K. Daniilidis and M. E. Spetsakis. Understanding noise sensitivity in structure from motion. In Y. Aloimonos, editor, Visual Navigation: From Biological Systems to Unmanned Ground Vehicles, Advances in Computer Vision, chapter 4. Lawrence Erlbaum Associates, Mahwah, NJ, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the constraint x 0 =y 0 = x 0 =y 0 ; that is, the projection of the actual translation and the projection of the estimated translation lie on a line passing through the image center. We refer to this second constraint as the line constraint. These results are in accordance with previous studies [2, 21], which found that the translational components along the x and y axes are confused with rotation around the y and x axes, respectively, and the line constraint under a set of restrictive assumptions. Epipolar Minimization on the Sphere The function representing deviation from the epipolar ....

K. Daniilidis and M. E. Spetsakis. Understanding noise sensitivity in structure from motion. In Y. Aloimonos, editor, Visual Navigation: From Biological Systems to Unmanned Ground Vehicles, chapter 4. Lawrence Erlbaum Associates, Mahwah, NJ, 1997.

....to noise in the input used (optic flow or correspondence) While many solutions have been proposed, they become problematic in the case of realistic scenes and most of them degrade ungracefully as the quality of the input deteriorates. This has motivated research on the stability of the problem; [7] contains an excellent survey of existing error analyses. We will discuss the most important results in Section 3 in more technical detail after some mathematical prerequisites are given in Section 2. In summary, it can be concluded that the majority of the existing analyses attempt to model the ....

....of parameters that are involved. As a result a fair number of observations and intuitive arguments have been developed by a multitude of authors over the years. Most important, a small number of studies have given rise to three crisp results regarding noise sensitivity in structure from motion [7]. These are: a) A translation can be easily confounded with a rotation in the case of a small field of view under the assumption of lateral motion and insufficient variation of depth [1, 6] Intuitively, translation along the x axis can be confused with rotation around the y axis and translation ....

K. Daniilidis and M. Spetsakis. Understanding noise sensitivity in structure from motion. In Y. Aloimonos, editor, Visual Navigation: From Biological Systems to Unmanned Ground Vehicles, chapter 4. Lawrence Erlbaum Associates, Hillsdale, NJ, 1996. In press.

....estimation. They developed a hierarchy of camera design with respect to the scene independence of the motion estimation due to the spacing of the view points and the stability of the estimation due to the field of view (see Fig. 2) A small field of view makes the motion estimation ill posed (see [13] for a study on this subject) thus for accurate and robust motion estimation the camera needs to have a wide field of view. One can see in the figure that the conventional pinhole camera is at the bottom of the hierarchy because the small field of view makes the motion estimation ill posed and ....

K. Daniilidis and M. Spetsakis, "Understanding noise sensitivity in structure from motion," in Visual Navigation: From Biological Systems to Unmanned Ground Vehicles, Y. Aloimonos, Ed., chapter 4. Lawrence Erlbaum Associates, Hillsdale, NJ, 1997.

....was derived in [2] Young and Chellappa [27] derived bounds for the estimation error for structure and motion parameters for two images under perspective projection as well as from a sequence of stereo images. The coupling of the translation and rotation for small field of view was studied in [3]. Zhang s work [28] on determining the uncertainty in estimation of the fundamental matrix is another important contribution in this area. Haralick showed how well known estimation techniques could be used to propagate additive random perturbations through different vision algorithms [9] Soatto ....

....in the computer vision community is the bias in the depth estimation. This has been observed by psycho physicists who note that it is hard to explain . the existence of systematic biases in observers magnitude estimation of perceived depth [23] Some authors, notably Daniilidis and Spetsakis [3] and Kanatani [12] have proved that there exists a bias in the translation and rotation estimates from stereo. In this paper, we study the problem of 3D modeling from a monocular video stream with a small baseline. There are two main sources of error for this problem: one arising from the ....

K. Daniilidis and M.E. Spetsakis. Understanding noise sensitivity in structure from motion. In VisNav93, 1993.

....is a geometric problem and it exists for both small and large baselines between the views, that is, for the case of continuous motion as well as in the case of discrete displacements of the cameras. The basic understanding of these difficulties has attracted only a few investigators over the years [3, 11, 12, 19, 21]. Having in mind the design of an optimal sensor, we are interested in how the stability of the estimation of motion changes with the field of view. In particular, we have compared the planar small field of view camera with a spherical camera [14] Since motion estimation amounts to solving some ....

K. Daniilidis and M. E. Spetsakis. Understanding noise sensitivity in structure from motion. In Y. Aloimonos, editor, Visual Navigation: From Biological Systems to Unmanned Ground Vehicles, Advances in Computer Vision, chapter 4. Lawrence Erlbaum Associates, Mahwah, NJ, 1997.

....for finding 3D motion from a video sequence. Almost all techniques are based on the socalled epipolar constraint, which shows how the motion of image points is related to 3D rigid motion and the scene. This constraint, at each image point r, is written as (t Theta r) Delta (r Theta r) 0 [3]. One is interested in the estimates of translation t and rotation which best satisfy the epipolar constraint at every point r according to some criteria of deviation. Usually the Euclidean norm is considered leading to the minimization of function. image Theta Gamma Delta ( r ....

....similar to one by Ouchi. scene, the motion field at the center of the image is very similar in the two cases. Thus, for example, translation along the x axis is confused with rotation around the y axis. The basic understanding of this confusion has attracted few investigators over the years [2, 3]. In [6, 8] a geometrical statistical analysis of the problem has been conducted. On the basis of (3) the expected value of E ep has been formulated as a five dimensional function of the motion parameters (two dimensions for t=jtj and three for ) Independent of specific estimators the ....

K. Daniilidis and M. E. Spetsakis. Understanding noise sensitivity in structure from motion. In Y. Aloimonos, editor, Visual Navigation: From Biological Systems to Unmanned Ground Vehicles, Advances in Computer Vision, chapter 4. Lawrence Erlbaum Associates, Mahwah, NJ, 1997.

....the VIRGO research network of the TMR Programme (EC Contract No ERBFMRX CT96 0049) the viewed scene. Since the dependence of the 2D image motion on the scene structure is nonlinear, small errors in the estimates of 2D motion can have a signicant impact on the accuracy of the recovered 3D motion [4]. In addition, the confounding of translation and rotation makes the problem of estimating unrestricted egomotion much harder compared to the problem of estimating pure translation or rotation [4] Due to its importance, many algorithms dealing with the problem of estimating egomotion have ....

.... in the estimates of 2D motion can have a signicant impact on the accuracy of the recovered 3D motion [4] In addition, the confounding of translation and rotation makes the problem of estimating unrestricted egomotion much harder compared to the problem of estimating pure translation or rotation [4]. Due to its importance, many algorithms dealing with the problem of estimating egomotion have appeared in the literature. The following paragraphs provide a short review of a few representative methods; more detailed discussions can be found in [7, 8, 10] Most of the methods reviewed here rely ....

[Article contains additional citation context not shown here]

K. Daniilidis and M.E. Spetsakis. Understanding Noise Sensitivity in Structure From Motion. In Y. Aloimonos, editor, Visual Navigation: From Biological Systems to Unmanned Ground Vehicles, chapter 4. Lawrence Erlbaum Associates, Hillsdale, NJ, 1997.

....work has been carried out while the author was with the Computer Science Dept, Univ. of Crete and the Inst. of Computer Science, FORTH, Heraklion, Crete, Greece. Funding was partially supplied by the VIRGO research network of the TMR Programme (EC Contract No ERBFMRX CT96 0049) covered 3D motion [5, 21]. In addition, the confounding of translation and rotation makes the problem of estimating unrestricted egomotion much harder compared to the problem of estimating pure translation or rotation [5] Due to its importance, many algorithms dealing with the problem of estimating egomotion have ....

.... VIRGO research network of the TMR Programme (EC Contract No ERBFMRX CT96 0049) covered 3D motion [5, 21] In addition, the confounding of translation and rotation makes the problem of estimating unrestricted egomotion much harder compared to the problem of estimating pure translation or rotation [5]. Due to its importance, many algorithms dealing with the problem of estimating egomotion have appeared in the literature. The following paragraphs provide a short review of a few representative methods; more detailed discussions can be found in [6, 7, 9] Most of the methods reviewed here rely ....

K. Daniilidis and M.E. Spetsakis. Understanding Noise Sensitivity in Structure From Motion. In Y. Aloimonos, editor, Visual Navigation: From Biological Systems to Unmanned Ground Vehicles, chapter 4. Lawrence Erlbaum Associates, Hillsdale, NJ, 1997.

....an error analysis needs to be performed. The problem of determining the epipolar geometry of a stereo rig is also known as the structure from motion problem in the motion analysis literature. A significant research effort has been devoted to the analysis of errors in structure from motion, e.g. [1, 18, 17, 52, 19, 31, 20], to cite just a few examples. One common characteristic of these approaches is that their error models are linear. The main reason for this is the mathematical and computational tractability of the linear model. A secondary reason is the lack of an adequate non linear error model. In this ....

K. Daniilidis and M. E. Spetsakis. Understanding noise sensitivity in structure from motion. In Y. Aloimonos, editor, Visual Navigation: From Biological Systems to Unmanned Ground Vehicles, pages 60--88, Lawrence Erlbaum Associates, Hillsdale, NJ, 1997.

....depth minimization from normal flow, given a rotational error the obtained solution will have no error in the translation. In other cases ambiguities remain. In the spherical eye case the analysis is simply performed for noiseless flow. A large number of error analyses have been carried out [2, 12, 13, 15, 29, 41, 43, 44] in the past for a camera type eye, while there is no published research of this kind for the full field of view case. None of the existing studies, however, has attempted a topographic characterization of the function to be minimized for the purpose of analyzing different motion techniques. All ....

K. Daniilidis and M. E. Spetsakis. Understanding noise sensitivity in structure from motion. In Visual Navigation: From Biological Systems to Unmanned Ground Vehicles, chapter 4. Lawrence Erlbaum Associates, Mahwah, NJ, 1997.

....point from the epipolar line is called the epipolar error. Minimization of epipolar errors is the basis of most 3D motion estimation algorithms. For the differential case of video, the epipolar constraint is obtained from the image motion equations as (t Theta r) Delta ( r Theta r) 0 [8]. One is interested in the estimates of translation t and rotation which best satisfy the epipolar constraint at every point r according to some criteria of deviation. Usually the Euclidean norm is considered leading to the minimization of function. M ep = Z Z image hi t Theta r j ....

....translate parallel to the scene, the motion field at the center of the image is very similar in the two cases. Thus, for example, translation along the x axis is confused with rotation around the y axis. The basic understanding of this confusion has attracted few investigators over the years. See [7,8] for a review. In this paper it is shown that the confusion exists no matter what estimator is used, proving that there is an inherent limitation to the estimation of 3D motion from data of only a limited field of view. To be more precise, a statistical analysis of all the possible computational ....

K. Daniilidis and M. E. Spetsakis. Understanding noise sensitivity in structure from motion. In Y. Aloimonos, editor, Visual Navigation: From Biological Systems to Unmanned Ground Vehicles, Advances in Computer Vision, chapter 4. Lawrence Erlbaum Associates, Mahwah, NJ, 1997.

....total least squares is known to perform very poorly if outliers are present, and these are difficult to detect from few measurements. The estimation and interpretation of optical flow from a statistical point of view has received attention before in the computational literature [Daniilidis, 1992; Daniilidis and Spetsakis, 1997; Heeger, 1988; Simoncelli et al. 1991; Szeliski, 1990; Weber and Malik, 1995] It has been pointed out in [Nagel, 1995; Nagel and Haag, 1998] that optical flow estimated using gradient methods is biased. In these studies the bias is interpreted only with regard to the underestimation in the ....

K. Daniilidis and M. E. Spetsakis. Understanding noise sensitivity in structure from motion. In Visual Navigation: From Biological Systems to Unmanned Ground Vehicles, chapter 4. Lawrence Erlbaum Associates, Mahwah, NJ, 1997.

....the computation of optical flow from noisy gradient measurements has a systematic bias dependent upon the gradient distribution of the image region. The estimation and interpretation of optical flow from a statistical point of view has received attention previously in the computational literature [1, 2, 3, 11, 14, 16]. Most closely related to this paper are the studies of Nagel and Haag [7, 8] who investigate and at1 tempt to compensate for the bias in a gradient based technique; however, they interpret the bias only with respect to the underestimation of the length of the flow, and do not discuss the effects ....

K. Daniilidis and M. E. Spetsakis. Understanding noise sensitivity in structure from motion. In Visual Navigation: From Biological Systems to Unmanned Ground Vehicles, chapter 4. Lawrence Erlbaum Associates, Mahwah, NJ, 1997.

....i BR component. 21 Recall that Step 2a1 neglects the constraints on the form of the Fa . 2.3.4 The Bas Relief Ambiguity: Step 2c A few inverse depth components are intrinsically more difficult to recover accurately than the rest. This is due to the well known bas relief ambiguity 22 [32, 1, 8, 24, 27, 58, 31, 57, 38, 40, 53, 7]. Any bias in an algorithm degrades the reconstructions of these components much more than that of the rest of the structure, so it is especially important to minimize the bias in computing them. The bas relief ambiguity reflects the fact that the image effects of rotating and translating are ....

K. Daniilidis and M. E. Spetsakis, "Understanding noise sensitivity in structure from motion," in Visual Navigation, Y. Aloimonos, ed, Lawrence Erlbaum, New Jersey, 60--88, 1997.

....to noise in the input used (optic flow or correspondence) While many solutions have been proposed, they become problematic in the case of realistic scenes and most of them degrade ungracefully as the quality of the input deteriorates. This has motivated research on the stability of the problem; [7] contains an excellent survey of existing error analyses. We will discuss the most important results in Section 3 in more technical detail after some mathematical prerequisites are given in Section 2. In summary, it can be concluded that the majority of the existing analyses attempt to model the ....

....of parameters that are involved. As a result a fair number of observations and intuitive arguments have been developed by a multitude of authors over the years. Most important, a small number of studies have given rise to three crisp results regarding noise sensitivity in structure from motion [7]. These are: a) A translation can be easily confounded with a rotation in the case of a small field of view under the assumption of lateral motion and insufficient variation of depth [1, 6] Intuitively, translation along the x axis can be confused with rotation around the y axis and ....

K. Daniilidis and M. Spetsakis. Understanding noise sensitivity in structure from motion. In Y. Aloimonos, editor, Visual Navigation: From Biological Systems to Unmanned Ground Vehicles, chapter 4. Lawrence Erlbaum Associates, Hillsdale, NJ, 1996. In press.

....constraint x 0 =y 0 = x 0 =y 0 ; that is, the projection of the actual translation and the projection of the estimated translation lie on a line passing through the image center. We refer to this second constraint as the line constraint. These results are in accordance with previous studies [2, 21], which found that the translational components along the x and y axes are confused with rotation around the y and x axes, respectively, and the line constraint under a set of restrictive assumptions. Epipolar Minimization on the Sphere The function representing deviation from the epipolar ....

K. Daniilidis and M. E. Spetsakis. Understanding noise sensitivity in structure from motion. In Y. Aloimonos, editor, Visual Navigation: From Biological Systems to Unmanned Ground Vehicles, chapter 4. Lawrence Erlbaum Associates, Mahwah, NJ, 1997.

....ffl x 0 , fi ffl = fl ffl y 0 . This means that the function has its global minimum along a line from which the FOE can be derived exactly, but the rotational component parallel to the translational component (x 0 ; y 0 ) cannot be determined. A large number of error analyses have been carried out [2, 11, 12, 14, 25, 39, 41, 44] in the past. None of the existing studies, however, has attempted a topographic characterization of the function to be minimized for the purpose of analyzing different motion techniques. All the studies consider optical flow or correspondence as image measurements and investigate minimizations ....

K. Daniilidis and M. E. Spetsakis. Understanding noise sensitivity in structure from motion. In Y. Aloimonos, editor, Visual Navigation: From Biological Systems to Unmanned Ground Vehicles, Chapter 4. Lawrence Erlbaum Associates, Hillsdale, NJ, 1997.

....In addition, total least squares is known to perform very poorly if outliers are present, and these are difficult to detect from a few measurements. The estimation and interpretation of optical flow from a statistical point of view has received attention previously in the computational literature [3, 5, 14, 36, 40, 43]. Most closely related to this paper is the study of Nagel [29] He considers an error model slightly more complicated than ours. In particular, he assumes Gaussian noise in the gray values, linearly varying flow, and he considers the dependence of the partial derivatives in the gray value ....

K. Daniilidis and M. E. Spetsakis. Understanding noise sensitivity in structure from motion. In Visual Navigation: From Biological Systems to Unmanned Ground Vehicles, chapter 4. Lawrence Erlbaum Associates, Hillsdale, NJ, 1997.

....manual calibration methods and was successfully tested on the task of back warping real images images onto virtual planes. 1. Introduction It is common sense in computer vision that increasing the field of view enhances several visual capabilities like ego motion estimation and localization [8, 3]. It has been further proved that uncertainty in egomotion estimates depends on the shape of the imaging surface (planar or spherical) in the spherical case it is lower than in the case of planar imaging surfaces [4] The advantages of omnidirectional sensing are obvious for applications like ....

K. Daniilidis and M. Spetsakis. Understanding noise sensitivity in structure from motion. In Y. Aloimonos, editor, Visual Navigation, pages 61--88. Lawrence Erlbaum Associates, Hillsdale, NJ, 1996.

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K. Daniilidis and M. E. Spetsakis. Understanding noise sensitivity in structure from motion. In Y. Aloimonos, editor, Visual Navigation: From Biological Systems to Unmanned Ground Vehicles, Advances in Computer Vision, chapter 4. Lawrence Erlbaum Associates, Mahwah, NJ, 1997.

No context found.

K. Daniilidis and M.E. Spetsakis. Understanding noise sensitivity in structure from motion. In VisNav93, 1993.

No context found.

K. Daniilidis and M. E. Spetsakis. Understanding noise sensitivity in structure from motion. In Y. Aloimonos, editor, Visual Navigation: From Biological Systems to Unmanned Ground Vehicles, Advances in Computer Vision, chapter 4. Lawrence Erlbaum Associates, Mahwah, NJ, 1997.

No context found.

K. Daniilidis and M. E. Spetsakis. Understanding noise sensitivity in structure from motion. In Visual Navigation: From Biological Systems to Unmanned Ground Vehicles, chapter 4. Lawrence Erlbaum Associates, Mahwah, NJ, 1997.

No context found.

K. Daniilidis and M. Spetsakis. Understanding noise sensitivity in structure from motion. In Visual Navigation: From Biological Systems to Unmanned Ground Vehicles, chapter 4, pages 61--88. Lawrence Erlbaum Associates, Hillsdale, NJ, 1997.

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K. Daniilidis and M. Spetsakis, Understanding noise sensitivity in structure from motion, in Visual Navigation (Y. Aloimonos, Ed.), pp. 61--88, Erlbaum, Hillsdale, NJ, 1996.

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