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## Interpolation revisited (2000)

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Venue: | IEEE Transactions on Medical Imaging |

Citations: | 194 - 33 self |

### Citations

643 | On the use of windows for harmonic analysis with the discrete fourier transform
- Harris
(Show Context)
Citation Context ... Thus, the classical solution is simply to truncate sinc itself by multiplying it with a finite-support window; this process is named apodization. A large catalog of apodizing windows is available in =-=[14]-=-, along with their detailed analysis. B. Dirichlet Apodization Dirichlet apodization is perhaps the laziest approach, since the window of total width is simply an enlarged version of , which requires ... |

368 | Cubic convolution interpolation for digital image processing
- Keys
- 1981
(Show Context)
Citation Context ...n in the linear case. F. Keys The principal reason for the popularity enjoyed by the family of Keys functions is the fact that they perform better than linear interpolation, while being interpolating =-=[15]-=-. Thus, they do not require the determination of interpolation coefficients, and the classical equation (1) can be used. These functions are made of piecewise cubic polynomials and depend on a paramet... |

225 | A pyramid approach to subpixel registration based on intensity
- Thevenaz, Unser
- 1998
(Show Context)
Citation Context ...plitude of the difference between two conditions (e.g., active versus inactive) is very small. Registration is used to align fMRI images before they are subtracted to reveal where and how they differ =-=[10]-=-. However, even with perfect knowledge of the ideal geometric transformation, a deficient interpolation can wash out these tiny differences. The essence of interpolation is to represent an arbitrary c... |

223 |
Contribution to the problem of approximation of equidistant data by analytic functions
- Schoenberg
- 1946
(Show Context)
Citation Context ...tes, no matter how many dimensions are involved. The price to pay is a severe loss of quality. B. B-Splines There is a whole family of basis functions made of B-splines . These functions are given by =-=[28]-=- where is the one-sided power function defined in (11). From Table I, it is easy to conclude that these B-spline functions enjoy the maximal order of approximation for a given support; conversely, the... |

182 |
Fourier analysis of the finite element variational method
- Strang, Fix
- 1971
(Show Context)
Citation Context ... we are interested in just the approximation order of a basis function , without caring much about the details of . In this case, the explicit computation of (8) is not necessary. Instead, Strang–Fix =-=[25]-=- have proposed a series of conditions that are equivalent to (9). The interest of these equivalent conditions is that they can be readily tested. They are valid for all basis functions with sufficient... |

158 | B-spline signal processing: Part i-theory
- Unser, Aldroubi, et al.
- 1993
(Show Context)
Citation Context ...nsists of a causal (with pole ) and of an anticausal (with pole ) IIR filter. With suitable parameters, we can then apply a very efficient algorithm which leads to a recursive in-place implementation =-=[19]-=-, [20]. Its computational load for the popular cubic B-spline is two additions and three multiplications per produced coefficient. E. Reconciliation Comparing (1) with (2), it appears that classical i... |

92 |
Sampling procedure in function spaces and asymptotic equivalence with Shannon's sampling theory
- Aldroubi, Unser
- 1994
(Show Context)
Citation Context ....s746 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 19, NO. 7, JULY 2000 Fig. 2. B-spline basis functions. Approximation kernel for several degrees. converge to a sinc function when its degree increases =-=[30]-=-, [31]. Throughout the paper, we have given numerous reasons why it is more profitable to approximate a true sinc by the use of noninterpolating basis functions rather than by apodization. Even for mo... |

71 | Evaluation and design of filters using a Taylor series expansion, IEEE Transaction on visualization and computer graphics 3(2
- Möller, Machiraju, et al.
- 1997
(Show Context)
Citation Context ...ture to the facets that compose the rendered object [8]. In addition, volume rendering may also require the computation of gradients, which is best done by taking the interpolation model into account =-=[9]-=-. In functional magnetic resonance imaging (fMRI), the relative Manuscript received July 22, 1999; revised May 12, 2000. The Associate Editor responsible for coordinating the review of this paper and ... |

64 | Approximation errors for quasi-interpolators and (multi)wavelet expansions
- Blu, Unser
- 1999
(Show Context)
Citation Context ...ets smaller, more details of can be captured; it is then reasonable to ask that the approximation error gets smaller, too. The following formula predicts the approximation error in the Fourier domain =-=[21]-=-–[23] where is the Fourier transform of the arbitrary function , and where is an interpolation error kernel that depends on the basis function only, and that is given by The equivalence holds for band... |

62 |
Threedimensional reconstruction of icosahedral particles—the uncommon
- Fuller, Butcher, et al.
- 1996
(Show Context)
Citation Context ...a sampling. This rescaling operation is desirable to build isometric volumes [4]–[6]. Another application of this transform arises in the three-dimensional (3-D) reconstruction of icosahedral viruses =-=[7]-=-. In volume rendering, it is common to apply by interpolation a texture to the facets that compose the rendered object [8]. In addition, volume rendering may also require the computation of gradients,... |

60 | Cardinal spline filters: stability and convergence t the ideal sinc interpolator, Signal Process
- Aldroubi, Unser, et al.
- 1992
(Show Context)
Citation Context ...IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 19, NO. 7, JULY 2000 Fig. 2. B-spline basis functions. Approximation kernel for several degrees. converge to a sinc function when its degree increases [30], =-=[31]-=-. Throughout the paper, we have given numerous reasons why it is more profitable to approximate a true sinc by the use of noninterpolating basis functions rather than by apodization. Even for moderate... |

47 |
On the comparison of interpolation methods
- Maeland
- 1988
(Show Context)
Citation Context ...polation methods, and we introduce generalized interpolation as a means to overcome the limitations of traditional interpolation. Interpolation is at the heart of various medical imaging applications =-=[1]-=-–[3]. In volumetric imaging, it is often used to compensate for nonhomogeneous data sampling. This rescaling operation is desirable to build isometric volumes [4]–[6]. Another application of this tran... |

43 | Quadratic interpolation for image resampling
- DODGSON
- 1997
(Show Context)
Citation Context ...aum loses an additional order of magnitude with . They are discontinuous for even degrees, and are for odd degrees. E. Dodgson Dodgson has proposed a basis function built out of quadratic polynomials =-=[12]-=-. Unfortunately, it fails to be a member of the family of Moms. While it has a support , its approximation order is , which is no higher than in the linear interpolation case that has the smaller supp... |

38 |
I.: On the approximation power of convolution-based least squares versus interpolation
- Unser, Daubechies
- 1997
(Show Context)
Citation Context ... set of shifted basis functions . This error is the smallest achievable in a quadratic sense, and requires that the set of discrete constraints expressed in (3) be replaced by a continuous constraint =-=[33]-=-. While it is also possible to find Moms functions such that their asymptotic approximation constant reaches its absolute minimum, experiments have shown that these particular functions are in fact le... |

37 |
Texture mapping 3D models of real-world scenes
- Weinhaus, Devarjan
- 1997
(Show Context)
Citation Context ...m arises in the three-dimensional (3-D) reconstruction of icosahedral viruses [7]. In volume rendering, it is common to apply by interpolation a texture to the facets that compose the rendered object =-=[8]-=-. In addition, volume rendering may also require the computation of gradients, which is best done by taking the interpolation model into account [9]. In functional magnetic resonance imaging (fMRI), t... |

33 |
Factorization theorems for univariate splines on regular grids
- Ron
- 1990
(Show Context)
Citation Context ...f a B-spline of order ,was first presented in [25] without proof and without specifying the properties of the distribution . A rigorous proof of an extended version of this theorem was later given in =-=[26]-=-, in a more mathematically abstract and general context than required for this paper. Our actual formulation is aimed at fitting the present task and audience; the corresponding proof will appear in a... |

23 | Ten good reasons for using spline wavelets
- Unser
- 1997
(Show Context)
Citation Context ...r a given support; conversely, they enjoy the minimal support for a given order of approximation. In addition, they are maximally continuous. They have many other interesting properties as well [28], =-=[29]-=-; most fall outside the scope of this paper, except perhaps the fact that a B-spline derivative can be computed recursively by Then, computing the exact gradient of a signal given by a discrete sequen... |

21 | Analysis of interpolation effects in the reslicing of functional MR-Images - Ostuni, Mattay, et al. - 1997 |

21 |
Image sampling, reconstruction, and the effect of sample-scene phasing
- Park, Schowengerdt
- 1982
(Show Context)
Citation Context ...erpolating basis functions [21]–[23]. In the restricting conditions where , for bandlimited functions and when the basis function is interpolating, this error kernel reduces to the kernel proposed in =-=[24]-=-. B. Order of Approximation A decrease in the sampling step will result in a decrease of the argument of in (7); thus, the error kernel must vanish at the origin to ensure that the approximation error... |

20 |
signal processing: Part II—Efficient design and applications
- B-spline
- 1993
(Show Context)
Citation Context ... of a causal (with pole ) and of an anticausal (with pole ) IIR filter. With suitable parameters, we can then apply a very efficient algorithm which leads to a recursive in-place implementation [19], =-=[20]-=-. Its computational load for the popular cubic B-spline is two additions and three multiplications per produced coefficient. E. Reconciliation Comparing (1) with (2), it appears that classical interpo... |

20 | Minimum Support Interpolators with Optimum Approximation
- Blu, Thévenaz, et al.
- 1998
(Show Context)
Citation Context ...ort is called Moms (or splines of minimal support in the terminology of [26]). It can be shown that any of these functions can be expressed as the weighted sum of a B-spline and its derivatives [26], =-=[32]-=-, such that the distribution of the decomposition theorem has a vanishing support Moms B-splines are those Moms functions that are maximally differentiable. There exist several other interesting class... |

14 |
Towards statistically optimal interpolation for 3-D medical imaging
- Parrott, Stytz, et al.
- 1993
(Show Context)
Citation Context ...tion methods, and we introduce generalized interpolation as a means to overcome the limitations of traditional interpolation. Interpolation is at the heart of various medical imaging applications [1]–=-=[3]-=-. In volumetric imaging, it is often used to compensate for nonhomogeneous data sampling. This rescaling operation is desirable to build isometric volumes [4]–[6]. Another application of this transfor... |

14 |
Polynomial splines and wavelets-A signal processing perspective
- Unser, Aldroubi
- 1992
(Show Context)
Citation Context ...additional properties such as easy analytical manipulation, several recursion relations, the -scale relation (of great importance for wavelets, a domain that has strong links with interpolation [29], =-=[35]-=-), minimal curvature for cubic B-splines, easy extension to inexact interpolation (smoothing splines, least-squares [33]), simplicity of their parametrization (a single number—their degree—is enough t... |

11 |
A new approach to the interpolation of sampled data
- Appledorn
- 1996
(Show Context)
Citation Context ... of those basis functions. The traditional view asks that they satisfy the interpolation property, and many researchers have put a significant effort in optimizing them under this specific constraint =-=[11]-=-–[17]. Over the years, these efforts have shown more and more diminishing returns. Here, instead, we introduce and advocate the use of generalized interpolation, which does away with the constraint of... |

8 |
Efficient Algorithms for Generating Interpolated (Zoomed
- Smith, Nichols
- 1988
(Show Context)
Citation Context ...ious medical imaging applications [1]–[3]. In volumetric imaging, it is often used to compensate for nonhomogeneous data sampling. This rescaling operation is desirable to build isometric volumes [4]–=-=[6]-=-. Another application of this transform arises in the three-dimensional (3-D) reconstruction of icosahedral viruses [7]. In volume rendering, it is common to apply by interpolation a texture to the fa... |

5 |
Impact of reorientation algorithms on quantitative myocardial SPECT perfusion imaging
- Haddad, Porenta
- 1998
(Show Context)
Citation Context ... various medical imaging applications [1]–[3]. In volumetric imaging, it is often used to compensate for nonhomogeneous data sampling. This rescaling operation is desirable to build isometric volumes =-=[4]-=-–[6]. Another application of this transform arises in the three-dimensional (3-D) reconstruction of icosahedral viruses [7]. In volume rendering, it is common to apply by interpolation a texture to th... |

5 |
Short kernel fifth-order interpolation
- GERMAN
- 1997
(Show Context)
Citation Context ...l these kernels (compare to the broader selection of performances to choose from in Fig. 2). One possible selection criterion is the constant of approximation it in Table II. ;wegive H. German German =-=[13]-=- has proposed an interpolating basis function with an approximation order . Its support is larger than necessary, which leaves some freedom to the designer. In this case, this freedom has been used to... |

4 |
Theory and design of local interpolators,” CVGIP: Graph
- Schaum
- 1993
(Show Context)
Citation Context ...hose basis functions. The traditional view asks that they satisfy the interpolation property, and many researchers have put a significant effort in optimizing them under this specific constraint [11]–=-=[17]-=-. Over the years, these efforts have shown more and more diminishing returns. Here, instead, we introduce and advocate the use of generalized interpolation, which does away with the constraint of inte... |

3 | In vito 3-D reconstruction of long bones using B-scan image processing - Migeon, Marche - 1997 |

3 |
Image reconstruction with symmetrical piecewise nth-order polynomial kernels
- Meijering, Zuidervel, et al.
- 1999
(Show Context)
Citation Context ...onsidered here, at the possible minor and very localized exception of nearest-neighbor interpolation. D. Regularity Some authors insist that the regularity of the basis function is an important issue =-=[16]-=-. This may be true when differentiation of is needed, but differentiating data more than, say, once or twice, is uncommon in everyday imaging applications. Often, at most the gradient is needed; thus,... |

3 |
Spline Fit Made Easy
- Liou
- 1976
(Show Context)
Citation Context ...wn a priori, this expression is nothing but a linear system of equations in terms of the unknown coefficients . We are now faced with a problem of the form , and a large part of the literature (e.g., =-=[18]-=-) is devoted to the development of efficient techniques for inverting the matrix in the context of specific basis functions . The problem is trivial when is interpolating, for then is the identity. An... |

1 | Complete parametrization of piecewise polynomial interpolators,”, submitted for publication
- Blu
(Show Context)
Citation Context ...tically abstract and general context than required for this paper. Our actual formulation is aimed at fitting the present task and audience; the corresponding proof will appear in a forthcoming paper =-=[27]-=-. A direct corollary of the decomposition theorem is that the support of any function of order must satisfy , because no distribution can have a negative support. However, it may happen that the suppo... |