## The Intrinsic Structure of Optic Flow Incorporating Measurement Duality (1997)

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Venue: | International Journal of Computer Vision |

Citations: | 20 - 13 self |

### BibTeX

@ARTICLE{Florack97theintrinsic,

author = {Luc Florack and Wiro Niessen and Mads Nielsen},

title = {The Intrinsic Structure of Optic Flow Incorporating Measurement Duality},

journal = {International Journal of Computer Vision},

year = {1997},

volume = {27},

pages = {263--286}

}

### Years of Citing Articles

### OpenURL

### Abstract

The purpose of this report 1 is to define optic flow for scalar and density images without using a priori knowledge other than its defining conservation principle, and to incorporate measurement duality, notably the scale-space paradigm. It is argued that the design of optic flow based applications may benefit from a manifest separation between factual image structure on the one hand, and goal-specific details and hypotheses about image flow formation on the other. The approach is based on a physical symmetry principle known as gauge invariance. Data-independent models can be incorporated by means of admissible gauge conditions, each of which may single out a distinct solution, but all of which must be compatible with the evidence supported by the image data. The theory is illustrated by examples and verified by simulations, and performance is compared to several techniques reported in the literature. 1 Introduction The conventional "spacetime" representation of a movie as...

### Citations

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Citation Context ...e without reference to subjective factors. The approach adopted in this article is inspired by original work of Arnspang [2, 3, 4, 5], which in turn pursues the classical approach of Horn and Schunck =-=[31]-=-. However, the novelty of this article is threefold: ffl Measurement duality is taken into account. More specifically we think of optic flow extraction as a measurement process on raw data that cannot... |

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Citation Context ...(but in principle ad hoc) representation of the factual homotopy is normal flow , sometimes called the optic flow field, to be distinguished from the physically induced image velocity or motion field =-=[29, 30]-=-. A gauge transformation, in casu any isophote automorphism, maps one admissible vector field onto another without affecting the data. The extra constraint introduced to single out a unique solution i... |

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Citation Context ...el [68] and by Werkhoven and Koenderink [83]. We also describe a quantitative study on a benchmark sequence, and compare performance with results from existing techniques as reported by Barron et al. =-=[7]-=-. We summarise results in Section 6 and discuss some aspects relevant for applications and future development. 2 Preliminaries This section serves to introduce some preliminary concepts that are essen... |

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Citation Context ...eld is not of a "speedometer" type (one velocity per base point), but is defined by virtue of a set of fiducial "point operators". In particular we consider the familiar Gaussian s=-=cale-space paradigm [35, 39, 84], and show-=- in which precise sense the problem of "deep structure" naturally carries over to optic flow. This problem is induced by the fact that there are 1 Accepted for publication in the Internation... |

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Citation Context ...lations naturally arise as a consequence of apparent conservation laws. It is for this reason that the concept of "optic flow" has been introduced by Gibson in the context of optical pilot n=-=avigation [26]-=-. Nowadays optic flow has become a familiar, yet still confusing concept in computer vision and image analysis. A meaningful definition depends very much on data-independent models. It is therefore of... |

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Citation Context ...ure [7]. In this section we evaluate zeroth and first order approximations for the two image sequences known as the Translating Tree Sequence (TTS) and the Diverging Tree Sequence (DTS), respectively =-=[7, 16]-=-: Figure 18. Notation: ~ u and ~ v denote horizontal and vertical components of the ideal velocity field, u; v are the zeroth order coefficients in the polynomial expansion, again without explicit ref... |

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Citation Context ...t potential image analysis. The topological dual G 0 (IR n ) is nothing but the well-known scale-space representation [35, 84], more precisely, its stratification into local jets of successive orders =-=[23, 27, 43, 48, 49, 69]-=-. It has a straightforward implementation up to some order and within physical scale limits (grain and scope). We will henceforth assume familiarity with the basics of scale-space theory [6, 19, 20, 2... |

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Citation Context ... problem", one that can be overcome as soon as "enough structure" is present in the image brightness, e.g. by taking into account "corner points", or the image's "higher =-=order differential structure" [60, 61, 62, 68, 75, 76, 79, 82, 83]-=-. That this is a misconception follows immediately from the trivial invariance of image structure under isophote automorphisms, making any tangential flow component conceivable (recall Figure 1). Data... |

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Citation Context ...a "naked" source field (a raw image sequence in digital format, say) and a conventionally designed class of probing devices, or filters. This state of affairs is familiar to mathematicians a=-=s duality [8, 9, 74, 78]-=-, and is a public fact to physicists. Needless to say that duality affects image analysis. It introduces an inevitable bias; we are interested in the source, not in the device. To some extent the bias... |

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Citation Context ...3]. If we adopt public consensus as our guideline, then optic flow must be a homotopy rather than a vector field; see Figure 1. The ambiguity of pointwise connections is known as the aperture problem =-=[28, 59, 81]. As soon -=-as one attempts to extend the optic flow definition beyond the intrinsically defined homotopy, one has to come up with a model in order to "solve the aperture problem". Different models yiel... |

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Citation Context ... Computer Vision. 1 no a priori preferred spatial and temporal inner scales. Scale preferences are most sensibly inferred a posteriori from image structure given a practical task and a suitable model =-=[55, 56, 58, 64, 65, 66]-=-. In particular, spatial and temporal scales can be adjusted to flow field, and vice versa, which seems natural if only for considerations of dimensional analysis: velocity and spacetime scales must b... |

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Citation Context ...s definition. Broad support exists with regard to the assertion that optic flow is constrained by some conservation principle. The classical example is the well-known "Optic Flow Constraint Equat=-=ion" [2, 3, 4, 5, 7, 31, 32, 60, 61, 68, 71, 75, 76, 79, 83]-=-. If we adopt public consensus as our guideline, then optic flow must be a homotopy rather than a vector field; see Figure 1. The ambiguity of pointwise connections is known as the aperture problem [2... |

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Citation Context ...by a data driven feedforward procedure, and is basically a kinematic reformatting of spatiotemporal image data. 2.3 Computational Problems There are many possible approaches to optic flow measurement =-=[7, 36]-=-, not all of which have been investigated in-depth. The classical approach based on the OFCE defines the optic flow vector field locally by means of a conservation law; invariant grey-values are attri... |

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Citation Context ...3]. If we adopt public consensus as our guideline, then optic flow must be a homotopy rather than a vector field; see Figure 1. The ambiguity of pointwise connections is known as the aperture problem =-=[28, 59, 81]. As soon -=-as one attempts to extend the optic flow definition beyond the intrinsically defined homotopy, one has to come up with a model in order to "solve the aperture problem". Different models yiel... |

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Citation Context ...or example, in the case of real-world movies, 1-st order properties of the vector field may reveal relevant information such as qualitative shape properties, surface slant [47], and time-to-collision =-=[51, 52]-=-. Unlike 1-st order, 2-nd order is quantitatively related to intrinsic surface properties of an object [50]. There is no a priori limit to the highest order that is still accessible and significant; t... |

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Citation Context ... problem", one that can be overcome as soon as "enough structure" is present in the image brightness, e.g. by taking into account "corner points", or the image's "higher =-=order differential structure" [60, 61, 62, 68, 75, 76, 79, 82, 83]-=-. That this is a misconception follows immediately from the trivial invariance of image structure under isophote automorphisms, making any tangential flow component conceivable (recall Figure 1). Data... |

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Citation Context ..., 48, 49, 69]. It has a straightforward implementation up to some order and within physical scale limits (grain and scope). We will henceforth assume familiarity with the basics of scale-space theory =-=[6, 19, 20, 22, 23, 33, 34, 35, 38, 40, 41, 42, 43, 44, 46, 53, 54, 56, 57, 58, 84]-=-. Especially its interpretation in the context of topological duality will turn out to be crucial for our optic flow definition. 2.2 Aperture Problem and Optic Flow Ambiguity It is taken for granted t... |

92 |
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Citation Context ...s definition. Broad support exists with regard to the assertion that optic flow is constrained by some conservation principle. The classical example is the well-known "Optic Flow Constraint Equat=-=ion" [2, 3, 4, 5, 7, 31, 32, 60, 61, 68, 71, 75, 76, 79, 83]-=-. If we adopt public consensus as our guideline, then optic flow must be a homotopy rather than a vector field; see Figure 1. The ambiguity of pointwise connections is known as the aperture problem [2... |

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Citation Context ..., 48, 49, 69]. It has a straightforward implementation up to some order and within physical scale limits (grain and scope). We will henceforth assume familiarity with the basics of scale-space theory =-=[6, 19, 20, 22, 23, 33, 34, 35, 38, 40, 41, 42, 43, 44, 46, 53, 54, 56, 57, 58, 84]-=-. Especially its interpretation in the context of topological duality will turn out to be crucial for our optic flow definition. 2.2 Aperture Problem and Optic Flow Ambiguity It is taken for granted t... |

90 |
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Citation Context ...(but in principle ad hoc) representation of the factual homotopy is normal flow , sometimes called the optic flow field, to be distinguished from the physically induced image velocity or motion field =-=[29, 30]-=-. A gauge transformation, in casu any isophote automorphism, maps one admissible vector field onto another without affecting the data. The extra constraint introduced to single out a unique solution i... |

89 |
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Citation Context ...t potential image analysis. The topological dual G 0 (IR n ) is nothing but the well-known scale-space representation [35, 84], more precisely, its stratification into local jets of successive orders =-=[23, 27, 43, 48, 49, 69]-=-. It has a straightforward implementation up to some order and within physical scale limits (grain and scope). We will henceforth assume familiarity with the basics of scale-space theory [6, 19, 20, 2... |

82 |
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Citation Context ...s usually too restrictive. For example, in the case of real-world movies, 1-st order properties of the vector field may reveal relevant information such as qualitative shape properties, surface slant =-=[47]-=-, and time-to-collision [51, 52]. Unlike 1-st order, 2-nd order is quantitatively related to intrinsic surface properties of an object [50]. There is no a priori limit to the highest order that is sti... |

81 |
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Citation Context ...e without reference to subjective factors. The approach adopted in this article is inspired by original work of Arnspang [2, 3, 4, 5], which in turn pursues the classical approach of Horn and Schunck =-=[31]-=-. However, the novelty of this article is threefold: Measurement duality is taken into account. More speci cally we think of optic ow extraction as a measurement process on raw data that cannot be dec... |

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Citation Context ...or example, in the case of real-world movies, 1-st order properties of the vector field may reveal relevant information such as qualitative shape properties, surface slant [47], and time-to-collision =-=[51, 52]-=-. Unlike 1-st order, 2-nd order is quantitatively related to intrinsic surface properties of an object [50]. There is no a priori limit to the highest order that is still accessible and significant; t... |

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Citation Context ...differentiation is anti-symmetric. Combined with the point concept this leads one to consider the Gaussian family , G(IR n ) ae S(IR n ), i.e. the class of all derivatives of the basic point operator =-=[37, 45]-=-. It is a complete family, and thus does not limit potential image analysis. The topological dual G 0 (IR n ) is nothing but the well-known scale-space representation [35, 84], more precisely, its str... |

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Citation Context ...test and illustration of the theory based on an analytically tractable stimulus. In Section 5 we make a conceptual comparison with similar models proposed in the literature, notably by Otte and Nagel =-=[68]-=- and by Werkhoven and Koenderink [83]. We also describe a quantitative study on a benchmark sequence, and compare performance with results from existing techniques as reported by Barron et al. [7]. We... |

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Citation Context ... Computer Vision. 1 no a priori preferred spatial and temporal inner scales. Scale preferences are most sensibly inferred a posteriori from image structure given a practical task and a suitable model =-=[55, 56, 58, 64, 65, 66]-=-. In particular, spatial and temporal scales can be adjusted to flow field, and vice versa, which seems natural if only for considerations of dimensional analysis: velocity and spacetime scales must b... |

24 |
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Citation Context ...adequacy of local semantics. For example, in the context of machine vision none of the assertions that motion is induced by projection of a shaded, sufficiently smooth, rigid surface patch, et cetera =-=[29, 31, 77, 80]-=-, is supported by the evidence; at best there is no contradiction. In medical imagery blood flow satisfies physical incompressibility and continuity constraints, whereas bone tissue induces rigid moti... |

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Citation Context ..., 48, 49, 69]. It has a straightforward implementation up to some order and within physical scale limits (grain and scope). We will henceforth assume familiarity with the basics of scale-space theory =-=[6, 19, 20, 22, 23, 33, 34, 35, 38, 40, 41, 42, 43, 44, 46, 53, 54, 56, 57, 58, 84]-=-. Especially its interpretation in the context of topological duality will turn out to be crucial for our optic ow de nition. 2.2 Aperture Problem and Optic Flow Ambiguity It is taken for granted that... |

21 |
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(Show Context)
Citation Context ...eld is not of a "speedometer" type (one velocity per base point), but is defined by virtue of a set of fiducial "point operators". In particular we consider the familiar Gaussian s=-=cale-space paradigm [35, 39, 84], and show-=- in which precise sense the problem of "deep structure" naturally carries over to optic flow. This problem is induced by the fact that there are 1 Accepted for publication in the Internation... |

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(Show Context)
Citation Context ..., 48, 49, 69]. It has a straightforward implementation up to some order and within physical scale limits (grain and scope). We will henceforth assume familiarity with the basics of scale-space theory =-=[6, 19, 20, 22, 23, 33, 34, 35, 38, 40, 41, 42, 43, 44, 46, 53, 54, 56, 57, 58, 84]-=-. Especially its interpretation in the context of topological duality will turn out to be crucial for our optic ow de nition. 2.2 Aperture Problem and Optic Flow Ambiguity It is taken for granted that... |

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The motion constraint equation for the optical flow
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Citation Context ...s definition. Broad support exists with regard to the assertion that optic flow is constrained by some conservation principle. The classical example is the well-known "Optic Flow Constraint Equat=-=ion" [2, 3, 4, 5, 7, 31, 32, 60, 61, 68, 71, 75, 76, 79, 83]-=-. If we adopt public consensus as our guideline, then optic flow must be a homotopy rather than a vector field; see Figure 1. The ambiguity of pointwise connections is known as the aperture problem [2... |