## Splines: A Perfect Fit for Signal/Image Processing (1999)

Venue: | IEEE SIGNAL PROCESSING MAGAZINE |

Citations: | 271 - 50 self |

### BibTeX

@ARTICLE{Unser99splines:a,

author = {Michael Unser},

title = {Splines: A Perfect Fit for Signal/Image Processing},

journal = {IEEE SIGNAL PROCESSING MAGAZINE},

year = {1999},

volume = {16},

pages = {22--38}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

3387 | A computational approach to edge detection
- Canny
- 1986
(Show Context)
Citation Context ...oposed using a smoothing spline technique [61, 62]. They showed the approach to be more or less equivalent to smoothing the image with a Gaussian filter in a preprocessing step (Canny's edge detector =-=[18]-=-). This analogy holds even further [96]: there is an exact equivalence between a smoothing spline edge detector and Deriche's recursive formulation of Canny's edge detector [28]. Finally, Mallat and Z... |

3246 | Snakes: Active contour models
- Kass, Witkin, et al.
- 1987
(Show Context)
Citation Context ...rate-distortion sense [44, 75]. Menet et al. proposed using B-splines snakes for extracting contours in images [52]. A snake is an energy minimization spline segment with external and internal forces =-=[43]-=-. It simulates an elastic material that can dynamically conform to local image features. The internal forces act as a regularization device by constraining the rigidity of the curve. Alternatively, th... |

2627 | A theory for multiresolution signal decomposition: the wavelet representation
- Mallat
- 1989
(Show Context)
Citation Context ...ause researchers in this field had become so accustomed to think in terms of bandlimited functions. Recently, thanks in part to a new (non-bandlimited) way of thinking brought forth by wavelet theory =-=[51]-=-, the situation has changed significantly. This paper attempts to fullfill three goals. The first one is to provide a tutorial on splines that is geared to a signal processing audience. The second one... |

2282 |
A wavelet tour of signal processing
- Mallat
- 1998
(Show Context)
Citation Context ...arkable facts connected to the above result. First, the two-scale equation (29) holds for any integer m — not just powers of two, as encountered in the multiresolution theory of the wavelet transfor=-=m [48, 109, 81]-=-. Second, the refinement filter is simply the (n+1)-fold convolution of the discrete rectangular impulse of width m; this can be the basis n for some very fast algorithms [101]. In the standard case w... |

1907 |
lectures on wavelets
- Daubechies, Ten
- 1992
(Show Context)
Citation Context ... to the only wavelets that have a closed-form formula (piecewise polynomial). All other wavelet bases are defined indirectly by an infinite recursion (or by an infinite product in the Fourier domain) =-=[23, 48, 81, 109].-=- It is therefore no coincidence that most of the earlier wavelet constructions were based on splines; for instance, the Haar wavelet transform (n=0) [34], the Franklin system (n=1), Strömberg's one-s... |

1732 | Orthonormal bases of compactly supported wavelets
- Daubechies
- 1988
(Show Context)
Citation Context ... down-side of semi-orthogonal wavelets is that some of the corresponding wavelet filters are IIR † . Researchers have also designed spline wavelets such that the corresponding wavelet filters are FI=-=R [21, 108]-=-. These biorthogonal wavelets are constructed using two multiresolutions instead of one, with the spline spaces on the synthesis side. The major difference with the semi-orthogonal case is that the un... |

1389 |
Spline Models for Observational Data
- Wahba
- 1990
(Show Context)
Citation Context ...his amounts to a small modification of the reduction filter h o [96]. This kind of algorithm provides an efficient filterbased implementation of the technique known as spline regression in statistics =-=[31, 2, 113]-=-. Most of the spline pyramids use symmetric filters that are centered on the origin (in fact, these are based on the centered B-splines rather that the causal ones that have been used here to simplify... |

1310 |
Embedded Image Coding using Zerotree of Wavelet Coefficients
- Shapiro
- 1993
(Show Context)
Citation Context ... compression Image compression is another area where splines can be helpful. Most of today state-of-theart methods use wavelets: the most prominent ones are Shapiro's embedded zero-tree wavelet coder =-=[77]-=- , and Said and Pearlman's SPIHT [69]. While there are many possible choices of wavelet filters, many researchers tend to favor the biorthogonal splines for the reasons mentioned before (symmetry, sho... |

1250 |
A Practical Guide to Splines
- Boor
- 1978
(Show Context)
Citation Context ...s been approached using a matrix framework, setting up a band-diagonal system of equations which is then solved using standard numerical techniques (forward/backward substitution or LU decomposition) =-=[25, 64]-=-. In the early 1990's, it was recognized that this problem (as well as many other related ones) could also be approached using simpler digital filtering techniques [33, 93, 97, 96]. To derive this typ... |

929 | A New Fast and Efficient Image Codec Based On Set Partitioning
- Said, Pearlman
- 1996
(Show Context)
Citation Context ...ther area where splines can be helpful. Most of today state-of-theart methods use wavelets: the most prominent ones are Shapiro's embedded zero-tree wavelet coder [77] , and Said and Pearlman's SPIHT =-=[69]-=-. While there are many possible choices of wavelet filters, many researchers tend to favor the biorthogonal splines for the reasons mentioned before (symmetry, short support, and excellent approximati... |

894 | De-noising by soft thresholding
- Donoho
- 1993
(Show Context)
Citation Context ...g techniques, which may be expressed in a regularization framework as well [19]. The main difference is that the smoothing spline is a linear estimator, while Donoho's wavelet shrinkage is non-linear =-=[30]-=-. The idea is simple and was pioneered Weaver et al. using orthogonal spline wavelets [114]: take the wavelet transform of a signal and set to zero the coefficients below some critical threshold while... |

742 |
Image coding using wavelet transform
- Antonini, Barlaud, et al.
- 1992
(Show Context)
Citation Context ...re many possible choices of wavelet filters, many researchers tend to favor the biorthogonal splines for the reasons mentioned before (symmetry, short support, and excellent approximation properties) =-=[9, 81]-=-. We should also mention some non wavelet-based systems: for example, the method of Toraichi et al., which uses quadratic spline interpolation [87], and Moulin's decomposition in terms of hierarchical... |

590 |
Communication in the presence of noise
- Shannon
(Show Context)
Citation Context ...s is provided by Shannon's sampling theory which describes an equivalence between a bandlimited function and its equidistant samples taken at a frequency that is superior or equal to the Nyquist rate =-=[76]-=-. Even though this theory has had an enormous impact on the field, it has a number of problems associated with it. First, it relies on the use of ideal filters which are devices not commonly found in ... |

545 |
Characterization of signal from multiscale edges
- Mallat, Zhong
- 1992
(Show Context)
Citation Context ... Deriche's recursive formulation of Canny's edge detector [28]. Finally, Mallat and Zhong used wavelets that are derivatives of B-splines for obtaining their multi-scale edge representation of images =-=[49]-=-. 6.7 Snakes and contour modeling In computer graphics, curves are often generated using B-splines [10]. This parametric representation is also well suited to the analysis of shapes and contours [32].... |

499 | Wavelets and subband coding
- Vetterli, Kovacevic
- 1995
(Show Context)
Citation Context ...arkable facts connected to the above result. First, the two-scale equation (29) holds for any integer m — not just powers of two, as encountered in the multiresolution theory of the wavelet transfor=-=m [48, 109, 81]-=-. Second, the refinement filter is simply the (n+1)-fold convolution of the discrete rectangular impulse of width m; this can be the basis n for some very fast algorithms [101]. In the standard case w... |

493 |
Multiresolution approximations and wavelet orthonormal bases of L2(R
- Mallat
- 1989
(Show Context)
Citation Context ...s displayed in Fig. 10a. 4.3 Spline wavelets The L 2 -spline pyramid that has been described above has all the required properties for a multiresolution analysis of L 2 in the sense defined by Mallat =-=[50, 51]-=-. In particular, the error bound (24) guarantees that we can approximate any L 2 -function as closely as we wish bys- 22 - letting the scale go to zero. In the wavelet terminology, the multiresolution... |

405 |
Spline Functions: Basic Theory
- Schumaker
- 1981
(Show Context)
Citation Context ...uld model the physical process of drawing a smooth curve (minimum curvature property). This created an intense interest in the subject and the applications were soon to follow in approximation theory =-=[24, 74]-=-, numerical analysis [64], and various other branches of applied mathematics [3]. With the advent of digital computers, splines caught the interest of engineers and had a tremendous impact on computer... |

254 |
Wavelets and Filter Banks: Theory and Design
- Vetterli, Herley
- 1992
(Show Context)
Citation Context ... down-side of semi-orthogonal wavelets is that some of the corresponding wavelet filters are IIR † . Researchers have also designed spline wavelets such that the corresponding wavelet filters are FI=-=R [21, 108]-=-. These biorthogonal wavelets are constructed using two multiresolutions instead of one, with the spline spaces on the synthesis side. The major difference with the semi-orthogonal case is that the un... |

252 |
Computational vision and regularization theory
- Poggio, Torre, et al.
- 1985
(Show Context)
Citation Context ...hing spline can be computed efficiently by recursive filtering [96].s- 25 - Introducing a regularization term as in (36) is a standard practice for dealing with many other types of ill-posed problems =-=[61],-=- including sparse and non-equally spaced data. The regularization parameter λ is typically used to control the smoothness of the solution. For m=1, the regularization will tend to privilege small val... |

244 | Using Canny’s criteria to derive a recursively implemented optimal edge detector. Int
- Deriche
- 1987
(Show Context)
Citation Context ...(Canny's edge detector [18]). This analogy holds even further [96]: there is an exact equivalence between a smoothing spline edge detector and Deriche's recursive formulation of Canny's edge detector =-=[28]-=-. Finally, Mallat and Zhong used wavelets that are derivatives of B-splines for obtaining their multi-scale edge representation of images [49]. 6.7 Snakes and contour modeling In computer graphics, cu... |

224 |
Wavelets and dilation equations, a brief introduction
- Strang
- 1989
(Show Context)
Citation Context ...d by the number of vanishing moments of the analysis wavelet ψ( x ) . An equivalent statement of the order property is that the translates of the function ϕ must reproduce the polynomials of degree =-=n [26, 79]. In general, t-=-he order property implies that one has the following asymptotic form of the approximation error (cf. [89])s- 26 - − L ( L) s− P s = C ⋅T ⋅ s , as T → 0 (37) ϕ, T ϕ, L { } ∈ where Pϕ, Ts... |

211 |
Zur theorie der orthogonalen funktionensysteme. Mathematische Annalen
- Haar
- 1910
(Show Context)
Citation Context ...e product in the Fourier domain) [23, 48, 81, 109]. It is therefore no coincidence that most of the earlier wavelet constructions were based on splines; for instance, the Haar wavelet transform (n=0) =-=[34], -=-the Franklin system (n=1), Strömberg's one-sided orthogonal splines [82], and the celebrated BattleLemarié wavelets [11, 47]. Since then, the family has grown and there are now several other subclas... |

204 | Nonlinear wavelet image processing: variational problems, compression and noise removal through wavelet shrinkage
- Chambolle, DeVore, et al.
- 1998
(Show Context)
Citation Context ... similar way by introducing more complex regularization terms [63]. Smoothing splines are closely related to wavelet denoising techniques, which may be expressed in a regularization framework as well =-=[19]-=-. The main difference is that the smoothing spline is a linear estimator, while Donoho's wavelet shrinkage is non-linear [30]. The idea is simple and was pioneered Weaver et al. using orthogonal splin... |

178 |
Contribution to the problem of approximation of equidistant data by analytic functions
- Schoenberg
- 1946
(Show Context)
Citation Context ...ill also bring out the connection with the multiresolution theory of the wavelet transform. Interestingly, splines are slightly older than Shannon's sampling theory. They were first described in 1946 =-=[70]. -=-In this landmark paper, Schoenberg laid the mathematical foundations for the subject; he showed how one could use splines to interpolate equally-spaced samples of a function—he also introduced the B... |

176 |
The theory of radial basis functions approximation
- Powell
- 1992
(Show Context)
Citation Context ...leads to another area of study called "thin-plates splines" [113]. Generalized splines and radial basis functions can also be defined in a similar way by introducing more complex regularizat=-=ion terms [63]-=-. Smoothing splines are closely related to wavelet denoising techniques, which may be expressed in a regularization framework as well [19]. The main difference is that the smoothing spline is a linear... |

160 |
A Fourier analysis of the finite element variational method
- Strang, Fix
- 1973
(Show Context)
Citation Context ...where sˆ( ω ) denotes the Fourier transform of s; this is nothing but the norm of the Lth derivative of s. The key result from the Strang-Fix theory of approximation is the following error bound (cf=-=. [80, 40]): ∀s∈W -=-L , s− Ps ≤C ⋅T ⋅ s 2 T L L ( L) where Ps T is the least-squares spline approximation of s at sampling step T and C L known constant; W L 2 (24) denotes the space of functions that are L times... |

138 |
The Theory of Splines and Their Applications
- Ahlberg, Nilson, et al.
- 1967
(Show Context)
Citation Context .... This created an intense interest in the subject and the applications were soon to follow in approximation theory [24, 74], numerical analysis [64], and various other branches of applied mathematics =-=[3]-=-. With the advent of digital computers, splines caught the interest of engineers and had a tremendous impact on computer-aided design [45, 29], and computer graphics [10]. However, there was little cr... |

134 |
Aldroubi: “A general sampling theory for nonideal acquisition devices
- Unser, A
- 1994
(Show Context)
Citation Context ...d wavelet sampling present interesting alternatives to the conventional approach dictated by Shannon's sampling theorem. These techniques can be adapted for dealing with non-ideal acquisition devices =-=[92]-=-, and multi-channel measurements [106]. With this more general view of sampling, it is tempting to modify the acquisition scheme so as to measure the coefficients of some signal expansion (i.e., to pe... |

125 |
Wavelets and Filter Banks
- Strang, Nguyen
- 1996
(Show Context)
Citation Context ...arkable facts connected to the above result. First, the two-scale equation (29) holds for any integer m — not just powers of two, as encountered in the multiresolution theory of the wavelet transfor=-=m [48, 109, 81]-=-. Second, the refinement filter is simply the (n+1)-fold convolution of the discrete rectangular impulse of width m; this can be the basis n for some very fast algorithms [101]. In the standard case w... |

123 | B-spline signal processing: Part I—theory - Unser, Aldroubi, et al. - 1993 |

122 |
Cubic splines for image interpolation and digital filtering
- Hou, Andrews
- 1978
(Show Context)
Citation Context ...dical imaging [60, 57], but also for multi-media and digital photography, which are rapidly expanding applications areas. The use of cubic splines in image processing was pioneered by Hou and Andrews =-=[36]-=-. The proposed approach was not yet very practical because the B-spline coefficients were determined by matrix inversion. The method was made much more efficient with the introduction of recursive fil... |

121 |
Cardinal Spline Interpolation
- Schoenberg
- 1973
(Show Context)
Citation Context ...stand for basis or basic) are the basic building blocks for splines. Their usefulness stems from the fact that they are compactly supported; in fact, they are the shortest possible polynomial splines =-=[72].-=- Here, we consider the center-symmetric Bspline of degree n, β n ( x), which is constructed from the (n+1)-fold convolution of a unit rectangular pulse (B-spline of degree 0). The simplest way to obt... |

120 | A pyramid approach to subpixel registration based on intensity
- Thevenaz, UE, et al.
- 1998
(Show Context)
Citation Context ... be quite robust, which means that the algorithm is much less likely to get trapped in a local optimum. A good illustration of these ideas is provided by the image registration algorithm described in =-=[85]-=-. This method makes use of the same high-quality spline model for all aspects of the computation: image pyramid, geometric transform, and computation of the gradient of the criterion that is optimized... |

114 | On Compactly Supported Spline Wavelets and a Duality Principle
- Chui, Wang
- 1992
(Show Context)
Citation Context ...e orthogonal. This gives flexibility and makes it possible to design wavelets with many interesting properties [5] and almost any desirable shape [1]. Of particular interest are the B-spline wavelets =-=[20, 94]-=-, which are compactly supported, and optimally localized in time and frequency; asymptotically, they achieve the lower limit specified by Heisenberg's uncertainty 2 1s- 23 - principle [94]. The only d... |

114 |
Package for calculating with B-splines
- Boor
(Show Context)
Citation Context ...uld model the physical process of drawing a smooth curve (minimum curvature property). This created an intense interest in the subject and the applications were soon to follow in approximation theory =-=[24, 74]-=-, numerical analysis [64], and various other branches of applied mathematics [3]. With the advent of digital computers, splines caught the interest of engineers and had a tremendous impact on computer... |

112 |
B-spline signal processing: part ii - efficient design and applications
- Unser, Aldroubi, et al.
- 1993
(Show Context)
Citation Context ...tution or LU decomposition) [25, 64]. In the early 1990's, it was recognized that this problem (as well as many other related ones) could also be approached using simpler digital filtering techniques =-=[33, 93, 97, 96]-=-. To derive this type of signal processing algorithm, we need to introduce the discrete Bn spline kernel bm , which is obtained by sampling the B-spline of degree n expanded by a factor of m: n n z n ... |

105 |
Fast algorithms for discrete and continuous wavelet transforms
- Rioul, Duhamel
- 1992
(Show Context)
Citation Context ...ng the continuous wavelet transform with integer scales [101]. This type of algorithm achieves the lowest O(1) complexity per computed coefficient. In contrast with other wavelet transform algorithms =-=[67]-=-, the B-spline approach is non-iterative across scale and therefore well suited to a parallel implementation. Splines are also used for computing wavelet transforms with arbitrary non-integer scales [... |

102 |
Comparision of interpolating methods for image resampling
- Parker, Kenyon, et al.
- 1983
(Show Context)
Citation Context ...f polynomial splines. 6.1 Zooming and visualization Image zooming and interpolation are perhaps the most obvious applications of splines. These manipulations are especially useful for medical imaging =-=[60, 57]-=-, but also for multi-media and digital photography, which are rapidly expanding applications areas. The use of cubic splines in image processing was pioneered by Hou and Andrews [36]. The proposed app... |

100 |
Splines and variational methods
- Prenter
- 1975
(Show Context)
Citation Context ...s of drawing a smooth curve (minimum curvature property). This created an intense interest in the subject and the applications were soon to follow in approximation theory [24, 74], numerical analysis =-=[64]-=-, and various other branches of applied mathematics [3]. With the advent of digital computers, splines caught the interest of engineers and had a tremendous impact on computer-aided design [45, 29], a... |

100 |
Fast B-spline Transforms for Continuous Image Representation and Interpolation
- Unser, Aldroubi, et al.
- 1991
(Show Context)
Citation Context ...tution or LU decomposition) [25, 64]. In the early 1990's, it was recognized that this problem (as well as many other related ones) could also be approached using simpler digital filtering techniques =-=[33, 93, 97, 96]-=-. To derive this type of signal processing algorithm, we need to introduce the discrete Bn spline kernel bm , which is obtained by sampling the B-spline of degree n expanded by a factor of m: n n z n ... |

80 | Simple regularity criteria for subdivision schemes
- Rioul
- 1992
(Show Context)
Citation Context ...re also the most regular ones if one takes the size of the refinement 1 filter into account [90]: their Sobolev regularity (r derivatives in L2 ) is rmax = n+ 2 [81] and their Hölder exponent is α ==-=n [66]. This lat-=-ter property means that the B-spline of degree n is "almost" n times continuously-differentiable; strictly speaking, the nth derivative of spline of degree n has some isolated points of disc... |

75 |
Sampling procedures in function spaces and asymptotic equivalence with Shannon’s sampling theory
- Aldroubi, Unser
- 1994
(Show Context)
Citation Context ... : 1 operation = 1 multiplication + 1 addition; this count assumes that the B-splines are denormalized. (17)s- 12 - 95]. The concepts are best explained from the general perspective of Hilbert spaces =-=[6]. For co-=-nvenience, we will use a slightly more general spline generating function which we represent as n ϕ( x) = ∑ p( k) β ( x−k) , (18) k∈l with the important restriction that the sequence p is such... |

73 |
Fractional splines and wavelets
- Unser, Blu
(Show Context)
Citation Context ...with the required order properties; i.e., the B-splines. 5.5 Fractional splines Interestingly, B-splines can be generalized to fractional orders (cf. the illustration on the cover page of this issue) =-=[102]. The -=-fractional splines are piecewise power functions with α 1 building blocks of the form ( x− xk) + , with α >−2 real. The corresponding B-splines provide a smooth transition between the polynomial... |

72 | Quantitative Fourier analysis of approximation techniques: Part I— interpolators and projectors
- Blu, Unser
- 1999
(Show Context)
Citation Context ...- − L ( L) s− P s = C ⋅T ⋅ s , as T → 0 (37) ϕ, T ϕ, L { } ∈ where Pϕ, Ts is the projection of s onto the space V = span ϕ ( x/ T −k) and where the − constant Cϕ, L T k Z can be d=-=etermined explicitly [89, 14]. T-=-his is essentially the same equation as − (24) with an equality instead of an upper bound; the asymptotic leading constant Cϕ, L therefore necessarily smaller than C L in (24). Among all known wave... |

69 |
On bandwidth
- Slepian
- 1976
(Show Context)
Citation Context ...ns in the signal domain very inefficient. While the first two problems can be dealt with by using approximations and introducing concepts such as an essential bandwidth and an essential time duration =-=[78]-=-, there is no way to address the last two issues other than changing basis functions. Our purpose here will be to provide arguments in favor of an alternative approach that uses splines, which is equa... |

68 | B-spline snakes: a flexible tool for parametric contour detection
- Brigger, Hoeg, et al.
(Show Context)
Citation Context ...orces act as a regularization device by constraining the rigidity of the curve. Alternatively, the smoothness of the curve can also be controlled directly by adapting the scale of the basis functions =-=[16]-=-.s6.8 Analog-to-digital conversion - 31 - Spline and wavelet sampling present interesting alternatives to the conventional approach dictated by Shannon's sampling theorem. These techniques can be adap... |

67 |
Spline functions and the problem of graduation
- Schoenberg
- 1964
(Show Context)
Citation Context ...he variance of the noise or the degree of smoothness of the signal as measured by (35). Here again, it can be shown that the optimal solution among all possible functions is a spline of degree n=2m-1 =-=[65, 71]-=-. Part of the argument follows from the first integral equation: any non-spline fit can be improved by using its spline interpolant which further reduces the second term in the criterion while keeping... |

65 |
A family of polynomial spline wavelet transforms
- Unser, Aldroubi, et al.
- 1993
(Show Context)
Citation Context ...and in their orthogonality properties. Corresponding to an orthogonal projection (and to the L 2 -pyramid above) is the class of semi-orthogonal wavelets which are orthogonal with respect to dilation =-=[98].-=- These wavelets span the same space as the Battle-Lemarié splines, but are not constrained to be orthogonal. This gives flexibility and makes it possible to design wavelets with many interesting prop... |

63 |
A block spin construction of ondelettes part i: Lemarie functions
- Battle
- 1987
(Show Context)
Citation Context ...uctions were based on splines; for instance, the Haar wavelet transform (n=0) [34], the Franklin system (n=1), Strömberg's one-sided orthogonal splines [82], and the celebrated BattleLemarié wavelet=-=s [11, 47]-=-. Since then, the family has grown and there are now several other subclasses of spline wavelets available; they differ in the type of projection used and in their orthogonality properties. Correspond... |

63 |
Interpolative multiresolution coding of advanced television with compatible subchannels
- Uz, Vetterli, et al.
- 1991
(Show Context)
Citation Context ...pline interpolation [87], and Moulin's decomposition in terms of hierarchical spline basis functions [53]. Pyramid coders, which extend Burt and Adelson's initial idea, should not be dismissed either =-=[107, 88, 39, 56]-=-; they can offer advantages, especially in higher dimensions where the overhead with respect to wavelets becomes negligible. Finally, splines provide a good solution for sub-pixel motion compensation.... |