An improved approach to the cubic-spline interpolation Tsung-Ching Lina, Shao-Hua Hong*b, Trieu-Kien Truonga,c, Lin Wangb a Department of Information Engineering, I-Shou University, Kaohsiung Country 840, Taiwan; b Department of Communication Engineering, Xiamen University, Xiamen, Fujian, 361005, China; c Department of Computer Science and Engineering, National Sun Yat-sen University, Kaohsiung Country 804, Taiwan ABSTRACT Cubic-spline interpolation (CSI) scheme is known to be designed to resample the discrete image data based on the leastsquares method with the cubic convolution interpolation (CCI) function. It is superior in performance to other interpolation functions for digital image processing. In this paper, an improved CSI scheme that combines the leastsquares method with an eight-point cubic interpolation kernel is developed in order to improve the performance of the original CSI scheme. Either the FFT/Winograd DFT or the fast direct computation algorithm can also be used to perform the circular convolution needed in this improved CSI scheme. Furthermore, its correlated image data and auto-correlated filter coefficients are also accurately calculated in this paper. Experimental results indicate that the proposed improved CSI scheme yields a much better quality of reconstructed image than existing interpolation algorithms. Keywords: Better quality, cubic-spline interpolation, eight-point cubic interpolation kernel, improved approach
1. INTRODUCTION The interpolation has been widely used for a variety of applications to signal and medical image processing. Interpolation functions such as linear interpolation, cubic convolution interpolation (CCI)1, and cubic B-spline interpolation2-3 have commonly been used in the image compression process. The main disadvantage of these interpolation schemes is that, in general, they are not designed to minimize the error between the original image and its reconstructed image. In 1981, based on the least-square method with linear interpolation function, Reed4 first developed a linear spline interpolation scheme for re-sampling discrete image data. In 2000, using an extension of the ideas of Reed, Truong et al. developed a considerable modified version of the linear spline interpolation algorithm, called the cubicspline interpolation (CSI) algorithm5. It combines the least-squares method with the four-point CCI function whose parameter is α = −0.5 for the decimation process, which achieves a better performance than the other existing interpolation algorithms1-4. Recently, based upon the principles of opportunity costs, the most suitable parameter α = −1 for the four-point parametric CSI is identified and a new four-point CSI scheme is therefore proposed6, which can achieve better performance with the same arithmetic operations in comparison with the original four-point CSI algorithm with parameter α = −0.5 . In 1999, Lehman et al.7 showed that if the size of the cubic interpolation kernel is increased, the improved quality of resampling can be achieved. In this paper, an improved approach to the CSI scheme, which is based on the least-squares method together with an eight-point cubic interpolation kernel, is proposed for the decimation and interpolation of image data. Based upon the ideas8, the extension of image can be applied to enable filtering at both boundaries of the image, and the correlated image data and auto-correlated filter coefficients are calculated in considerable detail. It follows from references5,9,10 that either the FFT/Winograd DFT or the fast direct computation algorithm can also be used to implement the circular convolution needed in this improved CSI scheme. The main goal of the proposed CSI scheme is to improve a quality the reconstructed image for higher compression ratios. Computer simulations indicate that it is far superior in performance to the conventional schemes. The rest of this work is organized as follows: in the next section, the encoding, decoding, and calculation of autocorrelation coefficients of the improved CSI scheme is described in considerable detail. The experimental results and discussion are presented in Section III. Finally, this paper concludes with a brief summary in Section VI. *
Email:
[email protected] Applications of Digital Image Processing XXXVI, edited by Andrew G. Tescher, Proc. of SPIE Vol. 8856, 885606 · © 2013 SPIE · CCC code: 0277-786X/13/$18 doi: 10.1117/12.2026867 Proc. of SPIE Vol. 8856 885606-1
Downloaded From: http://proceedings.spiedigitallibrary.org/ on 10/15/2014 Terms of Use: http://spiedl.org/terms
2. THE IMPROVED CSI SCHEME It is well-known that the philosophy of the CSI scheme is to recalculate the sampled values of the image data by means of the least-squares method that uses the CCI formula. In this section, an improved CSI scheme based on the leastsquares method associated with an eight-point cubic interpolation kernel is developed in order to improve the performance. From reference7, for the one-dimensional (1-D) CCI function ri ( t ) , where i is the size of CCI function, only 3i 2 + 1 equations can be obtained to determine the 2i parameters from the boundary conditions and i 2 − 1 degrees of
freedom are required to be determined. Thus, there are three degrees of freedom α1 , α 2 , and α 3 needed to be determined for the eight-point cubic interpolation kernel, given by ⎧α1 t 3 − (1 + α1 ) t 2 + 1, ⎪ 3 2 ⎪α 2 t − (α1 + 4α 2 − 2 ) t + ( 3α1 + 5α 2 − 6 ) t − ( 2α1 + 2α 2 − 4 ) , ⎪ r8 ( t ) = ⎨α 3 t 3 + (α1 − α 2 − 7α 3 − 2 ) t 2 − ( 5α1 − 5α 2 − 16α 3 − 10 ) t + 6 (α1 − α 2 − 2α 3 − 2 ) , ⎪ 3 2 ⎪(α1 − α 2 + α 3 − 2 ) t − 11(α1 − α 2 + α 3 − 2 ) t + 40 (α1 − α 2 + α 3 − 2 ) t − 48 (α1 − α 2 + α 3 − 2 ) , ⎪0 , ⎩
0 ≤ t
