B.K.P Horn and B.G. Schunck. Determining optical ow. Arti cial Intelligence, 17(1-3):185-203, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....poor recordings, poor recording conditions, restricted recording devices media, or because the objects appear identical anyway. The research elds concerned with these issues are among others object tracking [25] feature or token tracking [3] 13] 27] 38] and optical ow or motion estimation [12] [19] Applications range from surveillance [20] 30] 36] motion analysis, and structure from motion [31] 32] 35] 37] to (multi )target tracking [11] 21] 24] Here, we restrict ourselves to the case that for some reason the objects have indeed an identical appearance, which leaves us with ....

B.K.P. Horn and B.G. Schunck. Determining optical ow. Arti cial Intelligence, 17:185-203, 1981.

....Fluid Motion Estimation Application The motion estimator on which we focus on in this study [1] aims at recovering the appparent motion eld between two consecutive images of a sequence showing the evolution of a uid ow. It is a dedicated adaptation of an Horn and schunck optical ow estimator [4]. Such an estimator is de ned as the minimizer of a functionnal H = H 1 H 2 composed of two terms. The rst one encapsulates the data model term. This term gives an adequacy expression between the unkown (here the motion eld w) and the available observations (the luminance function f(x; t) ....

B. Horn and B. Schunck. Determining optical ow. Arti cial Intelligence, 17:185-203, 1981.

....of the warplet coecients, called the warpogram, thus undergoes a transport in the position scale parameter space. This fundamental transport equation obeyed by the warpogram is called Texture Gradient Equation. It can be seen as the analog of the Optical Flow Equation for motion estimation [13]. Whereas the velocity term in the Optical Flow Equation is related to the projection of the 3D velocity in the image, here, the velocity measures relative texture distorsion in the image. The texture distorsion is thus calculated by estimating the di erent terms of this equation. The next step ....

....s) v(u; s) log s w(u; s) 0 : 8) The velocity term v(u; s) can be interpreted as a texture gradient: it measures how the image energy moves across scales, according to the position in the image. The Texture Gradient Equation (8) is comparable to the Optical Flow Equation for motion estimation [13]. The conservation equation 1 0 1 0 0.2 0.4 0.6 0.8 1 4 2 log s 1 0 1 0 0.2 0.4 0.6 0.8 1 4 2 log s (b) Figure 3: a) Top: a realization of stationary process R(x) Bottom: scalogram EfjhR; u;s ij g. The horizontal and vertical axes represent u and log s, respectively. ....

B.K.P Horn and B.G. Schunck. Determining optical ow. Arti cial Intelligence, 17(1-3):185-203, 1981.

....vector eld of apparent velocities of light patterns in an image; see [26, 65] and the references therein. There are a number of approaches to motion eld estimation which are based on trying to compute the intensity ow. One of the most popular approaches in this area is due to Horn and Schunck [27] who base their method on the assumption that the image intensities of scene points are preserved over time together with a smoothness constraint. Their basic premise is that optical ow cannot be computed locally, since only one measurement is available at any point, while the optical ow vector ....

B. Horn and B. Schunck, \Determining optical ow," Arti cial Intelligence 23, pp. 185-203, 1981.

....side of (2.1) is f(x 1 (x; y) t; y 2 (x; y) t; t t) f(x; y; t) rf(x; y; t) 2.3) where = B B C C t. Substituting into (2.1) we obtain the gradient constraint equation: rf(x; y; t) 0: 2.4) The gradient constraint equation (2. 4) was rst proposed by Horn and Schunk [43] for computing optical ow elds and is the foundation of many optical ow and image registration algorithms. 2.2.2 The Aperture Problem We call attention to the fact that components of motion to rf(x; y; t) can not be computed from (2.4) This is an example fo the aperture problem, which refers ....

B. Horn and B. Schunk. Determining optical ow. Arti cial Intelligence, 17:185-203, 1981.

....decreases. The method does not depend on any particular algorithm, though the estimation of the IMI can be optimized for a particular method. We propose methods for estimating the MI using statistical sampling techniques. Using an example of reconstructing a scene from video using optical ow [9], 8] and Gaussian noise distribution, we show how the incremental MI can be computed from rst principles in terms of the input parameters. The paper is organized as follows. We start with an overview of error analysis methods in SfM and a brief survey of the use of information theory in computer ....

....satisfactory in many practical examples. In this paper, we focus on evaluating the quality of the reconstruction from video sequences with a small baseline. Since the motion between nearby frames of a video sequence is usually small, the SfM equations based on motion estimates from optical ow [9] is typically valid. However, since the motion is small, even a small amount of error in motion estimates can lead to large errors in structure estimates. This is the classical low signal to noise ratio case in signal processing. In our experiments, we have observed that the error can often be as ....

B.K.P. Horn and B.G. Schunck, \Determining optical ow," AI, vol. 17, pp. 185-203, 1981.

.... provides a practical description of a Bayesian multi scale gradient based optical ow estimation algorithm, based on work previously published in [6; 4; 7] 14.2 Di erential formulation Gradient formulations of optical ow begin with the di erential brightness constancy constraint equation [8]: g f g t = 0; 14.1) where rg and g t are the spatial image gradient and temporal derivative, respectively, of the image at a given spatial location and time (for notational simplicity, these parameters are omitted) The equation places a single linear constraint on the two dimensional ....

....399 vanishes, the equation provides no constraint on the velocity vector. This is sometimes called the blank wall problem. Typically, the aperture problem is overcome by imposing some form of smoothness on the eld of velocity vectors. Many formulations use global smoothness constraints [8], which require global optimization . Alternatively, one may assume locally constant velocity and combine linear constraints over local spatial (or temporal) regions [10] This is accomplished by writing a weighted sum of squares error function based on the constraints from each point within a ....

B K P Horn and B G Schunck. Determining optical ow. Arti cial Intelligence, 17:185-203, 1981.

....In the optic ow setting, f is the scalar valued image sequence and u describes the optic ow eld. vector valued di usion process optic ow regularizer t u i = x V u ix y V u iy V (rf; ru) homogeneous homogeneous t u i = u i (scalar case: Iijima 1959 [14] Horn Schunck 1981 [13]) linear isotropic image driven, isotropic g(jrf j ) scalar case: Fritsch 1992 [9] Alvarez et al. 1999 [1] linear anisotropic image driven, anisotropic 2 D(rf)ru i (scalar case: Iijima 1962 [15] Nagel 1983 [18] nonlinear isotropic ow driven, ....

B. Horn and B. Schunck, Determining optical ow, Arti cial Intelligence, 17 (1981), pp. 185-203.

....y; z) where (x; y) denotes the location and z 2 [0; Z] is the time. We are looking for the optic ow eld u(x;y;z) v(x;y;z) which describes the correspondence of image structures at di erent times. Variational methods constitute one possibility to solve the optic ow problem; see e.g. [14, 22, 8, 37]. In [36] a method is considered which is based on the following two assumptions: 1. Image structures do not change their grey value over time. Therefore, along their path (x(z) y(z) one obtains 0 = df(x(z) y(z) z) dz = f x u f y v f z : 19) 2. As second assumption we impose a ....

B. Horn and B. Schunck, Determining optical ow, Arti cial Intelligence, 17 (1981), pp. 185{ 203.

....image analysis problems at all levels. Most of the MRF models are for lowlevel processing. These include image restoration and segmentation [61,56,23,36,46,21,27,85,89,92] surface reconstruction [8,53,98,17,99,24,43] edge detection [103,131,44] texture analysis [23,30,40,34,35] optical ow [64,63,78,118,60], shape from X [9,68] active contours [74,3,123] deformable templates [97,95,70] data fusion [26] visual integration, and perceptual organization [2,125] The use of MRFs in high level, such as for object matching and recognition, has also emerged in recent years ....

....abruptly. For example, the surface of a table is at, a meadow presents a texture of grass, and a temporal event does not change abruptly over a short period of time. Indeed, we can always nd regularities of aphysical phenomenon with respect to certain properties. Since its early applications [53,64,68] aimed to impose constraints, in addition to those from the data, on the computation of image properties, the smoothness prior has been one of the most popular prior assumptions in low level problems. It has been developed into a general framework, called regularization [109,11] for ....

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B. K. P. Horn and B. G. Schunck. \Determining optical ow". Articial Intelligence, 17:185-203, 1981.

....move space. All the methods we develop use graph cuts as a powerful optimization technique. The energy minimization approach is very popular in the vision eld, and was widely applied to many vision problems. The amount of literature on the subject is immense, see for example Horn and Schunck [23] for optical ow; Geman and Geman [20] for image restoration; Blake and Zisserman [8] for surface reconstruction, Barnard [3] for stereo matching; Derin and Elliott [15] and Hofmann et al. 22] for texture segmentation; Torre and Poggio [41] for edge detection. For a brief overview of energy ....

....should have low cost, it is frequently hard to formalize these ideas concisely. A popular choice of prior which can be easily formalized expresses smoothness constraints on the labelings. The smoothness assumption is one of the oldest in vision (see Marr and Poggio [31] or Horn and Schunck [23]) It is suitable for problems where the quantity to be estimated varies smoothly everywhere or almost everywhere. In many vision problems the quantity varies smoothly everywhere except at object boundaries where it may change abruptly. Such an abrupt change is called a discontinuity. In this work ....

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B.K.P. Horn and B.G. Schunck. Determining optical ow. Articial Intelligence, 17:185-203, 1981.

....The overall smoothness is regulated by the constant . 2. 6 Direct Flow Regularisation Instead of performing a TLS analysis rst one might want to directly try to nd the ow eld by imposing the smoothness constraint, in analogy to the well known optical ow algorithm by Horn and Schunck [9]. This simple smoothness term is not generally advisable, mainly because problematic locations ( 4 0) are equally taken into account. As well, smoothing occurs across motion discontinuities. On the other hand, this regularisation works very well when a segmentation can be achieved by other ....

Horn B.K.P. and Schunck B.G. (1981), Determining optical ow, Articial Intelligence (AI), vol. 17, pp185-204.

....et al. 6] pointed out that, depending on its formulation, optic ow calculations may be ill conditioned or even ill posed. It is therefore natural to introduce additional smoothness constraints in order to stabilize or regularize the process. This way has been pioneered by Horn and Schunck [26] and improved by Nagel [35] and many others. Variationals approaches of this type calculate optic ow as the minimizer of an energy functional, which consists of a data term and a smoothness term. Formulations in terms of energy functionals allow a conceptually clear formalism without any hidden ....

....methods perform well [5, 21] The data term in the energy functional involves optic ow constraints such as the assumption that corresponding pixels in di erent frames should reveal the same grey value. The smoothness term usually requires that the optic ow eld should vary smoothly in space [26]. Such a term may be modi ed in an image driven way in order to suppress smoothing at or across image boundaries [1, 35] As an alternative, ow driven modi cations have been proposed which reduce smoothing across ow discontinuities [8, 12, 14, 30, 42, 45, 57] Most smoothness terms require only ....

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B. Horn, B. Schunck, Determining optical ow, Articial Intelligence, Vol. 17, 185{ 203, 1981.

....ow. Usual assumptions used in other methods are local or global rigidity or anity. The only unbiased local motion we can obtain is the motion orthogonal to the isophotes of the image. This type of velocity eld is called the normal ow. A large variety of methods exist for estimation of optic ow [1,2,4,8,12,19] (which is far from an exhaustive list) See also Barron et al. 3] for a discussion and evaluation of various methods. We wish to develop an algorithm, which can be used for doing motion analysis in experimental uid dynamics (see [20] For that reason it is important that it is least ....

....based on a method by Niessen et al. 18] to select the scale of model validity. In this paper we compute the normal ow by using the method proposed by Florack et al. 4] It is an incorporation of the so called Optic Flow Constraint Equation (OFCE) originally proposed by Horn and Schunck [8], into the linear Gaussian scale space formalism. In this method the normal ow is in general modeled by an M th order polynomial, but we choose the somewhat restrictive zeroth order model. This choice is based on the assumption that locally this model is a good approximation of the normal ow. ....

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B. K. P. Horn and B. G. Schunck. Determining optical ow. Articial Intelligence, 17:185-203, 1981.

....ts pixel p given the observed data. In image restoration, for example, D p (f p ) is normally (f p I p ) 2 , where I p is the observed intensity of p. The choice of E smooth is a critical issue, and many di erent functions have been proposed. For example, in some regularization based approaches [22, 34], E smooth makes f smooth everywhere. This leads to poor results at object boundaries. Energy functions that do not have this problem are called discontinuity preserving. A large number of discontinuity preserving energy functions have been proposed (see for example [21, 29, 42] The major ....

.... at in nity optimality properties. The quality of the solutions that these algorithms produce in practice under realistic cooling schedules is not clear. If the energy minimization problem is phrased in continuous terms, variational methods can be applied. These methods were popularized by [22]. Variational techniques use the Euler equations, which are guaranteed to hold at a local minimum. 4 To apply these algorithms to actual imagery, of course, requires discretization. Another alternative is to use discrete relaxation labeling methods; this has been done by many authors, including ....

B. K. P. Horn and B. Schunk. Determining optical ow. Articial Intelligence, 17:185{ 203, 1981.

....t is not uniquely determined by (2) since it is one equation for two unknown functions u and v. Thus additional constraints have to be imposed and there have been proposed several models in the literature. Variational optical ow computations started with the pioneering work of Horn and Schunck [19] who proposed to calculate an approximate solution of (2) that minimizes the functional J HS ( w) 1 2 Z (jru(x)j 2 jrv(x)j 2 ) dx : 3) Recently there has been a trend to use more sophisticated constraints to preserve edges and corners in the motion eld (see e.g. 23, 8, 27, 30, 2, ....

.... corresponding function (t) t q is Lipschitz continuous with Lipschitz constant L r = C(1 r q 1 ) Thus from Theorem 12 and Corollary 5 it follows that I [ w] attains a minimum on W 1;p( R 2 ) The case q = 1, p = 2 has been studied in [18] The case q = 2, p = 2 goes back to [19] and has been analyzed in [26] As long as W(P ) is quasiconvex, satis es some growth rate and is elliptic, Theorem 12 and Corollary 5 are valid and guarantee weak lower semicontinuity of I [ w] on W 1;p( R d ) and existence of a minimizer. In particular the general results are applicable ....

B. Horn and B. Schunck. Determining optical ow. Artif. Intell., 17:185{ 203, 1981.

....v) t is not uniquely determined by (2) since it is one equation for two unknown functions u and v. Thus additional constraints have to imposed and there have been proposed several models in the literature. Variational optical ow computations started with the pioneering work of Horn and Schunck [19] who proposed to calculate an approximate solution of (2) that minimizes the functional J HS ( w) 1 2 Z (jru(x)j 2 jrv(x)j 2 ) dx : 3) Recently there has been a trend to use more sophisticated constraints to preserve edges and corners in the motion eld (see e.g. 23, 8, 27, 30, 2, ....

.... The according function (t) t q is Lipschitz continuous with Lipschitz constant L r = C(1 r q 1 ) Thus from Theorem 12 and Corollary 5 it follows that I[ w] attains a minimum on W 1;p( R 2 ) The case q = 1, p = 2 has been studied in [18] The case q = 2, p = 2 goes back to [19] and has been analyzed in [26] As long as W(P ) is quasiconvex, satis es some growth rate and is elliptic, Theorem 12 and Corollary 5 are valid and guarantee weak lower semi continuity of I[ w] on W 1;p( R d ) and existence of a minimizer. In particular the general results are ....

B. Horn and B. Schunck. Determining optical ow. Artif. Intell., 17:185-203, 1981.

....only described as algorithmic details, but indeed they are often very crucial for the quality of the algorithm. Thus, it would be consequent to make the role of smoothing more explicit by incorporating it already in a continuous problem formulation. This way has been pioneered by Horn and Schunck [25] and improved by Nagel [34] and many others. Approaches of this type calculate optic ow as the minimizer of an energy functional, which consists of a data term and a smoothness term. Formulations in terms of energy functionals allow a conceptually clear formalism without any hidden model ....

....methods perform well [5, 20] The data term in the energy functional involves optic ow constraints such as the assumption that corresponding pixels in di erent frames should reveal the same grey value. The smoothness term usually requires that the optic ow eld should vary smoothly in space [25]. Such a term may be modi ed in an image driven way in order to suppress smoothing at or across image boundaries [1, 34] As an alternative, ow driven modi cations have been proposed which reduce smoothing across ow discontinuities [8, 12, 14, 29, 40, 43, 54] Most smoothness terms require only ....

[Article contains additional citation context not shown here]

B. Horn, B. Schunck, Determining optical ow, Articial Intelligence, Vol. 17, 185{ 203, 1981.

.... I t = 0; 1) where the subscripts x, y and t represent the partial derivatives. This equation, based on the assumption that the pixel gray level remains constant, relates the temporal and spatial changes of the image gray level I(x; y; t) at point (x; y) to the velocity (u; v) at the same point [14]. Equation (1) is not sucient for computing the image velocity (u; v) at each point since the velocity components are constrained by only one equation; this is the aperture problem. Therefore, most of the techniques use a regularity constraint that restricts the space of admissible solutions of ....

....regularization scheme that preserves ow discontinuities while insuring a unique solution of equation (1) 2.1. Quadratic and Stochastic Models With a quadratic regularization technique, constraining the space of admissible solutions of equation (1) leads to the minimization of the functional [14]: E(u; v) Z jruj 2 jrvj 2 dxdy (2) Z (I x u I y v I t ) 2 dxdy; where (u; v) are the ow eld components, is a Lagrange multiplier (or a regularization parameter) and r denotes the gradient operator. The quadratic regularizer Z jruj 2 jrvj 2 dxdy constrains the ....

[Article contains additional citation context not shown here]

B.K.P. Horn and G. Schunck. Determining optical ow. Articial Intelligence, 17:185-203, 1981.

....analysis procedes by rst computing local 2d velocities, and then by combining these local estimates to compute the global motion of an object. A well known problem with this approach is that local motion information is often ambiguous, a situation often referred to as the aperture problem [13, 6, 2, 8]. Consider the scene depicted in Figure 1. A local analyzer that sees only the vertical edge of a square can only determine the horizontal component of the motion. Whether the square translates horizontally to the right, diagonally up and to the right, or diagonally down and to the right, the ....

....rst decide which property of the image to track from one time to the next. One common, successful approach in machine vision is based on the assumption that the light re ected from a object surface remains constant through time, in which case one can track points of constant image intensity (e.g. [3, 6, 7]) Mathematically, this can be expressed in terms of a path, x(t) along which the image, I(x(t) t) remains constant: i.e. I(x(t) t) C ; 3) where C is a constant. Taking the temporal derivative of both sides of Equation 3, and assuming that the path x(t) is suciently smooth to be ....

B. K. P. Horn and B. G. Schunck. Determining optical ow. Articial Intelligence, 17(1-3):185-203, 1981.

....in quadtree architectures Willsky and his colleagues [10] have shown that MRFs can be approximated with hierarchical or multi resolution models. We have been experimenting with BBP on quadtree architectures. Figure 10 shows an example where we are interested in minimizing the Horn and Schunck [6] cost functional. This functional is the 2D analog of the Hildreth functional coherence is measured at all four nearest neighbors on a grid. Following [10] a b 0 10 20 30 40 50 60 70 0 0.2 0.4 0.6 0.8 1 MFA BBP c d Figure 9: a. a synthetic 2D gure ground problem. bright pixels ....

B. K. P. Horn and B. G. Schunck. Determining optical ow. Artif. Intell., 17(1-3):185-203, August 1981.

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B.K.P Horn and B.G. Schunck. Determining optical ow. Arti cial Intelligence, 17(1-3):185-203, 1981.

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B. K. P. Horn and B. G. Schunk, \Determining optical ow," Arti cial Intelligence, vol. 17, pp. 185-203, 1981.

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B. Horn and B. Schunck, #Determining optical #ow," Arti#cial Intelligence 17, pp. 185#203, August 1981.

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B. Horn and B. Schunck. Determining optical ow. Arti cial Intelligence, 17:185-203, 1981.

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