barcode generator vb.net download Surface Interpolation in Software

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This section shows how some of the surface representations presented above can be used to interpolate samples of a graph surface, such as depth measurements obtained with binocular stereo or active triangulation Surface interpolation may be necessary when depth measurements do not conform to the uniform grid format required for image processing It may be necessary to interpolate depth measurements onto a uniform grid before using image processing algorithms such as edge detection and segmentation
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Suppose that we have obtained samples Zk of a graph surface, Z = f(x, y), at scattered points (Xk' Yk) for k = 1, ,n using binocular stereo or active triangulation and we need to interpolate the depth measurements at grid locations [i,jj in the image plane The coordinates (Xj, Yi) at each location in the n x m grid (image array) are given by
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(1341) (1342)
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We need to interpolate the Z value at each point (Xj, Yi)' Use the scattered point coordinates and depth values (Xk' Yk, Zk) to create a triangular mesh Since the depth measurements are from a graph surface, each triangle has an explicit planar equation (1343) with coefficients calculated from the formulas presented in Section 1331 For each grid location, find the triangle that encloses point (Xj, Yi) and use
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CHAPTER 13 CURVES AND SURFACES
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the equation corresponding to this triangle to calculate the depth value at the grid location: (1344)
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Sometimes it is necessary to interpolate values that already lie on a rectilinear grid For example, rectification, discussed in Section 125, requires that the image intensity at a location between grid coordinates be interpolated from the pixel values at the four nearest grid locations Bilinear interpolation is an easy method for interpolating values on a rectilinear grid The function (1345) is called bilinear because if one variable is set to a constant, then the function is linear in the other variable In other words, each cross section of a bilinear surface patch taken parallel to a coordinate axis is a line segment For any rectangle in the plane with sides parallel to the coordinate axes, there is a unique bilinear polynomial that interpolates the corner values Suppose that we need to interpolate the value at point (x, y) between four values on a rectilinear grid The point is enclosed by a rectangle with sides parallel to the coordinate axes; the corners of the rectangle are the closest grid coordinates to the point at which the value is to be interpolated The corner coordinates are (Xl, YI), (Xl, Y2), (X2' YI), and (X2' Y2) with values Zn, Z12, Z21, and Z22, respectively, as shown in Figure 134 The coefficients of the bilinear interpolant are determined by the values at the four corners of the grid rectangle and are computed by plugging each grid coordinate and value into Equation 1345:
Zn Z12 Z21
+ a2 x I + a3YI + a4 X IYI al + a2 X l + a3Y2 + a4 X IY2 al + a2 x 2 + a3Yl + a4 X 2Yl al + a2 X 2 + a3Y2 + a4 X 2Y2
(1346) (1347) (1348) (1349)
and solving the four simultaneous equations for the coefficients of the interpolant:
X2Y2 Z n X2YIZl2 (X2 XIY2 Z 21
+ XIYIZ22
XI)(Y2 -
(1350)
136 SURFACE INTERPOLATION
Figure 134: The bilinear interpolant is used to estimate the value between samples on a rectilinear grid The values at the corners are known and uniquely determine the coefficients of the bilinear interpolant
-Y2 Z 11 a2 -X2 Z 11 a3
Zll -
+ YIZ12 + Y2 Z 21 Xl)(Y2 Yl)
YIZ22
(1351) (1352) (1353)
(X2 -
+ X2 Z 12 + XIZ21 Xl)(Y2 Z21 Yl)
XIZ22
(X2 a4 =
Z12 -
+ Z22
Yl)
(X2 -
Xd(Y2 -
The bilinear interpolant has a very simple form for the special case where the rectilinear grid is a square with unit spacing between the rows and columns Let the point (x, y) at which interpolation is to be performed be given as offsets (8x,8y) from the upper left corner of a grid square The bilinear interpolant is
This formula can be used to interpolate the pixel value at an image plane point that is between the pixel locations
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