Toward a Theory of Shape from Specular Flow

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equations can be solved analytically to yield dense surface known 11 While the exact mechanisms underlying these. shape and here we explore two such cases in detail We be results are not yet known it has been suggested that humans. gin our exploration in a two dimensional world Sec 3 in exploit the fact that image gradient directions are often cor. which the specular object is a plane curve the image plane related with second derivatives of the surface 11. is a line and the surrounding environment is a function Computationally the inference of shape in such general. defined on the unit circle In this case one can uniquely conditions is severely ill posed One can obtain additional. recover the surface convex or not by solving a separable constraints however through observations of dense spec. non linear ODE with initial conditions provided by an oc ular flow induced by relative motion of an object viewer. clusion boundary We then consider a three dimensional and or environment In a qualitative analysis Walden. world Sec 4 where the specular object is a surface and we and Dyer 26 show that specular flow is singular along. derive a coupled pair of non linear PDEs that relate specu parabolic curves when either the environment or viewer is. lar flow to surface shape We show how singularities of far from the surface and that singularities can drift from. these PDEs relate directly to the parabolic lines of the shape parabolic curves when both are nearby More quantitatively. where the specular flow generically grows unbounded and Roth and Black 21 present an optical flow algorithm that. how analytic reconstruction is feasible under a specific class estimates a specular flow field and simultaneously identi. of environment motions Based on the analytic approach we fies a surface from a parametric family of implicit functions. demonstrate numerical shape recovery using both 2D and e g spheres of varying radii. 3D experimental data It should be mentioned that previous work on the re. covery of specular shape also include 3D scanning sys. 2 Related work tems that use calibrated environments to obtain shape in. formation Example configurations include extended light. Most studies of the relationship between specular reflec sources with object or source motion 27 15 and one or. tions and surface shape consider environments that contain more views of a fixed object under one or more grid like. a single point light source In these cases one observes a environments 23 12 8 9. small number of specularities each of which induces a In contrast to previous work we seek quantitative shape. constraint between a surface point its normal and its local recovery for general surfaces that are not constrained to be. view and illumination directions In addition small changes convex or of a particular parametric form We consider. in viewpoint induce specular motion and by observing this completely unknown dense illumination environments and. specular motion relative to the motion of fixed surface tex surfaces that are absent of diffuse texture that could other. ture one can make local inferences about the sign of the wise assist in the reconstruction process The main contri. Gaussian curvature 4 5 28 6 In order to obtain more bution of our work is to show that shape can be recovered. quantitative surface information from sparse specular ob under these conditions. servations however one must employ significant regular. ization 25,3 Specular shape from specular flow in 2D. More information regarding surface shape can be ob. tained by observing the motion of sparse specularities over Before addressing the general three dimensional prob. extended motion sequences Qualitatively it is known that lem important insights can be gained from analyzing the. as the observer moves specularities are created and anni inference of specular shape in two dimensions i e surface. hilated in pairs at or in the near field case close to 7 profiles In this case surfaces are reduced to plane curves. parabolic surface points 17 19 More quantitatively the images and specular flow fields are one dimensional and. ory suggests that one can recover a complete surface profile the space of illumination directions is a circle Fig 2 Here. i e a curve by observing the specular motion induced by we show that under the conditions of our model i e far. continuous camera motion 28 Practical methods for do field illumination and observer one can analytically re. ing so however have been developed for convex or con cover an arbitrary continuous surface profile from the ob. cave surfaces and do not allow parabolic points 20 served specular flow. Specular shape inference in natural uncontrolled envi As is shown in Fig 2 the visible part of a smooth surface. ronments has received significantly less attention Since profile is assumed to be the graph of a function f x and. curved specular surfaces reflect illumination from all di the far field illumination environment E describes the. rections real world environments induce dense specular incident radiance which is independent of x At a point x. reflections that are qualitatively very different from the on the image plane we observe the radiance reflected from. sparse specularities described above For still images of this a point on the surface having normal orientation x and. type it has been observed that humans often but not al the radiance measured at I x is simply the value of the il. ways 24 infer accurate shape even when the illumination lumination environment E x in the mirror reflected di. environment and bounding contour of the surface are un rection x Since the viewing direction is aligned with. Surface height f x,0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1. Image coordinate x,Optical flow u x, Figure 2 The specular shape reconstruction problem in two dimensions. A surface profile f x a plane curve is illuminated by a far field illumi. nation environment E and is viewed orthographically to produce a 1D. image I x The sign convention for the angular dimension is shown in. 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1,Image coordinate x. 0 it follows that x 2 x Figure 3 Recovering a surface profile from specular flow A surface pro. file f x top blue solid curve is viewed under a rotating environment as. To recover shape from specular flow we seek a relation depicted in Fig 2 This induces a specular flow bottom that is singular at. ship between motion of the environment d dt and inflection points Using this flow we recover the surface by solving Eq 4. the induced motion field or specular flow on the image using the left hand surface boundary as an initial condition The surface is. plane u dx dt From the sign convention in Fig 2 it recovered top red dashed curve despite the singularities in specular flow. because reconstruction relies on the integration of inverse flow which is. follows that well defined everywhere,tan x 2 fx x, Taking temporal derivatives of this expression and using the.
fact that sec2 2 1 fx2 we obtain the desired relation v fx and it has a relatively simple analytic solution We. ship first obtain the first derivative of the surface using. 2 x 1 fx x 2 x d,fx x tan C, where x fxx 1 fx2 3 2 is the curvature at point x 2 xi u. Equation 1 is a generative equation for specular flow and. it shows that specular flow is well defined everywhere ex where is a dummy variable xi is some initial point on. cept at the projections of surface points having zero curva the image plane and C is an arbitrary constant that can be. ture Furthermore as is exemplified in Fig 3 projections determined using an initial condition C tan 1 fx xi. of these points behave as either sources or sinks of spec Once the first derivative is known the surface f can be re. ular flow in accordance with the pair wise specular birth covered through integration This introduces another arbi. and death that is expected at parabolic lines in three di trary constant which determines the absolute depth of the. mensions 17 19 In this example where the environment surface and can be set to zero. rotates in a counter clockwise manner the flow is divergent. expands outwards in both directions at the left inflection. point This point behaves as a flow source a point where 3 1 Object boundaries as initial conditions. new regions of the environment come into view on the im. When the surface profile is a smooth closed curve object. age plane By the same reasoning the right inflection point. boundaries occur where the surface normal is orthogonal to. is a sink because the flow is convergent there Their roles. the viewing direction and the derivative is therefore known. would change if one were to reverse the direction of envi. at these points see x0 and xf in Fig 2 Thus object. ronment rotation, boundaries provide a convenient source for initial condi. In order to recover the surface from the observed specu. tions Using the surface parameterization proposed above. lar flow we rearrange Eq 1 to obtain a Riccati equation. however the initial conditions at the left and right object. 2u x fxx fx2 0 2 boundaries are fx xo and fx xf which. are inconvenient for numerical purposes, This second order non linear ODE can be reduced to a To get around this we can re parameterize the surface. separable first order equation by making the substitution derivative fx using a stereographic projection 16 accord. ing to which we define, Substituting in Eq 2 yields an equation of the same form. u x qx q 0 4 f x y y,whose solution is, Figure 4 The specular shape reconstruction problem in three dimensions.
q x 2 tan C A surface f x y is illuminated by a far field illumination environment. 4 xo u and is viewed orthographically to produce a 2D image I x y The illu. mination sphere is parameterized using spherical coordinates. Here C is an arbitrary constant that can now be determined. using the initial condition provided by one of the object. boundaries These nice properties of the 2D SFSF equation fol. low directly from an image formation model that in. C tan 1 q xo tan 1 2 cludes a far field viewer and environment and relative. object environment motion As we show in Sect 4 2, To recover the surface f x the solution q x is trans many of these desirable properties carry over to the three. formed via Eq 3 and then integrated as before dimensional case as well. A demonstration of this procedure is shown in Fig 3. Here a sequence of 1D images is rendered under an envi. 4 Specular shape from specular flow in 3D, ronment that rotates in a counter clockwise direction The. environment is extracted from a great circle of the captured Much like the two dimensional case described in the pre. St Peter s environment map 18 Flow is estimated nu vious section we begin the three dimensional analysis by. merically and independently at each pixel using the optical considering a surface S x y x y f x y that is the. flow equation Ix u It 0 and the surface is recovered by graph of a bi variate function As before the surface is. solving Eq 4 using the left most point as an initial condi viewed orthographically from above and illuminated by a. tion far field environment see Fig 4,Let v 0 0 1 be the viewing direction n x y the. 3 2 Observations surface normal at surface point x y f x y and r x y. Since it enables the recovery of surface shape we refer the mirror reflection direction at the same point An image. to Eq 4 as the shape from specular flow SFSF equation of S x y on the orthographic image plane constitutes radi. in two dimensions It has a number of notable properties ance values of the distant illumination environment In the. The ODE can be solved analytically given an analytic ex 3D case this environment constitutes a sphere of directions. pression for the specular flow and a unique solution can be which we parameterize with two spherical angles zenith. readily obtained using an occluding contour or any other and azimuth In particular we represent reflection direc. point at which the first derivative is known as a boundary tions as and normal directions as both under. condition Since there is no aperture problem in two dimen the usual sign convention shown in Fig 4 As in the previ. sions specular flow can be estimated independently at ev ous section these directions are related by. ery image point from as few as two images Thus provided x y 2 x y. that the illumination environment exhibits sufficient angu x y x y. lar radiance variation we are able to completely recover a. two dimensional surface profile from as few as two frames In order to relate displacements on the image plane to those. Another important property is that the 2D SFSF equation on the illumination sphere we note that the reflection vec. enables the recovery of arbitrary smooth surfaces includ tor at each point can be expressed both in terms of surface. ing those with points of zero curvature As noted above the derivatives and spherical coordinates. specular flow approaches at the projection of an inflec. tion point Surface reconstruction requires the integration r sin cos sin sin cos. of the inverse flow however which is well defined every 2fx 2fy 1 fx2 fy2. where fx2 fy2 1, From this relationship we can deduce that 4 1 Behavior at parabolic points. 2 f While Eq 7 may be used to solve for an unknown shape. tan 1 f 2 f x y from a known specular flow u v it can be rear. 5 ranged through inversion of the Jacobian J to derive a gen. tan fx erative equation for an unknown specular flow u induced by. Toward a Theory of Shape from Specular Flow targets general surfaces in unknown real world environ ments Our approach is built on an image formation model that is complex enough to be practical but simple enough fortractableanalysis Themodelhastwoessential features 1 The environment and observer are far from the spec ular surface relative to the surface relief This im plies a parallel

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