TSWJ The Scientific World Journal 1537-744X Hindawi Publishing Corporation 602453 10.1155/2014/602453 602453 Research Article Monotone Data Visualization Using Rational Trigonometric Spline Interpolation Ibraheem Farheen 1 Hussain Maria 2 http://orcid.org/0000-0002-9335-3117 Hussain Malik Zawwar 3 Bellouquid A. Elsadany A. A. 1 National University of Computer and Emerging Sciences Lahore Pakistan nu.edu.pk 2 Department of Mathematics Lahore College for Women University Lahore 54600 Pakistan lcwu.edu.pk 3 Department of Mathematics University of the Punjab Lahore 54590 Pakistan pu.edu.pk 2014 3 4 2014 2014 03 01 2014 05 02 2014 3 4 2014 2014 Copyright © 2014 Farheen Ibraheem et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Rational cubic and bicubic trigonometric schemes are developed to conserve monotonicity of curve and surface data, respectively. The rational cubic function has four parameters in each subinterval, while the rational bicubic partially blended function has eight parameters in each rectangular patch. The monotonicity of curve and surface data is retained by developing constraints on some of these parameters in description of rational cubic and bicubic trigonometric functions. The remaining parameters are kept free to modify the shape of curve and surface if required. The developed algorithm is verified mathematically and demonstrated graphically.

1. Introduction

The technique or algorithm employed in creating images, diagrams, or animations for imparting a piece of information is termed as visualization. It has a key role to play in different fields like science, engineering, education, and medicine as it can aid experts in identifying and interpreting different patterns and artifacts in their data and provide a three-dimensional display of data for the solution of a wide range of problems.

The methods used to obtain visual representations from abstract data have been in practice for a long time. However, physical quantities often emanate distinctive features (such as positivity, convexity, and monotonicity) and it becomes imperative that the visual model must contain the shape feature to fathom the physical phenomenon, the scientific experiment, and the idea of the designer. Spline interpolating functions play elemental role in visualizing shaped data. This paper specifically addresses the problem of visualizing monotone curve and surface data.

Monotonicity is an indispensable characteristic of data stemming from many physical and scientific experiments. The relationship between the partial pressure of oxygen and percentage dissociation of hemoglobin, consumption function in economics, concentration of atrazine and nitrate in shallow ground waters, and approximation of couples and quasi couples are few phenomena which exhibit monotone trend.

Efforts have been put in by many researchers and a variety of approaches has been proposed to solve this eminent issue . Cripps and Hussain  visualized the 2D monotone data by Bernstein-Bézier rational cubic function. The authors in  converted the Bernstein-Bézier rational cubic function to C 1 cubic Hermite by applying the C 1 continuity conditions at the end points of interval. The lower bounds of weights functions were determined to visualize monotone data as monotone curve. Hussain and Sarfraz  have conserved monotonicity of curve data by rational cubic function with four shape parameters, two of which were set free and two were shape parameters. Data dependent constraints on shape parameters were developed which assure the monotonicity but one shape parameter is dependent on the other which makes it economically very expensive. Rational cubic function with two shape parameters suggested by Sarfraz  sustained monotonicity of curves but lacked the liberty to amend the curve which makes it inappropriate for interactive design. Piecewise rational cubic function was used by M. Z. Hussain and M. Hussain  to visualize 2D monotone data by developing constraints on the free parameters in the specification of rational cubic function. The authors also extended rational cubic function to rational bicubic partially blended function. Simple constraints were derived on the free parameters in the description of rational bicubic partially blended patches to visualize the 3D monotone data. Three kinds of monotonicity preservation of systems of bivariate functions on triangle were defined and studied by Floater and Peña . Sarfraz et al.  developed constraints in the specification of a bicubic function to visualize the shape of 3D monotone data.

This paper is a noteworthy addition in the field of shape preservation when the data under consideration admits monotone trend. The suggested algorithm offers numerous advantages over the prevailing ones. Orthogonality of sine and cosine function compels much smoother visual results as compared to algebraic spline. Derivative of the trigonometric spline is much lower than that of algebraic spline. Moreover, trigonometric splines play an instrumental role in robotic manipulator path planning.

The remainder of the paper is structured as follows. Section 2 is devoted to reviewing the rational trigonometric cubic function developed in . In Section 3, rational trigonometric cubic function is extended to rational trigonometric bicubic function. Section 4 aims to develop monotonicity preserving constraints for 2D data. Section 5 submits a solution to shape preservation of 3D monotone data. In Section 6, numerical examples have been demonstrated. Section 7 draws the conclusion and significance of this research.

2. Rational Trigonometric Cubic Function

In this section, rational trigonometric cubic function  is reviewed.

Let { ( x i , f i ) , i = 0,1 , 2 , , n } be the given set of data points defined over the interval [ a , b ] , where a = x 0 < x 1 < x 2 < < x n = b . Piecewise rational trigonometric cubic function is defined over each subinterval I i = [ x i , x i + 1 ] as (1) S i ( x ) = p i ( θ ) q i ( θ ) , (2) p i ( θ ) = α i f i ( 1 - sin θ ) 3 + { β i f i + 2 h i α i d i π } × sin θ ( 1 - sin θ ) 2 + { γ i f i + 1 - 2 h i δ i d i + 1 π } × cos θ ( 1 - cos θ ) 2 + δ i f i + 1 ( 1 - cos θ ) 3 , q i ( θ ) = α i ( 1 - sin θ ) 3 + β i sin θ ( 1 - sin θ ) 2 + γ i cos θ ( 1 - cos θ ) 2 + δ i ( 1 - cos θ ) 3 , where θ = ( π / 2 ) ( ( x - x i ) / h i ) , h i = x i + 1 - x i .

The rational trigonometric cubic function (1) is C 1 ; that is, it satisfies the following properties: (3) S ( x i ) = f i , S ( x i + 1 ) = f i + 1 , S ( x i ) = d i , S ( x i + 1 ) = d i + 1 . Here d i and d i + 1 are derivatives at the end points of the interval I i = [ x i , x i + 1 ] . The parameters α i and δ i are real numbers used to modify the shape of the curve.

3. Rational Trigonometric Bicubic Partially Blended Function

Let { ( x i , y j , F i , j ) , i = 0 , 1 , 2 , , n - 1 ; j = 0 , 1 , 2 , , m - 1 } be the 3D regular data set defined over the rectangular mesh I = [ a , b ] × [ c , d ] , let p : a = x 0 < x 1 < < x m = b be a partition of [ a , b ] , and let q : a = y 0 < y 1 < < y n be a partition of [ c , d ] . Rational trigonometric bicubic function which is an extension of rational trigonometric cubic function (1) is defined over each rectangular patch [ x i , x i + 1 ] × [ y j , y j + 1 ] , where i = 0 , 1 , 2 , , n - 1 ; j = 0 , 1 , 2 , , m - 1 , as (4) S ( x , y ) = - A F B T , where (5) F = ( 0 S ( x , y j ) S ( x , y j + 1 ) S ( x i , y ) S ( x i , y j ) S ( x i , y j + 1 ) S ( x i + 1 , y ) S ( x i + 1 , y j ) S ( x i + 1 , y j + 1 ) ) , A = [ - 1 a 0 ( θ ) a 1 ( θ ) ] , B = [ - 1 b 0 ( θ ) b 1 ( θ ) ] , a 0 = cos 2 θ , a 1 = sin 2 θ , b 0 = cos 2 φ , b 1 = sin 2 φ . S ( x , y j ) , S ( x , y j + 1 ) , S ( x i , y ) , and S ( x i + 1 , y ) are rational trigonometric bicubic functions defined on the boundary of rectangular patch [ x i , x i + 1 ] × [ y j , y j + 1 ] as (6) S ( x , y j ) = ( A 0 ( 1 - sin θ ) 3 + A 1 sin θ ( 1 - sin θ ) 2 + A 2 cos θ ( 1 - cos θ ) 2 + A 3 ( 1 - cos θ ) 3 ) × ( q 1 ( θ ) ) - 1 ,

where (7) A 0 = α i , j F i , j , A 1 = β i , j F i , j + 2 α i , j h i F i , j x π , A 2 = γ i , j F i + 1 , j - 2 δ i , j h i F i + 1 , j x π , A 3 = δ i , j F i + 1 , j , q 1 ( θ ) = α i , j ( 1 - sin θ ) 3 + β i , j sin θ ( 1 - sin θ ) 2 + γ i , j cos θ ( 1 - cos θ ) 2 + δ i , j ( 1 - cos θ ) 3 , (8) S ( x , y j + 1 ) = ( B 0 ( 1 - sin θ ) 3 + B 1 sin θ ( 1 - sin θ ) 2 + B 2 cos θ ( 1 - cos θ ) 2 + B 3 ( 1 - cos θ ) 3 ) × ( q 2 ( θ ) ) - 1 ,

where (9) B 0 = α i , j + 1 F i , j + 1 , B 1 = β i , j + 1 F i , j + 1 + 2 α i , j + 1 h i F i , j + 1 x π , B 2 = γ i , j + 1 F i + 1 , j + 1 - 2 δ i , j + 1 h i F i + 1 , j + 1 x π , B 3 = δ i , j + 1 F i + 1 , j + 1 , q 2 ( θ ) = α i , j + 1 ( 1 - sin θ ) 3 + β i , j + 1 sin θ ( 1 - sin θ ) 2 + γ i , j + 1 cos θ ( 1 - cos θ ) 2 + δ i , j + 1 ( 1 - cos θ ) 3 , (10) S ( x i , y ) = ( C 0 ( 1 - sin φ ) 3 + C 1 sin φ ( 1 - sin φ ) 2 + C 2 cos φ ( 1 - cos φ ) 2 + C 3 ( 1 - cos φ ) 3 ) × ( q 3 ( φ ) ) - 1 ,

where (11) C 0 = α ^ i , j F i , j , C 1 = β ^ i , j F i , j + 2 α ^ i , j h j F i , j y π , C 2 = γ ^ i , j F i , j + 1 - 2 δ ^ i , j h j F i , j + 1 y π , C 3 = δ ^ i , j F i , j + 1 , q 3 ( φ ) = α ^ i , j ( 1 - sin φ ) 3 + β ^ i , j sin φ ( 1 - sin φ ) 2 + γ ^ i , j cos φ ( 1 - cos φ ) 2 + δ ^ i , j ( 1 - cos φ ) 3 , (12) S ( x i + 1 , y ) = ( D 0 ( 1 - sin φ ) 3 + D 1 sin φ ( 1 - sin φ ) 2 + D 2 cos φ ( 1 - cos φ ) 2 + D 3 ( 1 - cos φ ) 3 ) × ( q 4 ( φ ) ) - 1 ,

where (13) D 0 = α ^ i + 1 , j F i + 1 , j , D 1 = β ^ i + 1 , j F i , j + 1 + 2 α ^ i + 1 , j h j F i + 1 , j y π , D 2 = γ ^ i + 1 , j F i + 1 , j + 1 - 2 δ ^ i + 1 , j h j F i + 1 , j + 1 y π , D 3 = δ ^ i + 1 , j F i + 1 , j + 1 , q 4 ( φ ) = α ^ i + 1 , j ( 1 - sin φ ) 3 + β ^ i + 1 , j sin φ ( 1 - sin φ ) 2 + γ ^ i + 1 , j cos φ ( 1 - cos φ ) 2 + δ ^ i + 1 , j ( 1 - cos φ ) 3 .

4. Monotone Curve Interpolation

Monotonicity is a crucial shape property of data and it emanates from many physical phenomenon, engineering problems, scientific applications, and so forth, for instance, dose response curve in biochemistry and pharmacology, approximation of couples and quasi couples in statistics, empirical option pricing model in finance, consumption function in economics, and so forth. Therefore, it is customary that the resulting interpolating curve must retain the monotone shape of data.

In this section, constraints on shape parameters in the description of rational trigonometric cubic function (1) have been developed to preserve 2D monotone data.

Let { ( x i , f i ) , i = 0,1 , 2 , , n } be the monotone data defined over the interval [ a , b ] ; that is, (14) f i < f i + 1 , Δ i = f i + 1 - f i h i > 0 , i = 0,1 , 2 , , n - 1 , d i > 0 , i = 0,1 , 2 , , n . The curve will be monotone if the rational trigonometric cubic function (1) satisfies the condition (15) S i ( x ) , x [ x i , x i + 1 ] , i = 0,1 , 2 , , n - 1 . Now, we have (16) S i ( x ) = π 2 h i ( q i ( θ ) ) 2 × { ( 1 - sin θ ) 4 sin θ cos θ B 0 + cos 2 θ ( 1 - cos θ ) 2 ( 1 - sin θ ) 2 B 1 + cos θ ( 1 - cos θ ) 3 ( 1 - sin θ ) 2 B 2 + cos θ ( 1 - sin θ ) 5 B 3 + sin θ cos 2 θ ( 1 - cos θ ) 2 × ( 1 - sin θ ) B 4 + sin θ cos θ ( 1 - sin θ ) ( 1 - cos θ ) 3 B 5 + sin θ ( 1 - cos θ ) 2 ( 1 - sin θ ) 3 B 6 + sin 2 θ ( 1 - sin θ ) 2 ( 1 - cos θ ) 2 B 7 + sin θ cos θ ( 1 - cos θ ) 4 B 8 + sin θ ( 1 - cos θ ) 5 B 9 + sin θ cos θ ( 1 - cos θ ) ( 1 - sin θ ) 3 B 10 + sin 2 θ cos θ ( 1 - cos θ ) ( 1 - sin θ ) 2 B 11 } , where (17) B 0 = 2 h i d i α i 2 π , B 1 = ( 3 α i γ i - β i δ i ) Δ i + 2 d i α i γ i π - ( 3 α i - β i ) 2 δ i d i + 1 π , B 2 = ( β i - 3 α i ) Δ i - 2 d i α i π , B 3 = 2 h i α i d i π , B 4 = 2 β i γ i Δ i - 4 δ i β i d i + 1 π - 4 α i γ i d i π , B 5 = β i Δ i - 2 α i d i π , B 6 = ( 3 δ i - γ i ) Δ i + 2 α i d i + 1 π , B 7 = β i ( γ i - 3 δ i ) Δ i + 2 α i γ i d i π + ( β i - 3 α i ) 2 δ i d i + 1 π , B 8 = 2 h i δ i d i + 1 π , B 9 = 2 h i δ i d i + 1 π , B 10 = γ i Δ i - 2 δ i d i + 1 π , B 11 = β i γ i Δ i - 2 β i δ i d i + 1 π - 2 α i γ i d i π . The denominator in (16) is a squared quantity, thus, positive. Hence, monotonicity of rational trigonometric cubic spline depends upon the positivity of numerator which can be attained if the coefficients B i , i = 0 , 1 , 2 , , 11 of the trigonometric basis functions are all positive. This yields the following result: (18) β i > 2 α i d i π Δ i , γ i > 2 δ i d i + 1 π Δ i .

The above discussion can be summarized as follows.

Theorem 1.

The C 1 piecewise trigonometric rational cubic function (1) preserves the monotonicity of monotone data if in each subinterval I i = [ x i , x i + 1 ] , the parameters β i and γ i satisfy the following sufficient conditions: (19) β i > 2 α i d i π Δ i , γ i > 2 δ i d i + 1 π Δ i . The above constraints can be rearranged as (20) β i = u i + max { 0 , 2 α i d i π Δ i } , u i > 0 ; γ i = v i + max { 0 , 2 δ i d i + 1 π Δ i } , v i > 0 .

Algorithm 2.

h h h h h h h h h

Step 1. Take a monotone data set { ( x i , f i ) : i = 0,1 , 2 , , n } .

Step 2. Use the Arithmetic Mean Method  to estimate the derivatives d i ’s at knots x i ’s (note: Step 2 is only applicable if data is not provided with derivatives).

Step 3. Compute the values of parameters β i ’s and γ i ’s using Theorem 1.

Step 4. Substitute the values of variables from Steps 1–3 in rational trigonometric cubic function (1) to visualize monotone curve through monotone data.

5. Monotone Surface Interpolation

Let { ( x i , y j , F i , j ) , i = 0 , 1 , 2 , , n - 1 ; j = 0 , 1 , 2 , , m - 1 } be the monotone data set defined over the rectangular mesh I = [ x i , x i + 1 ] × [ y j , y j + 1 ] such that (21) F i , j < F i + 1 , j , F i , j < F i , j + 1 , F i , j x > 0 , F i , j y > 0 , Δ i , j > 0 , Δ ^ i , j > 0 . Now, surface patch (4) is monotone if the boundary curves defined in (6)–(12) are monotone.

Now, S ( x , y j ) is monotone if S i ( x , y j ) > 0 , where (22) S i ( x , y j ) = π 2 h i ( q 1 ( θ ) ) 2 × { ( 1 - sin θ ) 4 sin θ cos θ R 0 + cos 2 θ ( 1 - cos θ ) 2 ( 1 - sin θ ) 2 R 1 + cos θ ( 1 - cos θ ) 3 ( 1 - sin θ ) 2 R 2 + cos θ ( 1 - sin θ ) 5 R 3 + sin θ cos 2 θ ( 1 - cos θ ) 2 ( 1 - sin θ ) R 4 + sin θ cos θ ( 1 - sin θ ) ( 1 - cos θ ) 3 R 5 + sin θ ( 1 - cos θ ) 2 ( 1 - sin θ ) 3 R 6 + sin 2 θ ( 1 - sin θ ) 2 ( 1 - cos θ ) 2 R 7 + sin θ cos θ ( 1 - cos θ ) 4 R 8 + sin θ ( 1 - cos θ ) 5 R 9 + sin θ cos θ ( 1 - cos θ ) ( 1 - sin θ ) 3 R 10 + sin 2 θ cos θ ( 1 - cos θ ) ( 1 - sin θ ) 2 R 11 } , with (23) R 0 = 2 h i F i , j x α i , j 2 π , R 1 = ( 3 α i , j γ i , j - β i , j δ i , j ) Δ i , j + 2 F i , j x α i , j γ i , j π - ( 3 α i , j - β i , j ) 2 δ i F i + 1 , j x π , R 2 = ( β i , j - 3 α i , j ) Δ i , j - 2 F i , j x α i , j π , R 3 = 2 h i α i , j F i , j x π , R 4 = 2 β i , j γ i , j Δ i , j - 4 δ i , j β i , j F i + 1 , j x π - 4 α i , j γ i , j F i , j x π , R 5 = β i , j Δ i , j - 2 α i , j F i , j x π , R 6 = ( 3 δ i , j - γ i , j ) Δ i , j + 2 α i , j F i + 1 , j x π , R 7 = β i , j ( γ i , j - 3 δ i , j ) Δ i , j + 2 α i , j γ i , j F i , j x π + ( β i , j - 3 α i , j ) 2 δ i , j F i + 1 , j x π , R 8 = 2 h i δ i , j F i + 1 , j x π , R 9 = 2 h i δ i , j F i + 1 , j x π , R 10 = γ i , j Δ i , j - 2 δ i , j F i + 1 , j x π , R 11 = β i , j γ i , j Δ i , j - 2 β i , j δ i , j F i + 1 , j x π - 2 α i , j γ i , j F i , j x π . Now the positivity of S i ( x , y j ) entirely depends on R i , i = 0,1 , 2 , 11 . The denominator in (22) is always positive. Since the parameter θ lies in first quadrant therefore the trigonometric basis functions will be positive also. This yields the following constraints on the free parameters: (24) β i , j > 2 α i , j F i , j x π Δ i , j , γ i , j > 2 δ i , j F i + 1 , j x π Δ i , j .

S ( x , y j + 1 ) is monotone if (25) S i ( x , y j + 1 ) > 0 ,

where (26) S i ( x , y j + 1 ) = π 2 h i ( q 2 ( θ ) ) 2 × { ( 1 - sin θ ) 4 sin θ cos θ T 0 + cos 2 θ ( 1 - cos θ ) 2 ( 1 - sin θ ) 2 T 1 + cos θ ( 1 - cos θ ) 3 ( 1 - sin θ ) 2 T 2 + cos θ ( 1 - sin θ ) 5 T 3 + sin θ cos 2 θ ( 1 - cos θ ) 2 ( 1 - sin θ ) T 4 + sin θ cos θ ( 1 - sin θ ) ( 1 cos θ ) 3 T 5 + sin θ ( 1 - cos θ ) 2 ( 1 - sin θ ) 3 T 6 + sin 2 θ ( 1 - sin θ ) 2 ( 1 - cos θ ) 2 T 7 + sin θ cos θ ( 1 - cos θ ) 4 T 8 + sin θ ( 1 cos θ ) 5 T 9 + sin θ cos θ ( 1 - cos θ ) ( 1 - sin θ ) 3 T 10 + sin 2 θ cos θ ( 1 - cos θ ) ( 1 - sin θ ) 2 T 11 } , with (27) T 0 = 2 h i F i , j + 1 x α i , j + 1 2 π , T 1 = ( 3 α i , j + 1 γ i , j + 1 - β i , j + 1 δ i , j + 1 ) Δ i , j + 1 + 2 F i , j + 1 x α i , j + 1 γ i , j + 1 π - ( 3 α i , j + 1 - β i , j + 1 ) 2 δ i , j + 1 F i + 1 , j + 1 x π , T 2 = ( β i , j + 1 - 3 α i , j + 1 ) Δ i , j + 1 - 2 F i , j + 1 x α i , j + 1 π , T 3 = 2 h i α i , j + 1 F i , j + 1 x π , T 4 = 2 β i , j + 1 γ i , j + 1 Δ i , j + 1 - 4 δ i , j + 1 β i , j + 1 F i + 1 , j + 1 x π - 4 α i , j + 1 γ i , j + 1 F i , j + 1 x π , T 5 = β i , j + 1 Δ i , j + 1 - 2 α i , j + 1 F i , j + 1 x π , T 6 = ( 3 δ i , j + 1 - γ i , j + 1 ) Δ i , j + 1 + 2 α i , j + 1 F i + 1 , j + 1 x π , T 7 = β i , j + 1 ( γ i , j + 1 - 3 δ i , j + 1 ) Δ i , j + 1 + 2 α i , j + 1 γ i , j + 1 F i , j + 1 x π + ( β i , j + 1 - 3 α i , j + 1 ) 2 δ i , j + 1 F i + 1 , j + 1 x π , T 8 = 2 h i δ i , j + 1 F i , j + 1 x π , T 9 = 2 h i δ i , j + 1 F i + 1 , j + 1 x π , T 10 = γ i , j + 1 Δ i , j + 1 - 2 δ i , j + 1 F i + 1 , j + 1 x π , T 11 = β i , j + 1 γ i , j + 1 Δ i , j + 1 - 2 β i , j + 1 δ i , j + 1 F i + 1 , j + 1 x π - 2 α i , j + 1 γ i , j + 1 F i , j + 1 x π . The denominator in (26) is always positive. Moreover, the trigonometric basis functions are also positive for 0 θ π / 2 . It follows that the positivity of S i ( x , y j + 1 ) entirely depends upon T i , i = 0,1 , 2 , 11 . This yields the following constraints on the free parameters: (28) β i , j + 1 > 2 α i , j + 1 F i , j + 1 x π Δ i , j + 1 , γ i , j + 1 > 2 δ i , j + 1 F i + 1 , j + 1 x π Δ i , j + 1 .

S ( x i , y ) is monotone if S i ( x i , y ) > 0 . We have (29) S i ( x i , y ) = π 2 h i ( q 3 ( φ ) ) 2 × { ( 1 - sin φ ) 4 sin φ cos φ U 0 + cos 2 φ ( 1 - cos φ ) 2 ( 1 - sin φ ) 2 U 1 + cos φ ( 1 - cos φ ) 3 ( 1 - sin φ ) 2 U 2 + cos φ ( 1 - sin φ ) 5 U 3 + sin φ cos 2 φ ( 1 - cos φ ) 2 ( 1 - sin φ ) U 4 + sin φ cos φ ( 1 - sin φ ) ( 1 cos φ ) 3 U 5 + sin φ ( 1 - cos φ ) 2 ( 1 - sin φ ) 3 U 6 + sin 2 φ ( 1 - sin φ ) 2 ( 1 - cos φ ) 2 U 7 + sin φ cos φ ( 1 - cos φ ) 4 U 8 + sin φ ( 1 cos φ ) 5 U 9 + sin φ cos φ ( 1 - cos φ ) ( 1 - sin φ ) 3 U 10 + sin 2 φ cos φ ( 1 - cos φ ) ( 1 - sin φ ) 2 U 11 } , where (30) U 0 = 2 h j F i , j y α ^ i , j 2 π , U 1 = ( 3 α ^ i , j γ ^ i , j - β ^ i , j δ ^ i , j ) Δ ^ i , j + 2 F i , j y α ^ i , j γ ^ i , j π - ( 3 α ^ i , j - β ^ i , j ) 2 δ ^ i , j F i , j + 1 y π , U 2 = ( β ^ i , j - 3 α ^ i , j ) Δ ^ i , j - 2 F i , j y α ^ i , j π , U 3 = 2 h j α ^ i , j F i , j y π , U 4 = 2 β ^ i , j γ ^ i , j Δ ^ i , j - 4 δ ^ i , j β ^ i , j F i , j + 1 y π - 4 α ^ i , j γ ^ i , j F i , j y π , U 5 = β ^ i , j Δ ^ i , j - 2 α ^ i , j F i , j y π , U 6 = ( 3 δ ^ i , j - γ ^ i , j ) Δ ^ i , j + 2 α ^ i , j F i , j + 1 y π , U 7 = β ^ i , j ( γ ^ i , j - 3 δ ^ i , j ) Δ ^ i , j + 2 α ^ i , j γ ^ i , j F i , j y π + ( β ^ i , j - 3 α ^ i , j ) 2 δ ^ i , j F i , j + 1 y π , U 8 = 2 h j δ ^ i , j F i , j + 1 y π , U 9 = 2 h j δ ^ i , j F i , j + 1 y π , U 10 = γ ^ i , j Δ ^ i , j - 2 δ ^ i , j F i , j + 1 y π , U 11 = β ^ i , j γ ^ i , j Δ ^ i , j - 2 β ^ i , j δ ^ i , j F i , j + 1 y π - 2 α ^ i , j γ ^ i , j F i , j y π . Since the denomoinator of (29) is always positive and trigonometric basis functions are positive for so the positivity of 0 φ π / 2 . It follows that the positivity of S i ( x i + 1 , y ) entirely depends upon U i , i = 0,1 , 2 , 11 . This yields the following constraints on the free parameters: (31) β ^ i , j > 2 α ^ i , j F i , j y π Δ ^ i , j , γ ^ i , j > 2 δ ^ i , j F i , j + 1 y π Δ ^ i , j .

S ( x i + 1 , y ) is monotone if S i ( x i + 1 , y ) > 0 . We have (32) S i ( x i + 1 , y ) = π 2 h i ( q 4 ( φ ) ) 2 × { ( 1 - sin φ ) 4 sin φ cos φ V 0 + cos 2 φ ( 1 - cos φ ) 2 ( 1 - sin φ ) 2 V 1 + cos φ ( 1 - cos φ ) 3 ( 1 - sin φ ) 2 V 2 + cos φ ( 1 - sin φ ) 5 V 3 + sin φ cos 2 φ ( 1 - cos φ ) 2 ( 1 - sin φ ) V 4 + sin φ cos φ ( 1 - sin φ ) ( 1 cos φ ) 3 V 5 + sin φ ( 1 - cos φ ) 2 ( 1 - sin φ ) 3 V 6 + sin 2 φ ( 1 - sin φ ) 2 ( 1 - cos φ ) 2 V 7 + sin φ cos φ ( 1 - cos φ ) 4 V 8 + sin φ ( 1 cos φ ) 5 V 9 + sin φ cos φ ( 1 - cos φ ) ( 1 - sin φ ) 3 V 10 + sin 2 φ cos φ ( 1 - cos φ ) ( 1 - sin φ ) 2 V 11 } , where (33) V 0 = 2 h j F i + 1 , j y α ^ i + 1 , j 2 π , V 1 = ( 3 α ^ i + 1 , j γ ^ i , j - β ^ i + 1 , j δ ^ i + 1 , j ) Δ ^ i + 1 , j + 2 F i + 1 , j y α ^ i + 1 , j γ ^ i + 1 , j π - ( 3 α ^ i + 1 , j - β ^ i + 1 , j ) 2 δ ^ i + 1 , j F i + 1 , j + 1 y π , V 3 = 2 h j α ^ i + 1 , j F i + 1 , j y π , V 4 = 2 β ^ i + 1 , j γ ^ i + 1 , j Δ ^ i + 1 , j - 4 δ ^ i + 1 , j β ^ i + 1 , j F i + 1 , j + 1 y π - 4 α ^ i + 1 , j γ ^ i + 1 , j F i + 1 , j y π , V 5 = β ^ i + 1 , j Δ ^ i + 1 , j - 2 α ^ i + 1 , j F i + 1 , j x π , V 6 = ( 3 δ ^ i + 1 , j - γ ^ i + 1 , j ) Δ ^ i , j + 2 α ^ i + 1 , j F i + 1 , j + 1 x π , V 7 = β ^ i + 1 , j ( γ ^ i + 1 , j - 3 δ ^ i + 1 , j ) Δ ^ i + 1 , j + 2 α ^ i + 1 , j γ ^ i + 1 , j F i , j y π + ( β ^ i + 1 , j - 3 α ^ i + 1 , j ) 2 δ ^ i , j F i + 1 , j + 1 y π , V 8 = 2 h j δ ^ i + 1 , j F i + 1 , j + 1 y π , V 9 = 2 h j δ ^ i + 1 , j F i + 1 , j + 1 x π , V 10 = γ ^ i + 1 , j Δ ^ i + 1 , j - 2 δ ^ i + 1 , j F i + 1 , j + 1 y π , V 11 = β ^ i + 1 , j γ ^ i + 1 , j Δ ^ i + 1 , j - 2 β ^ i + 1 , j δ ^ i + 1 , j F i + 1 , j + 1 y π - 2 α ^ i + 1 , j γ ^ i + 1 , j F i + 1 , j x π . Finally, S i ( x i + 1 , y ) is positive if V i , i = 0,1 , 2 , 11 are positive. This yields the following constraints on the free parameters: (34) β ^ i + 1 , j > 2 α ^ i + 1 , j F i + 1 , j y π Δ ^ i + 1 , j , γ ^ i + 1 , j > 2 δ ^ i + 1 , j F i + 1 , j + 1 y π Δ ^ i + 1 , j . The above discussion can be put forward as the following theorem.

Theorem 3.

The bicubic partially blended rational trigonometric function defined in (4) visualizes monotone data in view of the monotone surface if in each rectangular grid I = [ x i , x i + 1 ] × [ y j , y j + 1 ] , free parameters β i , j , γ i , j , β i , j + 1 , γ i , j + 1 , β ^ i , j , γ ^ i , j , β ^ i + 1 , j , γ ^ i + 1 , j satisfy the following constraints: (35) β i , j > 2 α i , j F i , j x π Δ i , j , γ i , j > 2 δ i , j F i + 1 , j x π Δ i , j , β i , j + 1 > 2 α i , j + 1 F i , j + 1 x π Δ i , j + 1 , γ i , j + 1 > 2 δ i , j + 1 F i + 1 , j + 1 x π Δ i , j + 1 , β ^ i , j > 2 α ^ i , j F i , j y π Δ ^ i , j , γ ^ i , j > 2 δ ^ i , j F i , j + 1 y π Δ ^ i , j , β ^ i + 1 , j > 2 α ^ i + 1 , j F i + 1 , j y π Δ ^ i + 1 , j , γ ^ i + 1 , j > 2 δ ^ i + 1 , j F i + 1 , j + 1 y π Δ ^ i + 1 , j . The above constraints are rearranged as (36) β i , j = l i , j + max { 0 , 2 α i , j F i , j x π Δ i , j } , l i , j > 0 , γ i , j = m i , j + max { 2 δ i , j F i + 1 , j x π Δ i , j } , m i , j > 0 , β i , j + 1 = n i , j + max { 2 α i , j + 1 F i , j + 1 x π Δ i , j + 1 } , n i , j > 0 , γ i , j + 1 = o i , j + max { 2 δ i , j + 1 F i + 1 , j + 1 x π Δ i , j + 1 } , o i , j > 0 , β ^ i , j = r i , j + max { 2 α ^ i , j F i , j y π Δ ^ i , j } , r i , j > 0 , γ ^ i , j = s i , j + max { 2 δ ^ i , j F i , j + 1 y π Δ ^ i , j } , s i , j > 0 , β ^ i + 1 , j = t i , j + max { 2 α ^ i + 1 , j F i + 1 , j y π Δ ^ i + 1 , j } , t i , j > 0 , γ ^ i + 1 , j = u i , j + max { 2 δ ^ i + 1 , j F i + 1 , j + 1 y π Δ ^ i + 1 , j } , u i , j > 0 .

Algorithm 4.

h h h h h h h h h h h

Step 1. Take a 3D monotone data set { ( x i , y j , F i , j ) , i = 0,1 , 2 , , n ; j = 0,1 , 2 , , m } .

Step 2. Use the Arithmetic Mean Method to estimate the derivatives F i , j x , F i , j y , F i , j x y at knots (note: Step 2 is only applicable if data is not provided with derivatives).

Step 3. Compute the values of parameters β i , j , γ i , j , β i , j + 1 , γ i , j + 1 , β ^ i , j , γ ^ i , j , β ^ i + 1 , j , γ ^ i + 1 , j using Theorem 3.

Step 4. Substitute the values of variables from Steps 1–3 in rational trigonometric cubic function (4) to visualize monotone surface through monotone data.

6. Numerical Example

This section illustrates the monotonicity preserving schemes developed in Sections 4 and 5 with the help of examples. The data in Table 1 is observed by exposing identical samples of hemoglobin to different partial pressures of oxygen which results in varying degree of saturation of hemoglobin with oxygen. The sample obtaining the highest amount is said to be saturated. The amount of oxygen combined with the remaining samples is taken as percentage of this maximum value. At a low partial pressure of oxygen, the percentage saturation of hemoglobin is very low; that is, hemoglobin is combined with only a very little oxygen. At high partial pressure of oxygen, the percentage saturation of hemoglobin is very high; that is, hemoglobin is combined with large amounts of oxygen, that is, a monotone relation, so the resulting curve must exhibit the same behavior. Figure 1 represents the curve created by assigning random values to free parameters in description of C 1 rational trigonometric cubic function (1) which does not retain the monotone nature of the data. This impediment is removed by applying monotonicity preserving schemes developed in Section 4 and is shown in Figure 2. It is evident from the figure that this curve preserves the monotone shape of hemoglobin dissociation curve. Similar investigation in Table 2 displays a series of results for percentage saturation of myoglobin and partial pressure of oxygen. Figure 3 is produced by assigning random values to free parameters in description of C 1 rational trigonometric cubic function (1) which fails to conserve the monotone trend of data. Algorithm 2 developed in Section 4 is applied to remove this drawback and Figure 4 displays the required result. Numerical results corresponding to Figures 2 and 4 are shown in Tables 3 and 4.

The varying ability of hemoglobin to carry oxygen.

Partial pressure of oxygen (kPa) 0 2 8 10 18
Saturation of hemoglobin (%) 0 70 91 91 110

The varying ability of myoglobin.

Partial pressure of oxygen (kPa) 0 4 6 8 10
Saturation of myoglobin (%) 0 100 100 100 115

Numerical results corresponding to Figure 2.

i 1 2 3 4 5
d i 50.9065 19.6819 0 0 5.7983
β i 35.01 7.17 0 0.01
γ i 0.1890 0.0100 0 0.7871

Numerical results corresponding to Figure 4.

i 1 2 3 4 5
d i 56.25 0 0 0 15
β i 2.8748 9.3179 0 0.01
γ i 0.01 0.01 0 0.6466

C 1 rational trigonometric cubic function with α i = 1.0 , β i = 0.5 , γ i = 1.0 , δ i = 2.0 .

C 1 monotone rational trigonometric cubic function with α i = 2.6 , δ i = 0.4 .

C 1 rational trigonometric cubic function with α i = 2.5 , β i = 0.5 , γ i = 0.5 , δ i = 2.0 .

C 1 monotone rational trigonometric cubic function with α i = 2.0 , δ i = 0.5 .

The 3D monotone data set in Tables 5 and 6 are generated from the following functions: (37) F ( x , y ) = x 2 25 + y 2 16 , F ( x , y ) = log ( x 2 + y 2 ) . respectively.

A 3D monotone data set.

y / x 1 2 3 4 5 6
1 0.3202 0.5385 0.7762 1.0198 1.2659 1.5133
2 0.4717 0.6403 0.8500 1.0770 1.3124 1.5524
3 0.6500 0.7810 0.9605 1.1662 1.3865 1.6155
4 0.8382 0.9434 1.0966 1.2806 1.4841 1.7000
5 1.0308 1.1180 1.2500 1.4142 1.6008 1.8028
6 1.2258 1.3000 1.4151 1.5620 1.7328 1.9209

A 3D monotone data set.

y / x 1 2 3 4 5 6
1 0.6931 1.6094 2.3026 2.8332 3.2581 3.6109
2 1.6094 2.0794 2.5649 2.9957 3.3673 3.6889
3 2.3026 2.5649 2.8904 3.2189 3.5264 3.8067
4 2.8332 2.9957 3.2189 3.4657 3.7136 3.9512
5 3.2581 3.3673 3.5264 3.7136 3.9120 4.1109
6 3.6109 3.6889 3.8067 3.9512 4.1109 4.2767

Figures 5 and 7 are produced by interpolating the monotone data sets in Tables 5 and 6, respectively, by C 1 rational trigonometric bicubic function for arbitrary values of free parameter. Monotone surfaces in Figures 6 and 8 are produced by interpolating the same data by the monotonicity preserving scheme developed in Section 5. Tables 7 and 8 enclose numerical results against Figures 6 and 8.

Numerical values corresponding to Figure 6.

( x i , y j ) 1 2 3 4 5 6
Numerical values of F i , j x
1 0.1382 0.0823 0.0555 0.0413 0.0328 0.0271
2 0.1649 0.1213 0.0921 0.0732 0.0603 0.0511
3 0.1832 0.1515 0.1233 0.1018 0.0858 0.0738
4 0.1904 0.1685 0.1448 0.1240 0.1071 0.0936
5 0.1938 0.1783 0.1593 0.1407 0.1243 0.1105
6 0.1962 0.1856 0.1709 0.1550 0.1396 0.1259

Numerical values of F i , j y
1 0.2087 0.2280 0.2406 0.2448 0.2467 0.2480
2 0.1481 0.1892 0.2184 0.2312 0.2377 0.2423
3 0.1068 0.1552 0.1926 0.2130 0.2247 0.2333
4 0.0813 0.1292 0.1686 0.1937 0.2097 0.2221
5 0.0649 0.1096 0.1481 0.1754 0.1943 0.2097
6 0.0538 0.0947 0.1310 0.1588 0.1794 0.1969

Numerical values of β i , j
1 10.9406 9.7062 9.0177 8.6526 8.4470
2 11.0996 10.3406 10.0079 9.8513 9.7684
3 11.6858 11.1996 10.8694 10.6747 10.5583
4 11.8607 11.5787 11.3235 11.1397 11.0149
5 11.9272 11.7583 11.5754 11.4218 11.3048
6

Numerical values of γ i , j
1 13.0594 14.2938 14.9823 15.3474 15.5530
2 12.3315 12.9236 13.3931 13.7011 13.8977
3 12.1426 12.4531 12.7625 13.0043 13.1785
4 12.0737 12.2518 12.4569 12.6399 12.7863
5 12.0728 12.2417 12.4246 12.5782 12.6952
6

Numerical values of β ^ i , j
1 11.4688 11.5120 11.8546 11.9391 11.9689
2 10.5384 10.8247 11.5416 11.7866 11.8858
3 9.7828 10.3810 11.2336 11.6016 11.7732
4 9.2668 10.1222 10.9942 11.4274 11.6537
5 8.9258 9.9673 10.8217 11.2811 11.5418
6

Numerical values of γ ^ i , j
1 12.5312 12.1490 12.0616 12.0312 12.0311
2 13.4616 12.4963 12.2213 12.1165 12.1142
3 14.2172 12.8787 12.4267 12.2357 12.2268
4 14.7332 13.2084 12.6331 12.3675 12.3463
5 15.0742 13.4662 12.8168 12.4961 12.4582
6

Numerical values of β i , j + 1
1 11.3239 10.5207 10.0947 9.8548 9.7100
2 12.0640 11.6759 11.4932 11.3965 11.3401
3 13.0662 12.6810 12.4538 12.3180 12.2329
4 13.5085 13.2108 12.9963 12.8508 12.7519
5 13.7180 13.5047 13.3254 13.1890 13.0885
6

Numerical values of γ i , j + 1
1 15.4850 16.2308 16.6264 16.8491 16.9836
2 14.0006 14.5092 14.8428 15.0559 15.1949
3 13.4909 13.8260 14.0880 14.2768 14.4104
4 13.2728 13.4950 13.6932 13.8518 13.9731
5 13.2618 13.4600 13.6264 13.7531 13.8464
6

Numerical values of β ^ i + 1 , j
1 9.4938 11.1409 12.5497 13.1538 13.4520
2 8.8689 10.3644 11.8756 12.6696 13.1076
3 8.6842 10.0813 11.4747 12.3097 12.8189
4 8.6336 10.0176 11.2653 12.0687 12.5971
5 8.6325 10.0429 11.1705 11.9192 12.4370
6

Numerical values of γ ^ i + 1 , j
1 14.5834 13.5377 13.2398 13.1262 13.1238
2 15.4020 13.9519 13.4623 13.2553 13.2457
3 15.9609 14.3091 13.6858 13.3981 13.3751
4 16.3304 14.5884 13.8848 13.5375 13.4964
5 16.5779 14.7990 14.0514 13.6641 13.6025
6

Numerical values corresponding to Figure 8.

( x i , y j ) 1 2 3 4 5 6
Numerical values of F i , j x
1 1.0279 0.4623 0.2308 0.1322 0.0843 0.0581
2 0.8047 0.4778 0.2939 0.1928 0.1341 0.0979
3 0.6119 0.4581 0.3270 0.2350 0.1731 0.1312
4 0.4778 0.4012 0.3180 0.2473 0.1928 0.1521
5 0.3889 0.3466 0.2939 0.2428 0.1987 0.1627
6 0.3168 0.2966 0.2667 0.2326 0.1991 0.1689

Numerical values of F i , j y
1 1.0279 0.8047 0.6119 0.4778 0.3889 0.3168
2 0.4623 0.4778 0.4581 0.4012 0.3466 0.2966
3 0.2308 0.2939 0.3270 0.3180 0.2939 0.2667
4 0.1322 0.1928 0.2350 0.2473 0.2428 0.2326
5 0.0843 0.1341 0.1731 0.1928 0.1987 0.1991
6 0.0581 0.0979 0.1312 0.1521 0.1627 0.1689

Numerical values of β i , j
1 13.4612 11.8021 10.5579 9.7618 9.2601
2 13.9316 11.8084 10.8374 10.3699 10.1191
3 13.8377 12.7622 11.9437 11.4236 11.0979
4 13.4933 12.9563 12.4102 11.9764 11.6602
5 13.2255 12.9325 12.5819 12.2566 11.9879
6

Numerical values of γ i , j
1 10.5388 12.1979 13.4421 14.2382 14.7399
2 10.5932 11.3237 12.0568 12.6377 13.0617
3 10.8043 11.1752 11.6161 12.0237 12.3602
4 10.9824 11.1929 11.4696 11.7539 12.0121
5 10.7745 11.0675 11.4181 11.7434 12.0121
6

Numerical values of β ^ i , j
1 13.4612 13.9316 13.8377 13.4933 13.2255
2 11.8021 11.8084 12.7622 12.9563 12.9325
3 10.5579 10.8374 11.9437 12.4102 12.5819
4 9.7618 10.3699 11.4236 11.9764 12.2566
5 9.2601 10.1191 11.0979 11.6602 11.9879
6

Numerical values of γ ^ i , j
1 10.5388 10.5932 10.8043 10.9824 10.7745
2 12.1979 11.3237 11.1752 11.1929 11.0675
3 13.4421 12.0568 11.6161 11.4696 11.4181
4 14.2382 12.6377 12.0237 11.7539 11.7434
5 14.7399 13.0617 12.3602 12.0121 12.0121
6

Numerical values of β i , j + 1
1 13.7691 12.3176 11.3888 10.8035 10.4245
2 13.7765 12.6436 12.0982 11.8056 11.6334
3 14.8893 13.9343 13.3275 12.9476 12.7025
4 15.1157 14.4785 13.9724 13.6036 13.3401
5 15.0879 14.6788 14.2994 13.9859 13.7398
6

Numerical values of γ i , j + 1
1 13.2144 14.5622 15.4247 15.9682 16.3201
2 12.2673 13.0616 13.6908 14.1502 14.4789
3 12.1065 12.5841 13.0257 13.3902 13.6766
4 12.1257 12.4254 12.7334 13.0131 13.2509
5 11.9898 12.3697 12.7220 13.0131 13.2416
6

Numerical values of β ^ i + 1 , j
1 7.0627 9.6496 12.0876 13.2188 13.7521
2 6.8759 8.4746 10.6260 11.9816 12.7945
3 7.0547 8.2958 10.0152 11.2619 12.1246
4 7.2590 8.4154 9.8191 10.8928 11.7015
5 7.4425 8.6142 9.8100 10.7305 11.4556
6

Numerical values of γ ^ i + 1 , j
1 13.2144 12.2673 12.1065 12.1257 11.9898
2 14.5622 13.0616 12.5841 12.4254 12.3697
3 15.4247 13.6908 13.0257 12.7334 12.7220
4 15.9682 14.1502 13.3902 13.0131 13.0131
5 16.3201 14.4789 13.6766 13.2509 13.2416
6

C 1 rational trigonometric bicubic function with α i , j = 14 , β i , j = 15 , γ i , j = 6 ,   δ i , j = 7 ,   α i , j + 1 = 8 ,   β i , j + 1 = 8 ,   γ i , j + 1 = 4 , δ i , j + 1 = 9 ,   α ^ i , j = 6 , β ^ i , j = 5 ,   γ ^ i , j = 4 ,   δ ^ i , j = 8 , α ^ i + 1 , j = 15 ,   β ^ i + 1 , j = 2 ,   γ ^ i + 1 , j = 12 ,   δ ^ i + 1 , j = 8 .

C 1 monotone rational trigonometric bicubic function with α i , j = 12 ,   δ i , j = 12 , α i , j + 1 = 14 ,   δ i , j + 1 = 13 ,   α ^ i , j = 12 ,   δ ^ i , j = 12 ,   α ^ i + 1 , j = 15 , δ ^ i + 1 , j = 13 .

C 1 rational trigonometric bicubic function with α i , j = 14 ,   β i , j = 8 ,   γ i , j = 5 ,    δ i , j = 7 ,   α i , j + 1 = 14 ,   β i , j + 1 = 8 ,   γ i , j + 1 = 6 ,   δ i , j + 1 = 13 ,   α ^ i , j = 12 , β ^ i , j = 9 ,   γ ^ i , j = 8 ,   δ ^ i , j = 8 ,   α ^ i + 1 , j = 15 ,   β ^ i + 1 , j = 9 ,   γ ^ i + 1 , j = 8 , δ ^ i + 1 , j = 8 .

C 1 monotone rational trigonometric bicubic function with α i , j = 12 ,   δ i , j = 12 ,   α i , j + 1 = 13 ,   δ i , j + 1 = 14 ,   α ^ i , j = 10 ,   δ ^ i , j = 10 ,   α ^ i + 1 , j = 15 ,   δ ^ i + 1 , j = 13 .

7. Conclusion

In this paper, monotonicity of data is retained by developing constraints on free parameters in the specification of rational trigonometric function and bicubic blended function. Authors in [7, 8] used algebraic function while the proposed algorithm applies trigonometric function which gives much smoother result due to orthogonality of sine and cosine function. Shape preserving techniques of Butt and Brodlie  required insertion of additional knots. In , developed scheme failed to maintain smoothness. The proposed technique is local, affirms smoothness, works well for data with derivatives, and does not require insertion of extra knots. Derivative of trigonometric spline is much lower than that of polynomial spline.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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