Quantitative Comparison of Sinc-Approximating Kernels for Medical Image Interpolation

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1 Quantitative Coparison of Sinc-Approxiating Kernels for Medical Iage Interpolation Erik H. W. Meijering, Wiro J. Niessen, Josien P. W. Plui, Max A. Viergever Iage Sciences Institute, Utrecht University Heidelberglaan 100, 3584 CX Utrecht, The Netherlands URL: E-ail: Abstract. Interpolation is required in any edical iage processing operations. Fro sapling theory, it follows that the ideal interpolation kernel is the sinc function, which is of infinite extent. In the attept to obtain practical and coputationally efficient iage processing algoriths, any sinc-approxiating interpolation kernels have been devised. In this paper we present the results of a quantitative coparison of 84 different sinc-approxiating kernels, with spatial extents ranging fro 2 to 10 grid points in each diension. The evaluation involves the application of geoetrical transforations to edical iages fro different odalities (CT, MR, and PET), using the different kernels. The results show very clearly that, of all kernels with a spatial extent of 2 grid points, the linear interpolation kernel perfors best. Of all kernels with an extent of 4 grid points, the cubic convolution kernel is the best (28% - 75% reduction of the errors as copared to linear interpolation). Even better results (44% - 95% reduction) are obtained with kernels of larger extent, notably the Welch, Cosine, Lanczos, and Kaiser windowed sinc kernels. In general, the truncated sinc kernel is one of the worst perforing kernels. 1 Introduction Interpolation of sapled data is required in a variety of edical iage processing operations, such as rotation, translation, deforation, or agnification, which are frequently applied for registration or visualization purposes. In any applications, it is of paraount iportance to liit as uch as possible the grey-value errors introduced by interpolation. For exaple, in iage registration, interpolation errors ay introduce artifacts in the optiization cost function, which ay lead to registration errors [1]. Furtherore, it has been pointed out that, especially in the case of functional iages, interpolation errors ay affect the interpretation of longitudinal studies [2]. It has also been shown that the errors ade by interpolation kernels influence the results of easureents carried out in axiu intensity projection iages [3]. Whereas fro sapling theory it follows that the ideal interpolation kernel is the sinc function, it is not the ideal kernel fro an ipleentational point of view, since this function is of infinite extent and has a very low rate of decay. In In Medical Iage Coputing and Coputer-Assisted Intervention MICCAI 1999, C. Taylor and A. Colchester (eds.), vol of Lecture Notes in Coputer Science, Springer-Verlag, Berlin, 1999, pp

2 PP-2 Meijering et al. the attept to obtain practical and coputationally efficient iage processing algoriths, any sinc-approxiating interpolation kernels have been devised. However, an extensive quantitative coparison of the perforance of these kernels when using the to apply geoetrical transforations to edical iages, has never been described. The purpose of this paper is to present the results of such a coparison. 2 Sinc-Approxiating Kernels In this section we briefly present the sinc-approxiating kernels ost frequently encountered in the literature. These can be divided into piecewise polynoial kernels and windowed sinc kernels. 2.1 Piecewise Polynoial Kernels A frequently used approach to obtain a sinc-approxiating kernel is to odel the shape of the sinc kernel by piecewise polynoials. The siplest approach in this respect is to use zeroth order polynoials, resulting in the so-called nearest neighbor kernel: { 1, if 1 2 h NN (x) = x< 1 2, 0, if x< 1 2 x 1 2. (1) Higher order piecewise polynoial kernels can be written in the for [4]: { a0j + a 1j x a nj x n, if j x <j+1, h(x) = 0, if x, (2) where n is the order of the polynoials, j =0, 1,..., 1, the paraeter N\{0} deterines the extent of the kernel, and n and are related by n =2 1. The (n +1) coefficients a ij can be solved by iposing constraints on the polynoials, derived fro the shape of the sinc kernel [4]. For n = 1, Eq. (2) boils down to the linear interpolation kernel, h Lin.The resulting kernels for n>1 can be shown to be functions of a free paraeter, α. In order to obtain a unique value for α, one additional constraint needs to be iposed. In the literature on cubic convolution (n = 3), several constraints have been proposed [5]: (i) The slope constraint, which iplies that α is chosen such that the slope of the kernel equals the slope of the sinc function at x =1. This value is denoted α ς. (ii) The continuity constraint. For n>1, the piecewise polynoial kernels are eleents of C n 2. The continuity constraint iplies that α is chosen such that the (n 1)th-order derivative of the kernel is continuous at x = 1. The resulting value is denoted α. (iii) The flatness constraint, which iplies that α is chosen such that the Fourier spectru of the kernel, H(f), is flat at f = 0. This value of α is denoted α. These three constraints can also be applied to the higher order schees. The resulting values of α for the cubic (h Cub ), quintic (h Qui ), septic (h Sep ), and nonic (h Non ) piecewise polynoial interpolation kernels, are presented in Table 1.

3 Coparison of Interpolation Kernels PP-3 Kernel α ς α α h Cub h Qui h Sep h Non Table 1. Values of the free paraeter α for the cubic (h Cub), quintic (h Qui), septic (h Sep), and nonic (h Non) piecewise polynoial interpolation kernels, resulting fro the slope constraint (α ς), continuity constraint (α ), and flatness constraint (α ). 2.2 Windowed Sinc Kernels Another approach to obtain a sinc-approxiating kernel is to ultiply the sinc function with a window function of liited extent: { w(x), if 0 x <, h(x) =ω(x)sinc(x), with ω(x) = 0, if x, (3) where ω : R R is the window function, and w : R R deterines the shape of the window in the interval (, ), with N\{0}. The window functions which were used in the quantitative coparison described in the next section are listed in Table 2. 3 Quantitative Coparison The sinc-approxiating kernels presented in the previous section were quantitatively copared by using the to apply several geoetrical transforations to a nuber of edical iages, and by analyzing the resulting interpolation errors in the transfored iages. In this section we present the evaluation strategy and the results. 3.1 Evaluation Strategy Medical iages were obtained fro a collection of 3-D brain datasets fro three different odalities, viz., coputed toography (CT), agnetic resonance iaging (MR; T1-weighted), and positron eission toography (PET). Fro every subset (odality), we selected five iages. The five CT iages were of size ties 28, 29, 33, 30, and 28 voxels, respectively, all with a voxel size of The five T1-weighted MR iages were all of size voxels, with a voxel size of The five PET iages were of size voxels, one with a voxel size of , and the others with a voxel size of

PP-4 Meijering et al. Window Definition Bartlett w Bar 1 x Blackan w Bla 0.42 + 0.50 cos ( πx Blackan-Harris w BHa 0.42323 + 0.

5cos ( ) πx w Kai I 0(β) 1 ( x ) 2 I 0 (α), β=α ) ( +0.08 cos 2πx ) ) ( +0.07922 cos 2πx ) ) ( ) + 1 sin π x π Table 2. Window functions and their definitions.

4 PP-4 Meijering et al. Window Definition Bartlett w Bar 1 x Blackan w Bla cos ( πx Blackan-Harris w BHa cos ( πx ( ) w Boh 1 x w Cos cos ( ) πx 2 Bohan Cosine Haing Hann Kaiser Lanczos w Lan sinc ( ) πx Rectangular w Rec 1 Welch w Wel 1 x2 2 cos ( πx w Ha cos ( ) πx w Han cos ( ) πx w Kai I 0(β) 1 ( x ) 2 I 0 (α), β=α ) ( cos 2πx ) ) ( cos 2πx ) ) ( ) + 1 sin π x π Table 2. Window functions and their definitions. Throughout this paper, the corresponding kernels are given the sae subscript. In the definition of the Kaiser window, α R + is a free paraeter, for which we used values of 5.0, 6.0, 7.0, and 8.0. I 0 is the zeroth order odified Bessel function of the first kind, which can accurately be approxiated by using its series expansion. For details, see Harris [6] or Wolberg [5]. Fig. 1. Exaples of the edical test-iages used in the experients. Top row (left-toright): a transversal slice of a 3-D CT, MR-T1, and PET dataset, respectively. Botto row: sagittal slices of the sae datasets. Note that for display purposes, the iages of the sagittal slices shown in this figure were scaled, thus correcting for the voxel anisotropy. Nearest neighbor interpolation was used for this purpose. In order to be able to study the perforance of the interpolation kernels in different slice directions, we selected, for each of the 3-D iages, one transversal and one sagittal slice. This resulted in a total of 30 different 2-D test-iages (see Fig. 1 for exaples).

5 Coparison of Interpolation Kernels PP-5 The test-iages were subjected to several geoetrical transforations, including rotations and subpixel translations, as these are frequently required in (ultiodality) registration. In the rotation experients, the test-iages were successively rotated over 0.7,3.2,6.5,9.3,12.1,15.2,18.4,21.3,23.7, 26.6,29.8,32.9,35.7,38.5,41.8,and44.3, which adds up to a total of 360. In the subpixel translation experients, the iages were successively shifted over 0.01, 0.04, 0.07, 0.11, 0.15, 0.18, 0.21, 0.24, 0.26, 0.29, 0.32, 0.35, 0.39, 0.43, 0.46, and 0.49 pixels, which adds up to a total of 4.0 pixels.forevery test-iage, both experients were carried out for all interpolation kernels. Of the two failies described in Section 2, we used all kernels for which 5, which aounts to a total of 84 kernels. In order to avoid border probles, the test-iages were irrored around the borders in each diension. For every cobination of test-iage, experient (rotation or translation), and interpolation kernel, the root-ean-square error (RMSE) of the grey-values in the processed iage was coputed. Since in these experients the grid points of the processed iages coincide with those of the original iages, a gold standard is available: for the rotation experients, the references iages are siply the original iages, and for the translation experients, the reference iages are obtained by translating the original iage by four pixels (which requires no interpolation). In order to avoid quantization errors, all coputations were carried out with double precision floating-point nubers (12 significant decials). 3.2 Results As can be concluded fro the literature, the linear interpolation kernel, h Lin,isby far the ost frequently used kernel [7]. Therefore, in this study, our ain interest was to investigate the perforance of alternative kernels copared to h Lin.Tothis end, we coputed for every test-iage and type of experient, the percentile RMSE of every interpolation kernel copared to h Lin.Since,inallcases,the percentile errors of the five iages fro a given cobination of odality and slice direction were very siilar, they were averaged. The results of the translation experients are presented in Table 3, and those of the rotation experients in Table 4. For every odality, type of experient, and kernel extent, only the top-3 best kernels are shown. 4 Discussion The perforance of interpolation kernels ay be assessed by subjective visual inspection of iage quality, after having used the kernels to perfor certain resapling operations [8, 5]. An alternative evaluation approach is to copare the spectral characteristics of the kernels to those of the ideal sinc kernel [9, 10], or to copare their abilities to reconstruct certain atheatical test-functions [11]. In the evaluation described in this paper, we have chosen to use a ore pragatic approach, in which the different interpolation kernels are used to apply actual geoetrical transforations, which are frequently required in e.g.

6 PP-6 Meijering et al. Mod Slc Extent =1 =2 =3 =4 =5 CT Tr h Lin h α Cub 33.4 h α=5 Kai 12.3 h α=6 Kai 6.3 h α=7 Kai 4.7 h NN h α=6 Kai 63.6 h BH h BH h BH3 5.6 h Wel h Ha h Bla 28.3 h Bla 16.6 h BH4 8.6 Sa h Lin h α Cub 70.1 h α=5 Kai 52.0 h Lan 42.7 h Lan 37.5 h Wel h α=5 Kai 76.2 h Lan 54.0 h Cos 42.9 h Cos 37.9 h NN h Ha 79.2 h Cos 54.1 h α=5 Kai 43.8 h α=5 Kai 39.0 MR Tr h Lin h α Cub 66.9 h α=5 Kai 47.1 h Lan 37.8 h Lan 31.5 h Wel h α=5 Kai 74.5 h Ha 49.9 h α=5 Kai 38.4 h Cos 31.9 h NN h Ha 78.2 h Lan 50.9 h Cos 38.5 h α=5 Kai 33.3 Sa h Lin h α Cub 71.7 h Cos 55.9 h Cos 46.4 h Lan 41.6 h Wel h α=5 Kai 79.3 h Lan 56.1 h Lan 46.4 h Cos 41.7 h Cos h Wel 79.8 h α=5 Kai 56.3 h Wel 47.8 h Wel 42.8 PET Tr h Lin h α Cub 36.5 h α=5 Kai 21.2 h α=6 Kai 16.9 h α=6 Kai 15.3 h Wel h α=6 Kai 62.5 h BH h BH h BH h NN h Ha 90.9 h Bla 31.8 h Bla 20.9 h Bla 17.0 Sa h Lin h α Cub 66.5 h α=5 Kai 47.9 h Lan 39.2 h Lan 34.6 h Wel h α=5 Kai 72.0 h Lan 48.7 h Cos 39.5 h Cos 34.7 h Cos h Ha 74.9 h Cos 49.0 h α=5 Kai 40.4 h Wel 35.8 Table 3. The percentile root ean square errors (RMSEs) resulting fro the different interpolation kernels in the translation experients, relative to the RMSEs of the linear interpolation kernel, h Lin. For every extent (), odality (Mod), and slice direction (Slc), either transversal (Tr) or sagittal (Sa), only the top-3 best kernels are shown. edical iage registration probles. The design of the evaluation was such that true gold standards were available, viz., the original iages. The results of the evaluation allow us to draw soe iportant conclusions. For =1,i.e., a spatial extent of 2 grid points, the best interpolation kernel is the linear interpolation kernel. For =2,i.e., a spatial extent of 4 grid points, the best approach is to use a cubic convolution kernel, although the optial valueforthefreeparaeter,α, ay differ for different types of iages. For the test-iages used in this study, cubic convolution resulted in a considerable (28% - 75%) reduction of interpolation errors, as copared to linear interpolation. Even better results (44% - 95% reduction) were obtained with kernels of larger extent. These latter results showed that, for 3, ost windowed sinc kernels give better results than piecewise polynoial kernels, although one ust be very careful in choosing a window function. Of the window functions incorporated in this study, the Welch, Cosine, Lanczos, and Kaiser windows appeared to be the best. It was also concluded that a truncated sinc kernel (resulting fro applying a rectangular window), was one of the worst perforing kernels. Finally, we notice that due to the anisotropic nature of 3-D edical datasets, the through-

7 Coparison of Interpolation Kernels PP-7 Mod Slc Extent =1 =2 =3 =4 =5 CT Tr h Lin h α Cub 24.7 h α=5 Kai 10.8 h α=6 Kai 5.7 h α=7 Kai 5.1 h NN h α=6 Kai 56.2 h BH h BH3 9.9 h BH3 5.5 h Wel h Ha h α Qui 22.5 h BH h BH4 7.0 Sa h Lin h α Cub 58.9 h α=5 Kai 45.1 h Wel 35.4 h Wel 30.2 h NN h α=6 Kai 76.0 h Wel 46.0 h Cos 36.5 h Cos 30.8 h Wel h Ha 83.4 h Cos 48.0 h α=5 Kai 38.6 h Lan 32.1 MR Tr h Lin h α Cub 59.2 h α=5 Kai 43.7 h Wel 35.2 h Wel 30.3 h NN h α=6 Kai 74.1 h Wel 46.1 h Cos 36.3 h Cos 30.8 h Wel h Ha 83.4 h Cos 50.4 h α=5 Kai 37.8 h Lan 32.0 Sa h Lin h α Cub 62.3 h Wel 49.4 h Wel 40.5 h Wel 36.3 h NN h α=5 Kai 76.5 h α=5 Kai 50.2 h Cos 41.2 h Cos 36.7 h Wel h Ha 80.8 h Cos 51.7 h Lan 42.9 h Lan 37.9 PET Tr h Lin h α Cub 30.1 h α=5 Kai 18.3 h α=6 Kai 15.7 h α=7 Kai 14.7 h NN h α=6 Kai 54.2 h BH h BH h BH h Wel h Ha h α Qui 28.5 h Bla 20.7 h Bla 16.8 Sa h Lin h α Cub 57.1 h α=5 Kai 42.1 h Wel 33.5 h Wel 29.1 h NN h α=6 Kai 72.6 h Wel 44.3 h Cos 34.8 h Cos 29.7 h Wel h Ha 79.8 h Cos 48.1 h α=5 Kai 36.5 h Lan 30.9 Table 4. The percentile root ean square errors (RMSEs) resulting fro the different interpolation kernels in the rotation experients, relative to the RMSEs of the linear interpolation kernel, h Lin. For every extent (), odality (Mod), and slice direction (Slc), either transversal (Tr) or sagittal (Sa), only the top-3 best kernels are shown. plane interpolation errors were considerably larger than the in-plane errors. This iplies that through-plane interpolation usually requires larger kernels in order for the errors to be coparable to in-plane linear interpolation errors. 5 Conclusions In this paper we have presented the results of a quantitative coparison of sincapproxiating kernels for edical iage interpolation. The evaluation involved the application of several geoetrical transforations (rotations and subpixel translations) to a nuber of edical iages fro different odalities (CT, MR, and PET), using the different interpolation kernels, and by coparing the resulting grey-value errors to those resulting fro linear interpolation. A total of 84 different kernels were evaluated, with spatial extents ranging fro 2 to 10 grid points in each diension. The results of the evaluation show very clearly that, of the kernels with a spatial extent of 2 grid points, the linear interpolation kernel is the best. Of the kernels with an extent of 4 grid points, the cubic convolution kernel is the

8 PP-8 Meijering et al. best (28% - 75% reduction of the errors as copared to linear interpolation). Even better results (44% - 95% reduction) were obtained with kernels of larger extent, notably the Welch, Cosine, Lanczos, and Kaiser windowed sinc kernels. The truncated sinc kernel was one of the worst perforing kernels. Acknowledgent The iages used in the experients described in this paper were obtained fro Vanderbilt University, and were originally used in the project Evaluation of Retrospective Iage Registration, National Institutes of Health, Project Nuber: 1 R01 NS , Principal Investigator: Prof. Dr. J. Michael Fitzpatrick, Vanderbilt University, Nashville, TN, USA. References 1. J. P. W. Plui, J. B. A. Maintz, & M. A. Viergever, Interpolation artefacts in utual inforation based iage registration, Coputer Vision and Iage Understanding, In Press. 2. J.L.Ostuni,A.K.S.Santha,V.S.Mattay,D.R.Weinberger,R.L.Levin,&J.A. Frank, Analysis of interpolation effects in the reslicing of functional MR iages, Journal of Coputer Assisted Toography, vol. 21, no. 5, 1997, pp S. Schreiner, C. B. Paschal, & R. L. Galloway, Coparison of projection algoriths used for the construction of axiu intensity projection iages, Journal of Coputer Assisted Toography, vol. 20, no. 1, 1996, pp E. H. W. Meijering, K. J. Zuiderveld, & M. A. Viergever, Iage reconstruction by convolution with syetrical piecewise nth-order polynoial kernels, IEEE Transactions on Iage Processing, vol. 8, no. 2, 1999, pp G. Wolberg, Digital Iage Warping, IEEE Coputer Society Press, Washington, USA, F. J. Harris, On the use of windows for haronic analysis with the discrete Fourier transfor, Proceedings of the IEEE, vol. 66, no. 1, 1978, pp G. J. Grevera & J. K. Udupa, An objective coparison of 3-D iage interpolation ethods, IEEE Transactions on Medical Iaging, vol. 17, no. 4, 1998, pp J.A.Parker,R.V.Kenyon,&D.E.Troxel, Coparisonofinterpolatingethods for iage resapling, IEEE Transactions on Medical Iaging, vol. 2, no. 1, 1983, pp S. K. Park & R. A. Schowengerdt, Iage reconstruction by paraetric cubic convolution, Coputer Vision, Graphics and Iage Processing, vol. 23, no. 3, 1983, pp E. Maeland, On the coparison of interpolation ethods, IEEE Transactions on Medical Iaging, vol. 7, no. 3, 1988, pp R. G. Keys, Cubic convolution interpolation for digital iage processing, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 29, no. 6, 1981, pp

Quantitative Evaluation of Convolution-Based Methods for Medical Image Interpolation

Quantitative Evaluation of Convolution-Based Methods for Medical Iage Interpolation Erik H. W. Meijering, Wiro J. Niessen, Max A. Viergever Medical Iage Analysis, vol. 5, no. 2, June 2001, pp. 111 126