## Regularised Range Flow (2000)

Venue: | In European Conference on Computer Vision, LCNS 1843/44 |

Citations: | 13 - 7 self |

### BibTeX

@INPROCEEDINGS{Spies00regularisedrange,

author = {Hagen Spies and Bernd Jähne and John L. Barron},

title = {Regularised Range Flow},

booktitle = {In European Conference on Computer Vision, LCNS 1843/44},

year = {2000},

pages = {785--799},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

Extending a differential total least squares method for range flow estimation we present an iterative regularisation approach to compute dense range flow fields. We demonstrate how this algorithm can be used to detect motion discontinuities. This can can be used to segment the data into independently moving regions. The different types of aperture problem encountered are discussed. Our regularisation scheme then takes the various types of flow vectors and combines them into a smooth flow field within the previously segmented regions. A quantitative performance analysis is presented on both synthetic and real data. The proposed algorithm is also applied to range data from castor oil plants obtained with the Biris laser range sensor to study the 3-D motion of plant leaves.

### Citations

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Citation Context ... moving in an environment observed by a stationary sensor. As with optical flow calculation we initially assume that the flow field can be approximated as being constant within a small local aperture =-=[7, 8]-=-. In a second processing step this is replaced by requiring the flow field to be smooth. The work presented here is related to previously reported model based range flowsestimation on non-rigid surfac... |

1736 | Schunck.: Determining optical flow
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Citation Context ... of performing a TLS analysis first one might want to directly try to find the flow field by imposing the smoothness constraint, in analogy to the well known optical flow algorithm by Horn and Schunk =-=[22].-=- As mentioned before this simple smoothness term is not generally advisable, mainly because problematic locations (λ4 > 0) are equally taken into account. In particular it smoothes across motion disc... |

231 |
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Citation Context ...dless of ω. The next section explains why and how to deal with such situations. 3.2 Normal Flows Let the eigenvalues of F be sorted: λ1 ≥ λ2 ≥ λ3 ≥ λ4. Thus if λ3 ≈ λ4 no unique solutio=-=n can be found [18]-=-. More general any vector in the nullspace of F is a possible solution. In this case it is desirable to use the solution with minimal norm. Towards this end the possible solutions are expressed as lin... |

40 | Rigid body motion from range image sequences
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Citation Context ...re techniques. 2 Constraint Equation Depth is taken as a function of space and time Z = Z(X, Y, T ). From the total derivative with respect to time we derive the range flow motion constraint equation =-=[3, 9] ZX �-=-� X + ZY ˙ Y − ˙ Z + ZT = 0 . (1) Here partial derivatives are denoted by subscripts and time derivatives by using a dot. We call the 3D motion vector range flow f and introduce the following abbr... |

34 | 3D pose tracking with linear depth and brightness constraints
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Citation Context ...(τ2 = 0.1) and d TLS full flow. fit fails. This is also the case for pure noise without any coherent motion. To quantify this we introduce a confidence measure � 0 if λ4 > τ2 or tr(D) < τ1 ω = =-=� �2 . (6) τ2��-=-�λ4 else τ2+λ4 Figure 1 shows the obtained confidence measure for a synthetic sequence of depth maps. Clearly motion discontinuities and pure noise can be identified. Also the estimated full flow i... |

30 |
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Citation Context ...n a second processing step this is replaced by requiring the flow field to be smooth. The work presented here is related to previously reported model based range flowsestimation on non-rigid surfaces =-=[9, 10]-=-. The 3D range flow can also be recovered from optical flow if other surface properties such as depth or correspondences are available [11]. Some other work includes 2D range flow obtainable from a ra... |

30 | Qualitative and quantitative car tracking from a range image sequence
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Citation Context ...her surface properties such as depth or correspondences are available [11]. Some other work includes 2D range flow obtainable from a radial range sensor [12] and car tracking in range image sequences =-=[13]-=-. The underlying constraint equation is introduced in Sect. 2. Then Sect. 3 recapitulates the TLS estimation technique, in particular it is described how sensible parameters can be estimated even if n... |

25 |
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Citation Context ...flow f and introduce the following abbreviation f = [U V W ] T = [ ˙ X ˙ Y − ˙ Z] T (1) then becomes . The range flow motion constraint ZXU + ZY V + W + ZT = [ZX ZY 1 ZT ] T ⎡ ⎤ U ⎢ V ⎥ �=-=�� W ⎦ = 0 . 1 (2)-=- As this gives only one constraint equation in three unknowns we need to make further assumptions, this is the aperture problem revisited. Equation (2) describes a plane in velocity space. If there ar... |

24 |
Estimation of motion from a pair of range images: A review, CVGIP 54(3
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Citation Context ...ponding to the smallest eigenvalue λ4 of the generalised structure tensor F = D T ⎡ ⎤ < ZXZX > < ZXZY > < ZX > < ZXZT > ⎢ D = ⎢ < ZY ZX > < ZY ZY > < ZY > < ZY ZT > ⎥ ⎣ < ZX > < ZY > < 1 =-=> < ZT > ⎦ . (4) < ZT ZX -=-> < ZT ZY > < ZT > < ZT ZT > Here < · > denotes local averaging using a Box or Binomial filter. The desired range flow is then given by f f = 1 ⎡ ⎤ e14 ⎣ ⎦ . (5) e44 As F is real and symmetri... |

22 | Range flow estimation
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Citation Context ...d with the estimation of local three-dimensional velocity from a sequence of depth maps. Previously we introduced a total least squares (TLS) algorithm for the estimation of this so called range flow =-=[1]-=-. It is shown that the result of this TLS algorithm can be used to detect boundaries between independently moving regions, which enables a segmentation. However, within these regions the computed flow... |

22 |
Hierarchical model-based motion estimation, in: ECCV’92
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Citation Context ... to be filled in, far more iterations are necessary. Convergence can be accelerated by starting with an interpolated full flow field instead of a zero flow field or by employing a hierarchical method =-=[23]-=-. As we are unable to make real test data with other than translational motion, we took the depth map from one scan and warped the data 1 with a known flow field. Figure 3a shows the depth map taken f... |

17 | On discontinuity-preserving optic flow
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Citation Context ... membrane model is only justified because we have already segmented the data into differently moving objects. If no such segmentation were available more elaborate schemes would have to be considered =-=[19, 20]. The above sm-=-oothness term usually considers only spatial neighbourhoods (∇ = [∂x, ∂y] T ), however this is easily extended to enforce temporal smoothness as well (∇ = [∂x, ∂y, ∂t] T ). Combining the... |

15 |
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Citation Context ...Box or Binomial filter. The desired range flow is then given by f f = 1 ⎡ ⎤ e14 ⎣ ⎦ . (5) e44 As F is real and symmetric the eigenvalues and eigenvectors can easily be computed using Jacobi-Ro=-=tations [17].-=- In order to save execution time we only compute range flow where the trace of the tensor exceeds a threshold τ1. This eliminates regions with insufficient magnitude of the gradient. The regularisati... |

14 | Performance evaluation of three active vision systems built at the national research council of canada
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Citation Context ... (Sect. 5.3). Results of applying our algorithm to sequences of range scans of plant leaves are given in Sect. 6. The work reported here was performed with data gathered by a Biris laser range sensor =-=[14]-=-. The algorithm introduced could, however, be equally well used on dense depth maps obtained from structured lighting, stereo or motion and structure techniques. 2 Constraint Equation Depth is taken a... |

8 | Model-based force-driven nonrigid motion recovery from sequences of range images without point correspondences, Image Vision Comput
- Tsap, Goldgof, et al.
- 1999
(Show Context)
Citation Context ...n a second processing step this is replaced by requiring the flow field to be smooth. The work presented here is related to previously reported model based range flowsestimation on non-rigid surfaces =-=[9, 10]-=-. The 3D range flow can also be recovered from optical flow if other surface properties such as depth or correspondences are available [11]. Some other work includes 2D range flow obtainable from a ra... |

7 | A total least squares framework for low-level analysis of dynamic scenes and processes
- Haußecker, Garbe, et al.
- 1999
(Show Context)
Citation Context ...xtension of the structure tensor algorithm for optical flow estimation [15, 8]. The method may also be viewed as a special case of a more general technique for parameter estimation in image sequences =-=[16]-=-. Assuming constant flow in a region containing n pixel we have n equations (2). With d = [ZX ZY 1 ZT ] T , u = [U V W 1] T and the data matrix D = [d1 . . . dn] T , the flow estimation in a total lea... |

6 |
Loan Matrix Computations (3rd Edition
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(Show Context)
Citation Context ... v k+1 = αA −1 ¯v k + ωA −1 P f . (20) Initialisation is done as v 0 = 0. The matrix A −1 only has to be computed once, existence of the inverse is guaranteed by the Sherman-Morrison-Woodbury=-= formula [21]-=-, see Appendix B. 4.1 Direct Regularisation Instead of performing a TLS analysis first one might want to directly try to find the flow field by imposing the smoothness constraint, in analogy to the we... |

3 |
Variational Methods for Adaptive Image Smoothing and Segmentation
- Schnorr
- 1999
(Show Context)
Citation Context ... membrane model is only justified because we have already segmented the data into differently moving objects. If no such segmentation were available more elaborate schemes would have to be considered =-=[19, 20]. The above sm-=-oothness term usually considers only spatial neighbourhoods (∇ = [∂x, ∂y] T ), however this is easily extended to enforce temporal smoothness as well (∇ = [∂x, ∂y, ∂t] T ). Combining the... |

2 |
Recovering Motion Parameters from a 2D Range Image Sequence
- Gonzalez
- 1996
(Show Context)
Citation Context ...w can also be recovered from optical flow if other surface properties such as depth or correspondences are available [11]. Some other work includes 2D range flow obtainable from a radial range sensor =-=[12]-=- and car tracking in range image sequences [13]. The underlying constraint equation is introduced in Sect. 2. Then Sect. 3 recapitulates the TLS estimation technique, in particular it is described how... |

1 |
Cortelazzo and A. Vettore: Estimating 3-D Roto-translations from Range Data by a Frequency Domain Technique
- Lucchese, M
- 1997
(Show Context)
Citation Context ... ZX > < ZY > < 1 > < ZT > ⎦ . (4) < ZT ZX > < ZT ZY > < ZT > < ZT ZT > Here < · > denotes local averaging using a Box or Binomial filter. The desired range flow is then given by f f = 1 ⎡ ⎤ e14=-= ⎣ ⎦ . (5)-=- e44 As F is real and symmetric the eigenvalues and eigenvectors can easily be computed using Jacobi-Rotations [17]. In order to save execution time we only compute range flow where the trace of the t... |