Positive blending Hermite rational cubic spline fractal interpolation surfaces

Fractal interpolation provides an efficient way to describe data that have smooth and non-smooth structures. Based on the theory of fractal interpolation functions (FIFs), the Hermite rational cubic spline FIFs (fractal boundary curves) are constructed to approximate an original function along the grid lines of interpolation domain. Then the blending Hermite rational cubic spline fractal interpolation surface (FIS) is generated by using the blending functions with these fractal boundary curves. The convergence of the Hermite rational cubic spline FIS towards an original function is studied. The scaling factors and shape parameters involved in fractal boundary curves are constrained suitably such that these fractal boundary curves are positive whenever the given interpolation data along the grid lines are positive. Our Hermite blending rational cubic spline FIS is positive whenever the corresponding fractal boundary curves are positive. Various collections of fractal boundary curves can be adapted with suitable modifications in the associated scaling parameters or/and shape parameters, and consequently our construction allows interactive alteration in the shape of rational FIS.

[1]  A. K. B. Chand,et al.  Cubic hermite and cubic spline fractal interpolation functions , 2012 .

[2]  Douglas P. Hardin,et al.  Fractal Interpolation Functions from $R^n$ into $R^m$ and their Projections , 1993 .

[3]  Leoni Dalla,et al.  BIVARIATE FRACTAL INTERPOLATION FUNCTIONS ON GRIDS , 2002 .

[4]  Michael F. Barnsley,et al.  The calculus of fractal interpolation functions , 1989 .

[5]  A. K. B. Chand,et al.  Generalized Cubic Spline Fractal Interpolation Functions , 2006, SIAM J. Numer. Anal..

[6]  M. Navascués,et al.  Natural bicubic spline fractal interpolation , 2008 .

[7]  Bruce A. Campbell,et al.  Shadows on a Planetary Surface and Implications for Photometric Roughness , 1998 .

[8]  M. Navascués,et al.  Shape preservation of scientific data through rational fractal splines , 2014 .

[9]  A. K. B. Chand,et al.  Hidden Variable Bivariate Fractal Interpolation Surfaces , 2003 .

[10]  Robert Malysz,et al.  The Minkowski dimension of the bivariate fractal interpolation surfaces , 2006 .

[11]  A. Chand,et al.  Generalized hermite fractal interpolation , 2009 .

[12]  Arya Kumar Bedabrata Chand,et al.  Monotonicity Preserving Rational Quadratic Fractal Interpolation Functions , 2014, Adv. Numer. Anal..

[13]  Vasileios Drakopoulos,et al.  Image Compression Using Recurrent bivariate Fractal Interpolation Surfaces , 2006, Int. J. Bifurc. Chaos.

[14]  Nailiang Zhao,et al.  Construction and application of fractal interpolation surfaces , 1996, The Visual Computer.

[15]  Heping Xie,et al.  The Study on Bivariate Fractal Interpolation Functions and Creation of Fractal Interpolated Surfaces , 1997 .

[16]  A. Chand.  NATURAL CUBIC SPLINE COALESCENCE HIDDEN VARIABLE FRACTAL INTERPOLATION SURFACES , 2012 .

[17]  Michael F. Barnsley,et al.  Fractal functions and interpolation , 1986 .

[18]  Muhammad Sarfraz,et al.  Data visualization using rational spline interpolation , 2006 .

[19]  Lucia Romani,et al.  Rational Interpolants with Tension Parameters , 2003 .

[20]  Jochen W. Schmidt,et al.  Positivity of cubic polynomials on intervals and positive spline interpolation , 1988 .

[21]  Pantelis Bouboulis,et al.  Fractal interpolation surfaces derived from fractal interpolation functions , 2007 .

[22]  G. P. KAPOOR,et al.  Smoothness of Coalescence Hidden-Variable Fractal Interpolation Surfaces , 2009, Int. J. Bifurc. Chaos.

[23]  A. K. B. Chand,et al.  A constructive approach to cubic Hermite Fractal Interpolation Function and its constrained aspects , 2013 .

[24]  W. G. Price,et al.  The approximation properties of some rational cubic splines , 1999, Int. J. Comput. Math..

[25]  Ken Brodlie,et al.  Visualization of surface data to preserve positivity and other simple constraints , 1995, Comput. Graph..

[26]  J. W. Schmidt,et al.  Nonnegative interpolation by biquadratic splines on refined rectangular grids , 1994 .

[27]  M. A. Navascués,et al.  Smooth fractal interpolation , 2006 .

[28]  Pantelis Bouboulis,et al.  Closed fractal interpolation surfaces , 2007 .

[29]  Yizhuo Feng,et al.  Fractal interpolation surfaces with function vertical scaling factors , 2012, Appl. Math. Lett..

[30]  M. Navascués.  Fractal Polynomial Interpolation , 2005 .