<p> Univariate cubic <i>L</i><sup>1</sup> spline fits have been successful to preserve the shapes of 2D data with abrupt changes. The reason is that the minimization of <i>L</i><sup>1</sup> norm of the data is considered, as opposite to <i>L</i><sup>2</sup> norm. While univariate <i>L</i><sup>1</sup> spline fits for 2D data are discussed by many, bivariate <i>L</i><sup>1</sup> spline fits for 3D data are yet to be fully explored. This thesis aims to develop bicubic <i>L</i><sup>1</sup> spline fits for 3D data approximation. This can be achieved by solving a bi-level optimization problem. One level is bivariate cubic spline interpolation and the other level is <i> L</i><sup>1</sup> error minimization. In the first level, a bicubic interpolated spline surface will be constructed on a rectangular grid with necessary first and second order derivative values estimated by using a 5-point window algorithm for univariate <i>L</i><sup> 1</sup> interpolation. In the second level, the absolute error (i.e. <i> L</i><sup>1</sup> norm) will be minimized using an iterative gradient search. This study may be extended to higher dimensional cubic <i>L</i><sup> 1</sup> spline fits research.</p><p>
| Identifer | oai:union.ndltd.org:PROQUEST/oai:pqdtoai.proquest.com:10751900 |
| Date | 16 June 2018 |
| Creators | Zaman, Muhammad Adib Uz |
| Publisher | Northern Illinois University |
| Source Sets | ProQuest.com |
| Language | English |
| Detected Language | English |
| Type | thesis |
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