetic Algorithms (GA)
Genetic Algorithm (GA) is primitively suggested by Holland (1975). It represents a family of parallel adaptive search techniques. The techniques are based on the procedure of natural selection. GAs are practiced to produce globally optimized solutions in quick and effective way. Even in large solution spaces they perform outstandingly well as compared to other traditional optimization techniques. For survival in the large solution spaces GA strongly relies on three of its system operations; the selection, crossover and mutation. Through selection a pair of bit strings (parent bit strings) is chosen from the solution space or initial population which will further divided into two or more segments through the crossover operator. Crossover then combines these segments to produce new pair of bit strings (off springs) for next population. Mutation helps to reduce the possibility of GA to fall into local optimum. It carries out random changes in the slat of bit strings through the operations of bit shifting, rotation and inversion. Figure 4 illustrates both the crossover and mutation operators for some selected chromosomes (bunch of bit strings).
Since the this work is aimed to obtain an optimal solution for resulting interpolated images produced by using the rational cubic spline with control parameters and ̃ for and , GA is used here to search for the appropriate values of control parameters. Now before starting the procedure for GA, several terms with system parameters are needed to be fixed in advance.

A. System Parameters:
Here, population size is fixed at 30 with maximum number of iteration (generations) of GA is fixed as 10. Initial population is selected randomly which contains 30 chromosomes (bit strings) where each single gene represents the value for control parameters and ̃ .

B. Objective Function:
The proposed objective function is formulated by the sum square error which is defined for the image spatial data. For the original image spatial data metric and resulting image spatial data matrix , the proposed objective function is defined as: C. Stopping Criterion: The process will be stopped if no encouraging change in values of objective function is observed for a definite number of iterations.

Proposed Image Interpolation Scheme
An image interpolation scheme is designed using the rational cubic spline (15) with control parameters and ̃ for and , in its description. The scheme is made up of several steps which are elucidated here one by one. Firstly the spatial data of a selected original image is collected through any existing image decoding technique. In the next step, all the system parameters of GA are initialized to reach the optimized values of parameters and ̃ in the description of rational cubic spline (15). The initial population is taken randomly with potential compound of values of control parameters. Each control parameter corresponds to a single gene is a bit string or a chromosome. In the formulation if a gene is equal to 1, a value would be associated to the corresponding control parameter and if a gene is equal to 0, no value would be assigned to the corresponding control parameter. Successive implementation of GA search operations to the selected population, lead us to optimal values of and ̃ such that the fitness function (17) achieves its minimum value. Finally the spatial image data is interpolated using the optimized rational cubic spline.

Results and Discussion
In this section, experiments are done to evaluate the objective and subjective performance of the proposed image interpolation scheme. Here one traditional objective IQA metric, Peak-Signal-to-Noise-Ration (PSNR) with three structure similarity subjective IQA   Table 1 depicts the PSNR, FSIM, SSIM and MS-SSIM values for all three colored natural image produced by using proposed image interpolation schemes and all three existing schemes are considered for comparison. From the outcomes shown in the table one can easily notice that the proposed scheme is better than the other interpolation schemes. Moreover, Figure 5 shows the SSIM map of 'Flower', 'Plane' and 'Pepper' with their original and resulting interpolated images.

Conclusion
A local support rational cubic spline with control parameter is investigated for problems related to still image interpolation. A soft computing technique, GA is utilized here to optimize rational spline. It helps find the appropriate values of respective control parameters. An image interpolation scheme is designed using the two dimensional optimal rational spline interpolatory function. FSIM, SSIM and MS-SSIM indices along with traditional PSNR are exercised to examine the quality of resulting interpolated images. The results show that our proposed interpolation scheme performs better as compared to the three existing schemes. The processing time for the images Flower, Plane and pepper in seconds are 34.415, 34.592 and 32.271 respectively in between two iterations of the proposed algorithm.