## Discrete derivative approximations with scale-space properties: A basis for low-level feature extraction (1993)

Venue: | J. Math. Imaging Vision |

Citations: | 32 - 13 self |

### BibTeX

@ARTICLE{Lindeberg93discretederivative,

author = {Tony Lindeberg},

title = {Discrete derivative approximations with scale-space properties: A basis for low-level feature extraction},

journal = {J. Math. Imaging Vision},

year = {1993}

}

### OpenURL

### Abstract

It is developed how discrete derivative approximations can be de ned so that scale-space properties hold exactly also in the discrete domain. Starting from a set of natural requirements on the rst processing stages of a visual system, the visual front end, an axiomatic derivation is given of how amulti-scale representation of derivative approximations can be constructed from a discrete signal, so that it possesses an algebraic structure similar to that possessed by the derivatives of the traditional scale-space representation in the continuous domain. A family of kernels is derived which constitute discrete analogues to the continuous Gaussian derivatives. The representation has theoretical advantages to other discretizations of the scalespace theory in the sense that operators which commute before discretization commute after discretization. Some computational implications of this are that derivativeapproximations can be computed directly from smoothed data, and that this will give exactly the same result as convolution with the corresponding derivative approximation kernel. Moreover, a number of normalization conditions are automatically satis ed. The proposed methodology leads to a conceptually very simple scheme of computations for multi-scale low-level feature extraction, consisting of four basic steps � (i) large support convolution smoothing, (ii) small support di erence computations, (iii) point operations for computing di erential geometric entities, and (iv) nearest neighbour operations for feature detection. Applications are given demonstrating how the proposed scheme can be used for edge detection and junction detection based on derivatives up to order three.

### Citations

3367 | A Computational Approach to Edge Detection
- Canny
- 1986
(Show Context)
Citation Context ...presented methodology, experimental results are presented of using these operations for a few different visual tasks. A straightforward edge detection scheme is described, which is similar to Canny's =-=[6]-=- method, but does not need any direct 4 A rule of thumb sometimes used in this context is that when derivatives of order two and higher are computed from raw image data, then the amplitude of the ampl... |

1085 | The Laplacian pyramid as a compact image code
- Burt, Adelson
- 1983
(Show Context)
Citation Context ...ive approximations and discrete approximations to differential geometric descriptors using the proposed framework. 12 Similar operators have also been used in pyramid representations; see, e.g., Burt =-=[5]-=-, and Crowley [7]. 13 The same problem arises also if the computations are performed in the Fourier domain, since at least one inverse FFT transformation will be needed for each derivative approximati... |

940 | Theory of edge detection
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Citation Context ...s, forming the difference, and then dividing by the scale difference. In the literature, this method is usually referred to as the difference of Gaussians (DOG) approach; see, e.g., Marr and Hildreth =-=[28]-=-. Note, however, that when the scale difference tends to zero, the result of this operation is not guaranteed to converge to the actual result, of say convolving the original signal with the sampled L... |

694 |
The structure of images
- Koenderink
- 1984
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Citation Context ...his intrinsic property is the basic reason to why a multi-scale representation is of crucial importance when describing the structure of an image. A methodology proposed by Witkin [34] and Koenderink =-=[15]-=- to obtain such a multi-scale representation is by embedding the signal into a one-parameter family of derived signals, the scale-space, where the parameter t, called scale parameter 1 , describes the... |

560 |
Scale-space filtering
- Witkin
- 1983
(Show Context)
Citation Context ...e ranges of scale. This intrinsic property is the basic reason to why a multi-scale representation is of crucial importance when describing the structure of an image. A methodology proposed by Witkin =-=[34]-=- and Koenderink [15] to obtain such a multi-scale representation is by embedding the signal into a one-parameter family of derived signals, the scale-space, where the parameter t, called scale paramet... |

417 |
Scale space and edge detection using anisotropic diffusion
- Perona, Malik
- 1987
(Show Context)
Citation Context ... generalization to consider is non-linear diffusion, although further work may be needed in order to develop the notion of anisotropic smoothing as introduced into computer vision by Perona and Malik =-=[31]-=- and Nordstrom [30]. A Appendix A.1 Summary of Main Results from 2D Discrete Scale-Space Theory Below are stated for reference purpose some basic definitions and key results from the (zeroorder) scale... |

259 | Representation of local geometry in the visual system - Koenderink, Doorn - 1987 |

243 | Using Canny’s criteria to derive a recursively implemented edge detector
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Citation Context ... this edge detection scheme to an image of a table scene at a number of different scales, while Figure 7 shows a simple comparison with a traditional implementation of the Canny-Deriche edge detector =-=[9]-=-. Of course, it is not easy to make a fair comparison between the two methods, since the Canny-Deriche method is pixel oriented and uses a different smoothing filter. Moreover, the usefulness of the o... |

157 |
Digital step edges from zero crossing of second directional derivatives
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- 1984
(Show Context)
Citation Context ...ctions 12 were based on the relations to standard discrete operators used in numerical analysis; see, e.g., Dahlquist et al [8]. Other design criteria may lead to other operators, see, e.g., Haralick =-=[13]-=-, Meer and Weiss [29], and Vieville, Faugeras [33]. Nevertheless, the algebraic scale-space properties are preserved whatever linear operators are used. 4 Computational Implications 4.1 Derivative App... |

152 | T.A.: Scaling theorems for zero crossings
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Citation Context ...ufficiency lead to a formulation in terms of the diffusion equation, both in one and two dimensions. A similar result, although based on slightly different assumptions, was given by Yuille and Poggio =-=[35]-=- regarding the zero-crossings of the Laplacian. Babaud et al [2] gave a particular proof in the one-dimensional case and showed that natural constraints on a smoothing kernel necessarily implied that ... |

144 |
Uniqueness of the gaussian kernel for scale-space filtering
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- 1986
(Show Context)
Citation Context ...ion, both in one and two dimensions. A similar result, although based on slightly di erent assumptions, was given by Yuille and Poggio [35] regarding the zero-crossings of the Laplacian. Babaud et al =-=[2]-=- gave a particular proof in the one-dimensional case and showed that 3snatural constraints on a smoothing kernel necessarily implied that the smoothing kernel had to be a Gaussian. Lindeberg [20] show... |

129 |
Gray-level corner detection
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- 1982
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Citation Context ...tector can be obtained by very simple means. 6.2 Junction Detection An entity commonly used for junction detection is the curvature of level curves in intensity data; see, e.g., Kitchen and Rosenfeld =-=[14]-=-, or Koenderink and Richards [17]. In terms of derivatives of the intensity function with respect to the (x; y)-, and (u; v)-coordinates respectively, it can be expressed ass= L 2 y L xx \Gamma 2L x L... |

98 | Scale-space for discrete signals
- Lindeberg
- 1990
(Show Context)
Citation Context ...d et al [2] gave a particular proof in the one-dimensional case and showed that natural constraints on a smoothing kernel necessarily implied that the smoothing kernel had to be a Gaussian. Lindeberg =-=[20]-=- showed that a variation-diminishing property of not introducing new local extrema (or equivalently of not introducing new zero-crossings) in the smoothed signal with increasing scale, combined with a... |

85 | Steerable-scalable kernels for edge detection and junction analysis
- Perona
- 1992
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Citation Context ...n for the nth order directional derivative @ n �� ff of a function L in any direction ff, @ n �� ff L = (cos ff @ x + sin ff @ y ) n L: (59) In the terminology of Freeman and Adelson [12], and=-= Perona [32], these ke-=-rnels are trivially "steerable" (as is the directional derivative of any continuously differentiable function). Figure 4 gives an illustration of what might happen if the sampling problems a... |

54 |
Receptive field families
- Koenderink, Doorn
- 1990
(Show Context)
Citation Context ...space representation satisfy the diffusion equation, it follows that they will exhibit the same scale-space properties over scales as the smoothed grey-level signal; see also Koenderink and van Doorn =-=[18]-=-, who showed that scale invariance applied to operators derived from the scale-space representation leads to the derivatives of the Gaussian kernels. 2.2 Discrete Signals Concerning the discretization... |

50 | On scale selection for differential operators
- Lindeberg
- 1993
(Show Context)
Citation Context ...arts remains constant over scales. Such kernels have been used in edge detection, by for example Korn [19] and Zhang and Bergholm [36], for automatic scale selection in feature detection by Lindeberg =-=[26], and in shape f-=-rom texture by Lindeberg and Garding [27]. Then, the following relations are useful; Z 1 ��=0 (@ x g)(��; t) d�� = \Gammag(0; t) = 1 p 2��t ; 1 X n=0 (ffi x T )(n; t) = \GammaT (0; t):... |

42 | Shape from texture from a multi-scale perspective
- Lindeberg, Garding
- 1993
(Show Context)
Citation Context ... used in edge detection, by for example Korn [19] and Zhang and Bergholm [36], for automatic scale selection in feature detection by Lindeberg [26], and in shape from texture by Lindeberg and Garding =-=[27]. Then, the foll-=-owing relations are useful; Z 1 ��=0 (@ x g)(��; t) d�� = \Gammag(0; t) = 1 p 2��t ; 1 X n=0 (ffi x T )(n; t) = \GammaT (0; t): (53) In practice, to give the equivalent effect of norma... |

38 |
Scale-space behaviour of local extrema and blobs
- Lindeberg
- 1992
(Show Context)
Citation Context ...ner; the implicit function theorem can be used for defining paths across scales and deriving closed form expressions for the drift velocity of feature points with respect to scalespace smoothing; see =-=[23] and [25] for detail-=-s. For example, for a curved edge given by non-maximum suppression, i.e., L �� v��v = 0, the drift velocity in the normal direction of the curve, (ff �� u ; ff �� v ) = (L 2 �� v L... |

33 |
Two-dimensional curvature operators
- JJ, Richards
- 1988
(Show Context)
Citation Context ...mple means. 6.2 Junction Detection An entity commonly used for junction detection is the curvature of level curves in intensity data; see, e.g., Kitchen and Rosenfeld [14], or Koenderink and Richards =-=[17]-=-. In terms of derivatives of the intensity function with respect to the (x; y)-, and (u; v)-coordinates respectively, it can be expressed ass= L 2 y L xx \Gamma 2L x L y L xy + L 2 x L yy (L 2 x + L 2... |

29 |
Towards a symbolic representation of intensity changes in images
- Korn
- 1988
(Show Context)
Citation Context ...seful to normalize the kernels used for derivative computations so that the integral of positive parts remains constant over scales. Such kernels have been used in edge detection, by for example Korn =-=[19]-=- and Zhang and Bergholm [36], for automatic scale selection in feature detection by Lindeberg [26], and in shape from texture by Lindeberg and Garding [27]. Then, the following relations are useful; Z... |

28 |
Fast computation of the difference of low-pass transform
- Crowley, Stern
- 1984
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Citation Context ...s and discrete approximations to differential geometric descriptors using the proposed framework. 12 Similar operators have also been used in pyramid representations; see, e.g., Burt [5], and Crowley =-=[7]-=-. 13 The same problem arises also if the computations are performed in the Fourier domain, since at least one inverse FFT transformation will be needed for each derivative approximation. 4.2 Normaliza... |

28 |
Discrete Scale-Space Theory and the Scale-Space Primal Sketch. Ph. D. dissertation, Dept. of Numerical Analysis and Computing Science
- Lindeberg
- 1991
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Citation Context ...curvature at a number of different scales, and then taking the absolute value in every point. In order to enhance the maxima in the output, a certain type of blob detection, grey-level blob detection =-=[22, 21]-=-, has been applied to the curvature data. Basically, each grey-level blob corresponds to one local extremum and vice versa. Observe that at fine scales mainly blobs due to noise are detected, while at... |

19 |
Topological and Geometrical Aspects of Image Structure
- Blom
- 1992
(Show Context)
Citation Context ...ually multiplied by the gradient magnitude raised to some power, n. A natural choice is n = 3. This leads to a polynomial expression. Moreover, the resulting operator becomes skew invariant; see Blom =-=[3]. ~s= L 3 ��-=- vs= L 2 �� v L �� u��u = L 2 y L xx \Gamma 2L x L y L xy + L 2 x L yy : (64) Figure 6 displays the result of computing this rescaled level curve curvature at a number of different scales,... |

19 | Active detection and classification of junctions by foveation with a head-eye system guided by the scale-space primal sketch
- Brunnstrom, Lindeberg, et al.
- 1992
(Show Context)
Citation Context ...ar) computation of the differential descriptor, generates output results much more useful for further processing than the single scale extraction of grey-level blobs illustrated in Figure 6; see also =-=[4]-=- and [27]. A more general method for scale selection is decribed in [26]. 7 Summary and Discussion The main subject of this paper has been to describe a canonical way to discretize the primary compone... |

19 |
Smoothed differentiation filters for images
- Meer, Weiss
(Show Context)
Citation Context ...on the relations to standard discrete operators used in numerical analysis; see, e.g., Dahlquist et al [8]. Other design criteria may lead to other operators, see, e.g., Haralick [13], Meer and Weiss =-=[29]-=-, and Vieville, Faugeras [33]. Nevertheless, the algebraic scale-space properties are preserved whatever linear operators are used. 4 Computational Implications 4.1 Derivative Approximations Directly ... |

18 |
The design and use of steerable filters for image analysis
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- 1991
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Citation Context ...l-known expression for the nth order directional derivative @ n �� ff of a function L in any direction ff, @ n �� ff L = (cos ff @ x + sin ff @ y ) n L: (59) In the terminology of Freeman and =-=Adelson [12], and Pero-=-na [32], these kernels are trivially "steerable" (as is the directional derivative of any continuously differentiable function). Figure 4 gives an illustration of what might happen if the sa... |

15 |
Biased anisotropic diffusion—A unified regularization and diffusion approach to edge detection
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Citation Context ...consider is non-linear diffusion, although further work may be needed in order to develop the notion of anisotropic smoothing as introduced into computer vision by Perona and Malik [31] and Nordstrom =-=[30]-=-. A Appendix A.1 Summary of Main Results from 2D Discrete Scale-Space Theory Below are stated for reference purpose some basic definitions and key results from the (zeroorder) scale-space theory for t... |

14 |
Scale selection for dierential operators
- Lindeberg
- 1994
(Show Context)
Citation Context ...arts remains constant over scales. Such kernels have been used in edge detection, by for example Korn [19] and Zhang and Bergholm [36], for automatic scale selection in feature detection by Lindeberg =-=[26]-=-, and in shape from texture by Lindeberg and Garding [27]. Then, the following relations are useful� Z 1 =0 (@xg)( � t) d = ;g(0� t) = 1 p � 2 t 1X n=0 ( xT )(n� t) =;T (0� t): (53) In practice, to gi... |

12 | Eklundh, “On the computation of a scale-space primal sketch - Lindeberg, O - 1991 |

11 |
Robust and fast computation of unbiased intensity derivatives in images
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- 1992
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Citation Context ...discrete operators used in numerical analysis; see, e.g., Dahlquist et al [8]. Other design criteria may lead to other operators, see, e.g., Haralick [13], Meer and Weiss [29], and Vieville, Faugeras =-=[33]-=-. Nevertheless, the algebraic scale-space properties are preserved whatever linear operators are used. 4 Computational Implications 4.1 Derivative Approximations Directly from Smoothed Grey-Level Data... |

10 | A (1987) Representation of Local Geometry - Koenderink, Doorn |

10 |
An extension of Marr’s signature based edge classification other methods determining diffuseness and height of edges and bar edge width”, IEEE 4 conference on computer vision
- Zhang, Bergholm
- 1993
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Citation Context ...ls used for derivative computations so that the integral of positive parts remains constant over scales. Such kernels have been used in edge detection, by for example Korn [19] and Zhang and Bergholm =-=[36], for automa-=-tic scale selection in feature detection by Lindeberg [26], and in shape from texture by Lindeberg and Garding [27]. Then, the following relations are useful; Z 1 ��=0 (@ x g)(��; t) d�� =... |

7 |
Smoothed Di erentiation Filters for Images
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- 1992
(Show Context)
Citation Context ...on the relations to standard discrete operators used in numerical analysis� see, e.g., Dahlquist et al [8]. Other design criteria may lead to other operators, see, e.g., Haralick [13], Meer and Weiss =-=[29]-=-, and Vieville, Faugeras [33]. Nevertheless, the algebraic scale-space properties are preserved whatever linear operators are used. 4 Computational Implications 4.1 Derivative Approximations Directly ... |

5 | Scale-space for N-dimensional discrete signals
- Lindeberg
- 1994
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Citation Context ...dimension a fruitful requirement turns out to be as follows; if at some level a point is a local maximum (minimum), then its value must not increase (decrease) when the scale parameter increases, see =-=[20, 24]-=-. In the continuous case, this condition, which is similar to the maximum principle for parabolic differential equations, is equivalent to the causality requirement used by Koenderink [15] for derivin... |

4 |
Handbook of Mathematical Functions, Applied Mathematical Series, 55 (Washington: National Bureau of Standards) (1964). Iнституту фiзики конденсованих систем НАН України розповсюджуються серед наукових та iнформацiйних установ. Вони також доступнi по елект
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Citation Context ... T : Z \Theta R+ ! R denotes the discrete analogue of the Gaussian kernel T (n; t) = e \Gammat I n (t); (10) and I n are the modified Bessel functions of integer order, see e.g. Abramowitz and Stegun =-=[1]-=-. This scale-space family can equivalently be defined from the solution of the semi-discretized version of the diffusion equation (@ t L)(x; t) = 1 2 (r 2 3 L)(x; t) = 1 2 (L(x \Gamma 1; t) \Gamma L(x... |

4 |
Ter Haar Romeny B.M., Koenderink J.J., and Viergever M. Scale and the differential structure of images
- Florack
- 1992
(Show Context)
Citation Context ..., feature detectors formulated in terms of differential singularities by definition commute with a number of elementary transformations of the spatial and intensity domains, 16 See also Florack et al =-=[10]-=- concerning necessity results regarding a similar (but not identical) set of transformations. and it does not matter whether the transformation is performed before or after the smoothing step. Above, ... |

4 | Scale-space behaviour and invariance properties of differential singularities - Lindeberg - 1994 |

4 |
Active detection and classi cation of junctions by foveation with a head-eye system guided by the scale-space primal sketch
- Brunnstrom, Lindeberg, et al.
- 1992
(Show Context)
Citation Context ...ear) computation of the di erential descriptor, generates output results much more useful for further processing than the single scale extraction of grey-level blobs illustrated in Figure 6� see also =-=[4]-=- and [27]. A more general method for scale selection is decribed in [26]. 7 Summary and Discussion The main subject of this paper has been to describe a canonical way to discretize the primary compone... |

4 |
Fast computation of the Di#erence of Low Pass Transform
- Crowley
- 1984
(Show Context)
Citation Context ...ns and discrete approximations to di erential geometric descriptors using the proposed framework. 12 Similar operators have also been used in pyramid representations� see, e.g., Burt [5], and Crowley =-=[7]-=-. 13 The same problem arises also if the computations are performed in the Fourier domain, since at least one inverse FFT transformation will be needed for each derivative approximation. 12s4.2 Normal... |

3 |
The design and use of steerable lters for image analysis, enhancement, and wavelet representation
- Freeman, Adelson
- 1990
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Citation Context ... L x i y j using the well-known expression for the nth order directional derivative @ n of a function L in any direction , @ n L = (cos @x +sin @y) n L: (59) In the terminology of Freeman and Adelson =-=[12]-=-, and Perona [32], these kernels are trivially \steerable" (as is the directional derivative ofany continuously di erentiable function). Figure 4 gives an illustration of what might happen if the samp... |

1 |
Biased Anisotropic Di usion|A Uni ed Regularization and Di usion Approach to Edge Detection
- Nordstrom
- 1990
(Show Context)
Citation Context ... consider is non-linear di usion, although further work may be needed in order to develop the notion of anisotropic smoothing as introduced into computer vision by Perona and Malik [31] and Nordstrom =-=[30]-=-. A Appendix A.1 Summary of Main Results from 2D Discrete Scale-Space Theory Below are stated for reference purpose some basic de nitions and key results from the (zeroorder) scale-space theory for tw... |