## Visual reconstruction (1987)

Citations: | 736 - 2 self |

### BibTeX

@MISC{Blake87visualreconstruction,

author = {Andrew Blake and Andrew Zisserman},

title = {Visual reconstruction},

year = {1987}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

3716 |
Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images
- Geman, Geman
- 1984
(Show Context)
Citation Context ...drawn into the ground state, the positions of discontinuities will be stable in the required manner. And this is precisely what is achieved by certain statistical algorithms (Kirkpatrick et al. 1982, =-=Geman and Geman 1984-=-), and the deterministic “Graduated Non-convexity” (GNC) algorithm, proposed in this book. Some examples of the operation of the GNC algorithm, reconstructing various kinds of visual data, are shown i... |

3397 | Communicating Sequential Processes
- Hoare
- 1985
(Show Context)
Citation Context ...theories of Psychophysics (e.g. Julesz 1971, Marr 1976b) Cognitive Science (e.g. Hinton and Sejnowski 1983, Hopfield 1984), Pattern Recognition (e.g. Rosenfeld et al. 1976) and Computer Science (e.g. =-=Brookes et al. 1984-=-, Milner 1980). In vision, there have been cooperative algorithms for optical flow computation (Horn and Schunk 1981), analysis of shading (Woodham 1977, Ikeuchi and Horn 1981), analysis of motion (Ul... |

2608 |
Dynamic Programming
- Bellman
- 2003
(Show Context)
Citation Context ...obtaining the convex F ∗ is generally applicable to integer programming problems, but at the price of being a rather poor approximation compared with the F ∗ used in GNC. Finally dynamic programming (=-=Bellman and Dreyfus 1962-=-), previously used for curve detection by Montanari (1971), has been successfully applied to the weak string by Papoulias (1985). 6.3.2 Simulated annealing Simulated annealing is a powerful, general m... |

1572 |
Neural networks and physical systems with emergent collective computational abilities
- Hopfield
(Show Context)
Citation Context ...ally feasible can be designed for them. Furthermore, how far can energy functions be built or refined by inductive learning, as exhibited to some degree in neural networks (Hinton and Sejnowski 1983, =-=Hopfield 1982-=-, Wallace 1985). 8.2 Hardware Implementation It is clear from the parallel nature of the GNC algorithm (when simultaneous updating is used) that it maps efficiently onto massively parallel machines, w... |

1402 |
Robot Vision
- Horn
- 1986
(Show Context)
Citation Context ...end itself more naturally to VLSI implementation. Another intriguing possibility is that of parallel analogue hardware. Classical quadratic schemes can of course be realised with analogue components (=-=Horn 1986-=-, Poggio et al. 1985). Is this still true when weak continuity constraints are in force? We have done some simulations that strongly suggest feasibility. The benefits in terms of simplicity and speed ... |

1138 | Spatial interaction and the statistical analysis of lattice systems - Besag - 1974 |

1084 | A Practical Guide to Splines - Boor - 2001 |

588 |
Neurons with graded response have collective computational properties like those of two-state neurons
- Hopfield
- 1984
(Show Context)
Citation Context ... achieve a useful effect.6 Chapter 1 Networks of this kind have received much attention in theories of Psychophysics (e.g. Julesz 1971, Marr 1976b) Cognitive Science (e.g. Hinton and Sejnowski 1983, =-=Hopfield 1984-=-), Pattern Recognition (e.g. Rosenfeld et al. 1976) and Computer Science (e.g. Brookes et al. 1984, Milner 1980). In vision, there have been cooperative algorithms for optical flow computation (Horn a... |

586 |
Receptive fields and functional architecture in 2 nonstriate visual areas (18 and 19) of cat
- Hubel, Wiesel
- 1965
(Show Context)
Citation Context ...alian vision is understood in some detail (Marks et al. 1964); and the elegant, orderly, spatial correspondence of feature detectors in the brain with the array of cells in the retina, is well known (=-=Hubel and Wiesel 1968-=-). There has also been much dialogue between psychophysics and neurophysiology/neuroanatomy. Examples are the discovery of spatial bandpass channels (Campbell and Robson 1968, Braddick et al. 1978), a... |

577 |
A Treatise on the Theory of Bessel Functions
- Watson
- 1962
(Show Context)
Citation Context ... configuration u is given by ∫ u(x) = where the Green’s function is G(x, x ′ ) = 1 K0 2πλ2 G(x, x ′ )d(x ′ ) dA (4.31) ( ) ′ |x − x | λ (4.32) and K0 is a modified Bessel function of the second kind (=-=Watson 1952-=-). The energy is ∫ E = d(x)(d(x) − u(x)) dA (4.33) This is a straightforward analogue of the 1D method. Further details are given in appendix A. The Green’s function has the important asymptotic prope... |

537 |
Scale-Space Filtering
- Witkin
- 1983
(Show Context)
Citation Context ...t coarse and fine scales. But under linear filters such as the gaussian and its derivatives, complex structure arises, which is difficult to track and harder still to interpret (Asada and Brady 1986, =-=Witkin 1983-=-). A weak string scale-space for a silhouette taken from a real image, is shown Figure 2.14: Scale-space filtering. The hand drawn curve (a) segmented at coarse scale (b) and reconstructed by fitting ... |

520 | Differential Geometry of curves and surfaces - Carmo - 1976 |

481 |
Neural computations of decisions in optimization problems
- Hopfield, Tank
- 1985
(Show Context)
Citation Context ...course with totally different extrema. Hopfield’s procedure uses F (p) with an intermediate value of p. Recent work suggests that, as in GNC, sweeping p from 1 to 0 may be useful in the neural model (=-=Hopfield and Tank 1985-=-). Whereas such a strategy will be proven, in the next chapter, to be effective in GNC, there are as yet no corresponding proofs for neural networks. The main difference from GNC is that the convex ap... |

465 |
The Interpretation of Visual Motion
- Ullman
- 1979
(Show Context)
Citation Context ... practice by executing an algorithm that implements them, on a computer. All this has led to considerable enrichment of studies of human vision (e.g. Marr and Poggio 1979, Mayhew 1982, Hildreth 1984, =-=Ullman 1979-=-b, Koenderinck and van Doorn 1976). 3. Complete, though simple, vision systems can be built and tested. The restriction to study the visual systems that nature has kindlyModelling Piecewise Continuit... |

348 |
Scene labeling by relaxation operation
- Rosenfeld, Hummel, et al.
- 1976
(Show Context)
Citation Context ...orks of this kind have received much attention in theories of Psychophysics (e.g. Julesz 1971, Marr 1976b) Cognitive Science (e.g. Hinton and Sejnowski 1983, Hopfield 1984), Pattern Recognition (e.g. =-=Rosenfeld et al. 1976-=-) and Computer Science (e.g. Brookes et al. 1984, Milner 1980). In vision, there have been cooperative algorithms for optical flow computation (Horn and Schunk 1981), analysis of shading (Woodham 1977... |

294 |
The interpretation of a moving retinal image
- Longuet-Higgins, Prazdny
- 1980
(Show Context)
Citation Context ...however, the28 Chapter 2 sudden change in depth at the occluding edge, falling off one step onto the next, is quite unambiguous. (Motion parallax similarly facilitates perception of occluding edges (=-=Longuet-Higgins and Prazdny 1980-=-)). 2.2.4 Surface descriptions There are two distinct types of usage of information about visible surfaces: reasoning about visible objects, and path planning or collision avoidance. In the first, the... |

286 |
Principles of Neurodynamics
- Rosenblatt
- 1962
(Show Context)
Citation Context ...ear system 4 . So purely linear systems are inadequate for visual reconstruction. There must be some non-linearity, even if it is just a thresholding operation. This is what occurs in the Perceptron (=-=Rosenblatt 1962-=-, Minsky and Papert 1969), a simple, neuron-like switching element that computes a weighted 4 A linear system is one that simply outputs a weighted sum of its inputs.Modelling Piecewise Continuity 11... |

243 |
Machine perception of three-dimensional solids
- Roberts
- 1965
(Show Context)
Citation Context ...). These instances are but parts of a very large body of knowledge of biological vision. Over the last two decades, computers have introduced a new strand into the study of vision. The earliest work (=-=Roberts, 1965-=-) produced systems able to recognise simple objects and manipulate them in a controlled way (Ambler et al. 1975). These systems were, of course, vastly inferior to the biological systems studied by th... |

239 |
Computational Vision and Regularization Theory
- Poggio, Koch, et al.
- 1985
(Show Context)
Citation Context ...more naturally to VLSI implementation. Another intriguing possibility is that of parallel analogue hardware. Classical quadratic schemes can of course be realised with analogue components (Horn 1986, =-=Poggio et al. 1985-=-). Is this still true when weak continuity constraints are in force? We have done some simulations that strongly suggest feasibility. The benefits in terms of simplicity and speed are tempting, and th... |

218 |
Application of fourier analysis to the visibility of gratings
- Campbell, Robson
- 1968
(Show Context)
Citation Context ...n the retina, is well known (Hubel and Wiesel 1968). There has also been much dialogue between psychophysics and neurophysiology/neuroanatomy. Examples are the discovery of spatial bandpass channels (=-=Campbell and Robson 1968-=-, Braddick et al. 1978), and understanding the perception of coloured light (Livingstone and Hubel 1984, Jameson and Hurvich 1961) and surface colour (Land 1983, Zeki 1983). These instances are but pa... |

207 |
Theory of Elasticity
- Landau, Lifshitz
- 1986
(Show Context)
Citation Context ...y can be creased without any associated energy increase. In order to detect creases, a 2nd order surface must be used - one which has a high energy density where it is tightly curved. The thin plate (=-=Landau and Lifschitz 1959-=-, Grimson 1981, Terzopoulos 1983) has this property: intuitively it is easy to crease a sheet of elastic (a membrane) but hard to crease a sheet of steel. In this chapter, it is shown that the plate (... |

190 | Horn,“Numerical shape from shading and occluding boundaries
- Ikeuchi, B
- 1981
(Show Context)
Citation Context ...and Computer Science (e.g. Brookes et al. 1984, Milner 1980). In vision, there have been cooperative algorithms for optical flow computation (Horn and Schunk 1981), analysis of shading (Woodham 1977, =-=Ikeuchi and Horn 1981-=-), analysis of motion (Ullman 1979a), computation of lightness (Horn 1974, Blake 1985c) and reconstruction of stereoscopically viewed surfaces (Grimson 1981, Terzopoulos 1983). The implementation of w... |

189 | The curvature primal sketch
- Asada, Brady
- 1986
(Show Context)
Citation Context ...ary role. Commonly, a curve is converted to tangent angle/arc-length (θ, s) form, and filtered to detect corners (step discontinuities in θ) and possibly also curvature discontinuities (Perkins 1978, =-=Asada and Brady 1986-=-, Ramer 1975, Zucker et al. 1977, Zucker 1982, Blake et al. 1986a). This may be done at a variety of spatial scales in order to obtain both coarse and fine views of the curve’s shape. Corners, for ins... |

186 |
Symbolic reasoning among 3-D models and 2-D images
- Brooks
- 1981
(Show Context)
Citation Context ...shape of the patch. This is nothing to be ashamed of - descriptions of this sort are already a powerful handle for matching visible surfaces to one another (Pollard et al. 1987) and to stored models (=-=Brooks 1981-=-). New sources of information, such as analysis of surface shading, might add usefully to such descriptions - but that possibility must be left for discussion elsewhere (Ikeuchi and Horn 1981, Ikeuchi... |

173 |
1977] - Multi level adaptive solutions to boundary-value problems
- Brandt
(Show Context)
Citation Context ...ons do, however, exist: “simultaneous over-relaxation” and “chequerboard SOR”. 6.2.2 Multi-grid algorithms Terzopoulos greatly enhanced the basic SOR algorithm by making use of multi-grid techniques (=-=Brandt 1977-=-), in which relaxation takes place simultaneously, on coarse and fine arrays. Use of four arrays, with density inThe Discrete Problem 123 Nodes: i ∈ {0, ..., N}, j ∈ {0, ..., N}. Iterate n = 1, 2, ..... |

165 |
Numerical Solution of Partial Differential Equations: Finite Difference Methods
- Smith
- 1985
(Show Context)
Citation Context ...-relaxation (SOR): SOR can be made to converge much faster than Jacobi. If the decay time for Jacobi is γ then that for SOR is reduced to √ γ/8 - provided the following optimal SOR parameter is used (=-=Smith 1978-=-): ( w = 2 1 + √ 1 + 2γ 1 + γ )−1 . (7.40) Assuming λ (µ 2 ) to be somewhat greater than 1, this is approximated by ( w = 2 1 + √ ) −1 2/γ . (7.41) Experiments confirm that this value of w does indeed... |

152 |
PMF: a stereo correspondence algorithm using a disparity gradient limit
- Pollard, Mayhew, et al.
- 1985
(Show Context)
Citation Context ... stereo data. A stereo pair (a) of a foam block, with a step discontinuity across the middle, that is all but invisible monocularly. Stereo correspondence using a state-of-the-art matching algorithm (=-=Pollard et al. 1985-=-) produces depths along sparse contours (b). The reconstructed surface is shown with its contour of discontinuity (c).Applications of Piecewise Continuous Reconstruction 31 Figure 2.11: Isometric plo... |

121 |
Convex Function
- Roberts, Varberg
- 1973
(Show Context)
Citation Context ... ∑ by “balancing” the positive second derivatives in the first term D = (ui − di) 2 against the negative second derivatives in the g∗ terms. The i balancing procedure is to test the Hessian matrix H (=-=Roberts and Varberg 1976-=-) of F ∗ : if H is positive definite then F ∗ (u) is a convex function1 of u. The Hessian H of F ∗ is ∂2 F ∗ Hij = ∂ui∂uj = 2Ii,j + ∑ g ∗′′ (uk − uk−1)Qk,iQk,j, (7.1) where I is the identity matrix an... |

108 |
Determining lightness from an image
- Horn
- 1974
(Show Context)
Citation Context ...een cooperative algorithms for optical flow computation (Horn and Schunk 1981), analysis of shading (Woodham 1977, Ikeuchi and Horn 1981), analysis of motion (Ullman 1979a), computation of lightness (=-=Horn 1974-=-, Blake 1985c) and reconstruction of stereoscopically viewed surfaces (Grimson 1981, Terzopoulos 1983). The implementation of weak continuity constraints can be achieved very naturally too, we shall s... |

98 | Anatomy and physiology of a color system in the primate visual cortex
- Livingstone, Hubel
- 1984
(Show Context)
Citation Context ...physics and neurophysiology/neuroanatomy. Examples are the discovery of spatial bandpass channels (Campbell and Robson 1968, Braddick et al. 1978), and understanding the perception of coloured light (=-=Livingstone and Hubel 1984-=-, Jameson and Hurvich 1961) and surface colour (Land 1983, Zeki 1983). These instances are but parts of a very large body of knowledge of biological vision. Over the last two decades, computers have i... |

85 | The singularities of the visual mappings - Koenderink, Doorn - 1976 |

73 |
Multilevel computational processes for visual surface reconstruction
- Terzopoulos
- 1983
(Show Context)
Citation Context ...ading (Woodham 1977, Ikeuchi and Horn 1981), analysis of motion (Ullman 1979a), computation of lightness (Horn 1974, Blake 1985c) and reconstruction of stereoscopically viewed surfaces (Grimson 1981, =-=Terzopoulos 1983-=-). The implementation of weak continuity constraints can be achieved very naturally too, we shall see, by cooperative networks. 1.2 Continuity and cooperativity 1.2.1 Cooperativity in physical models ... |

65 |
From images to surfaces
- Grimson
- 1981
(Show Context)
Citation Context ...maps may be produced by stereopsis, in which images obtained from two slightly different viewpoints (e.g. two eyes) are compared and matched (Marr and Poggio 1979, Mayhew and Frisby 1981, Baker 1981, =-=Grimson 1981-=-); triangulation is then used to compute the depths. Alternatively depths may be obtained by appropriate processing of optic flow (Bruss 1983) or, artificially, from an optical rangefinder. • Sets of ... |

61 |
Recent advances in retinex theory and some implications for cortical computations: Color vision and the natural image
- Land
- 1983
(Show Context)
Citation Context ...atial bandpass channels (Campbell and Robson 1968, Braddick et al. 1978), and understanding the perception of coloured light (Livingstone and Hubel 1984, Jameson and Hurvich 1961) and surface colour (=-=Land 1983-=-, Zeki 1983). These instances are but parts of a very large body of knowledge of biological vision. Over the last two decades, computers have introduced a new strand into the study of vision. The earl... |

54 |
An operator which locates edges in digitized pictures
- Hueckel
- 1971
(Show Context)
Citation Context ...to 2D the use of blurring that we have already seen in 1D (Haralick 1980, Canny 1983, Marr and Hildreth 1980). The second kind use regression to fit step-shaped templates, locally, to intensity data (=-=Hueckel 1971-=-, O’Gorman 1978, Leclerc 1985, Gennert 1986). Where the template fits well, there must be a step discontinuity in the data. The third kind, also uses regression, but acts globally across the data, wit... |

53 |
The visual ambiguity of a moving plane
- Longuet-Higgins
- 1984
(Show Context)
Citation Context ...ontinuity - continuity along curves and surface features (Mayhew and Frisby 1981). Analysis of optical flow also appears to require assumptions of continuity, either in regions (Horn and Schunk 1981, =-=Longuet-Higgins 1984-=-) or along curves (Hildreth 1984). Computation of lightness, the perceptual correlate of surface reflectivity (i.e. surface colour), needs constraints on continuity both of the reflectivity itself, an... |

47 |
Boundary conditions for lightness computation in Mondrian world
- Blake
- 1985
(Show Context)
Citation Context ...tive algorithms for optical flow computation (Horn and Schunk 1981), analysis of shading (Woodham 1977, Ikeuchi and Horn 1981), analysis of motion (Ullman 1979a), computation of lightness (Horn 1974, =-=Blake 1985-=-c) and reconstruction of stereoscopically viewed surfaces (Grimson 1981, Terzopoulos 1983). The implementation of weak continuity constraints can be achieved very naturally too, we shall see, by coope... |

47 |
A four mechanism model for threshold spatial
- Wilson, McFarlane, et al.
- 1983
(Show Context)
Citation Context ...ly effective scales would be determined by the data. 8.4.1 Psychophysical models The literature on Gaussian multi-channel models of contrast sensitivity is well-established (Campbell and Robson 1968, =-=Wilson and Bergen 1979-=-, Watt and Morgan 1985). It would be interesting to know however, whether such models successfully predict systematic localisation error for asymmetric stimuli (figure 4.10). A pair of such stimuli, p... |

40 | An extremum principle for shape from contour - Brady, Yuille - 1983 |

40 |
Curve-fitting with piecewise parametric cubics
- Plass, Stone
- 1983
(Show Context)
Citation Context ...ts must be done automatically. An algorithm to do that might consist of constructing an initial spline fit, and then adding knots until the regression error measure reached an acceptably small value (=-=Plass and Stone 1983-=-). This would ensure a spline that closely fitted the data points. For visual reconstruction that is not enough. The requirement for stability has already been discussed, which means that the multiple... |

38 |
Determining optical How
- Horn, Schunk
- 1981
(Show Context)
Citation Context ...rticular, by figural continuity - continuity along curves and surface features (Mayhew and Frisby 1981). Analysis of optical flow also appears to require assumptions of continuity, either in regions (=-=Horn and Schunk 1981-=-, Longuet-Higgins 1984) or along curves (Hildreth 1984). Computation of lightness, the perceptual correlate of surface reflectivity (i.e. surface colour), needs constraints on continuity both of the r... |

36 |
Visual hyperacuity: Spatiotemporal interpolation in human vision
- Fahle, Poggio
- 1981
(Show Context)
Citation Context ...dense, using a continuous membrane (i.e. minimising (4.36) with very large α) at small scale λ. 4.5.1 Hyperacuity The problem of handling sparse data is closely related to the problem of hyperacuity (=-=Fahle and Poggio 1984-=-, Krotkov 1986) - that is, obtaining 3 Actually this problem is ill-posed as it stands - it has no continuous solution. To be technically correct, u(x) should be constrained to be constant along some ... |

33 |
A Regularized Solution to Edge Detection
- Poggio, Voorhees, et al.
- 1985
(Show Context)
Citation Context ...satile still are splines (de Boor 1978) - sequences of polynomials joined smoothly together. There is an interesting connection between cubic splines and elastic systems like the sheet in figure 1.1 (=-=Poggio et al. 1984-=-, Terzopoulos 1986). A flexible rod, such as draughtsmen commonly use to draw smooth curves is an elastic system. If it is loaded or clamped at several points, it takes up a shape - its minimum energy... |

29 |
A theory of the primitive spatial code in human vision
- Watt, Morgan
- 1985
(Show Context)
Citation Context ...d be determined by the data. 8.4.1 Psychophysical models The literature on Gaussian multi-channel models of contrast sensitivity is well-established (Campbell and Robson 1968, Wilson and Bergen 1979, =-=Watt and Morgan 1985-=-). It would be interesting to know however, whether such models successfully predict systematic localisation error for asymmetric stimuli (figure 4.10). A pair of such stimuli, positioned appropriatel... |

27 |
Segmentation of textured images using Gibbs random fields
- Derin, Cole
- 1986
(Show Context)
Citation Context ...he most disturbing thing is that one is forced to accept that the surface model is a probabilistic one, and therefore includes an element of randomness. This may be appropriate for modelling texture (=-=Derin and Cole 1986-=-), but in a model of smooth surfaces it has rather counter-intuitive consequences, illustrated in figure 1.5. A “1st order” MRF6 , for instance, ranks a noisy but horizontal plane more probable than a... |

25 | Integrals of Bessel function - Luke - 1962 |

24 |
Variational calculus with elementary convexity
- Troutman
- 1983
(Show Context)
Citation Context ...it can be shown (Blake 1984) that the integrand E is a non-convex function of ux, ... . So even with a fixed set of discontinuities, it is not known whether there is a uniquely optimal u to be found (=-=Troutman 1983-=-). This problem is dealt with, in an approximate way 1 , by using a 1st order plate, as already proposed for the non-invariant case. First, estimates for ux(x, y), uy(x, y) are obtained by fitting an ... |

23 | Bessel Functions for Engineers - McLachlan - 1955 |

22 |
Relaxation and its role in vision
- Hinton
- 1978
(Show Context)
Citation Context ... is to reach a satisfactory formalisation of “continuity almost everywhere”. We do that here by borrowing the idea of a “weak constraint” - a constraint that can be broken occasionally - from Hinton (=-=Hinton 1978-=-). With an appropriate class of continuous surface patches, this leads to “weak continuity constraints” (Blake 1983b) - preferring continuity, but grudgingly allowing occasional discontinuities if tha... |

19 |
Local Control of Bias and Tension in Betasplines
- Barsky, Beatty
- 1983
(Show Context)
Citation Context ...ly, a mixture of first and second order energies may be used, to give: ∫ { E = (u − d) 2 + λ 2 u ′2 4 ′′2 + µ u } dx + P. (5.2) When u is continuous, this is somewhat akin to “splines under tension” (=-=Barsky and Beatty 1983-=-), and is used for reconstruction by Terzopoulos (1983, 1985).Properties of the Weak Rod and Plate 99 Now that the energy is second order it is possible to include penalties both for steps and crease... |