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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Perceptual organization is related to Gestalt notions of ``good continuation.'' Observe how the ``Figure 8'' appears as a single curve, with smooth connections across the crossing point, and not as the non-generic arrangement of the two shapes in the middle. Such notions of orientation good continuation hold for textures as well; notice how this example appears to continue behind the occluders. }}{3}}
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\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces The geometry of co-circularity for curve detection in images. (left) Measurements of orientation differ at points along a curve. To determine whether they are consistent, nearby tangents are transported along the osculating circle approximation to the curve. If the transported tangents agree they are consistent; otherwise not. (right) To accomplish this transport operation in images, tangent position, orientation, and curvature must be discretized. This shows those nearby tangents that are consistent with a horizontal tangent at the center; that is, those tangent which, if transported along a (discretized) approximation to the osculating circle would support the central, horizontal tangent. (The width of the curve for this example is taken to be 3 pixels.) In the language of relaxation labeling, this is called an excitatory compatibility field. Note that the osculating circle and parabola approximations agree to within a fraction of a pixel over this neighborhood. }}{6}}
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\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The need for higher-dimensional spaces than the image arises in representing non-simple or piecewise-regular curves. Since {\em  a priori} a curve could be passing through any pixel at any orientation, it is natural to represent the (discretized) circle (the space of all unit vectors) $S^1$ at each (discretized) position (left). When the non-simple ``figure 8'' is lifted into the resultant space, the lift is a simple curve in $\@mathbb {R}^2 \times S^1$ (right). The (position, orientation) space, which is abstract from the image, is sufficent to represent all possible curves in the image. }}{7}}
\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces (a) Cartoon of the stereo relaxation process. A pair of space tangents associated with the Frenet approximation around the point with tangent $\@mathbf {e}_j$. Each of these tangents projects to a (left,right) image tangent pair; compatibility between the space tangents thus corresponds to compatibility over (left,right) image tangent pairs. The projected tangents are shown as thick lines. One left image tangent is redrawn in the right image (as a thin line) to illustrate positional disparity ($\Delta d$)and orientation disparity ($\Delta \theta $). The compatibility between the tangent pair $(i)$ and the pair $(j)$ is denoted $r_{ij}$. Of course, for the full system the complete Frenet 2-frames are used to infer the Frenet 3-frame attached to the space curve. (b) Just as the osculating circle provided a local model for transport for image curves, a section of a helix provides a local model for a space curve. The $(T,N)$ components of the Frenet 3-frame define the osculating plane, which rotates as the frame is moved along the space curve.}}{8}}
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\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces 3D reconstruction of a twig pair.(a) Left image (b) Right image; note in the highlighted region that subtleties in using the ordering constraint arise. Furthermore, occlusion of branches gives rise to discontinuities in orientation. Representing such discontinuities as multiple tangents facilitates proper matching. (c) Reconstruction. Depth scale is shown at right (units: meters).}}{9}}
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\@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces Displacement (transport) of a Frenet frame within a vector field or an oriented texture amounts to rotation, but differs for different displacements. The covariant derivative specifies the frame's initial rate of rotation for any direction vector $\mathaccentV {vec}17E{\@mathbf {V}}$. The four different cases in this figure illustrate how this rotation depends on $\mathaccentV {vec}17E{\@mathbf {V}}$ both quantitatively (i.e,, different magnitudes of rotation) and qualitatively (i.e., clockwise, counter-clockwise, or zero rotation). A pure displacement in the tangential direction ($\mathaccentV {hat}05E{\@mathbf {E}}_{T}$) specifies one rotation component (the tangential curvature) and a pure displacement in the normal direction ($\mathaccentV {hat}05E{\@mathbf {E}}_{N}$) specifies the other (normal curvature) component. }}{11}}
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\@writefile{lof}{\contentsline {figure}{\numberline {9}{\ignorespaces Texture compatability fields are discretizations of a helicoid approximation to a flow lifted into $\@mathbb {R}^2 \times S^1$. Three examples are shown: {\bf  (A)} both curvatures are zero; this is the analog to a straight line for curves; {\bf  (B)} tangential curvature is zero and normal curvature is positive; this shows a local portion of a texture flow in which the integral curves converge to a (singular) point, as lines converge to a point in the distance; and {\bf  (C)} both the tangential and the normal curvatures are positive. This is the general case: notice how singular points (where all orientations are possible) arise. These are indicated as multiple line segments displayed at the same position. }}{12}}
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\@writefile{lof}{\contentsline {figure}{\numberline {10}{\ignorespaces Examples of texture patterns rich in orientation. (A) A woven texture with two dominant orientations. This is an extension of (B) two overlapping textures, which are naturally separated when lifted into $\@mathbb {R}^2 \times S^1$ in (C). The bottom panels illustrate how a noisy pattern (D) is refined using the geometric compatibilities in Fig.\nobreakspace  {}9\hbox {} to (E), thereby enforcing a Gestalt-like good contination of the flows.}}{13}}
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\@writefile{lof}{\contentsline {figure}{\numberline {11}{\ignorespaces The HSV color representation in ${\@mathcal S}^{1} \times [0,1]^{2}$ and the color wheel.}}{14}}
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\@writefile{lof}{\contentsline {figure}{\numberline {12}{\ignorespaces A flow perspective on color images is provided by their hue fields. These are typically piecewise smooth. Most importantly, hue can vary smoothly even within perceptually coherent objects. {\bf  (top)} A natural image of an apple with varying hue. Notice that the everyday expression of ``red apple'' is limited. The corresponding hue field changes smoothly across the image of the apple's surface. {\bf  (bottom)} A 3D representation of the hue filed, where hue is represented as height. Identifying the top face with the bottom (since hue is a circle) leads to the (position, hue) space. }}{15}}
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