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Fig. 8. Fit error, region size, and surface order versus iteration number.

Figure 8 Fit error, region size, and surface order versus iteration number.

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Fig. 1. Eight fundamental surface types from surface curvature sign.
Fig. 1. Eight fundamental surface types from surface curvature sign.
In our case | F | = 4 and the set of approximating func tions F can be written in the form of a single equation: Planar surfaces are obtained by restricting the parameter vector space ®'° to three-dimensional subspace where only 49, 419, @o, may be nonzero. Biquadratic surfaces are restricted to a six-dimensional subspace, and bicubic surfaces to a ten-dimensional subspace. A least-squares solver computes the parameter vector d and the RMS fit error € from the digital surface data over a region quickly and efficiently. Moreover, a QR least-squares solution ap- proach allows surface region fits to be updated recursively during the region growing process as new data points are added [42], [22] for better computational efficiency.
In our case | F | = 4 and the set of approximating func tions F can be written in the form of a single equation: Planar surfaces are obtained by restricting the parameter vector space ®'° to three-dimensional subspace where only 49, 419, @o, may be nonzero. Biquadratic surfaces are restricted to a six-dimensional subspace, and bicubic surfaces to a ten-dimensional subspace. A least-squares solver computes the parameter vector d and the RMS fit error € from the digital surface data over a region quickly and efficiently. Moreover, a QR least-squares solution ap- proach allows surface region fits to be updated recursively during the region growing process as new data points are added [42], [22] for better computational efficiency.
Fig. 2. Conventional edge approach versus stimulus bound approach.
Fig. 2. Conventional edge approach versus stimulus bound approach.
Fig. 3. Surface type labels from surface curvature sign.
Fig. 3. Surface type labels from surface curvature sign.
Fig. 4. Contraction profiles for five surface type regions. gions are possible, but this method is simple and poten- tially very fast.
Fig. 4. Contraction profiles for five surface type regions. gions are possible, but this method is simple and poten- tially very fast.
Fig. 5. Runs test and regions test ideas for noiseless data examples. also shown in Fig. 5. The regions test is performed as follows: 1) For each original phe) value i in the region Ri lying above the fitted surface f (m*, a‘; x, y), turn a pixel on in a positive residual error image; for each pixel below the surface, turn a pixel on in a negative error image (all pixels initially off in both images). 2) Perform one 3 x 3 erosion on each error image and count the pixels left in the largest connected region in each image. 3) If either count is greater than 7, percent of the size of the current region |Ri|, then increase the fit order m*. The experi- mental results used a regions test threshold of r, = 0.9 + 0.20in. percent, which was determined empirically.
Fig. 5. Runs test and regions test ideas for noiseless data examples. also shown in Fig. 5. The regions test is performed as follows: 1) For each original phe) value i in the region Ri lying above the fitted surface f (m*, a‘; x, y), turn a pixel on in a positive residual error image; for each pixel below the surface, turn a pixel on in a negative error image (all pixels initially off in both images). 2) Perform one 3 x 3 erosion on each error image and count the pixels left in the largest connected region in each image. 3) If either count is greater than 7, percent of the size of the current region |Ri|, then increase the fit order m*. The experi- mental results used a regions test threshold of r, = 0.9 + 0.20in. percent, which was determined empirically.
Fig. 6. Segmentation results for coffee cup range image.
Fig. 6. Segmentation results for coffee cup range image.
Fig. 7. Range contour lines for original and reconstructed images.
Fig. 7. Range contour lines for original and reconstructed images.
Fig. 10. Segmentation results for computer keyboard range image.
Fig. 10. Segmentation results for computer keyboard range image.
Fig. 11. Segmentation results for ring on steps range image.
Fig. 11. Segmentation results for ring on steps range image.
Fig. 12. Range contour lines for original and reconstructed images.
Fig. 12. Range contour lines for original and reconstructed images.
Fig. 13. Segmentation results for auto part range image.
Fig. 13. Segmentation results for auto part range image.
Fig. 14. Segmentation results for cube with three holes range image.
Fig. 14. Segmentation results for cube with three holes range image.
Fig. 15. Segmentation results for road scene range image.
Fig. 15. Segmentation results for road scene range image.
Fig. 16. Segmentation results for road scene intensity image.
Fig. 16. Segmentation results for road scene intensity image.
Fig. 18. Segmentation results for house scene intensity image Fig. 17. Segmentation results for space shuttle intensity image.
Fig. 18. Segmentation results for house scene intensity image Fig. 17. Segmentation results for space shuttle intensity image.
The transformations from orthogonal axis angular co- ordinates to Cartesian coordinates are given by the fol- lowing: Note the symmetry between the horizontal and vertical angles. The inverse transformations are given by
The transformations from orthogonal axis angular co- ordinates to Cartesian coordinates are given by the fol- lowing: Note the symmetry between the horizontal and vertical angles. The inverse transformations are given by
Note that horizontal angle @ is not a function of the ver- tical Cartesian coordinate y and that the vertical angle ¢ is not a function of the horizontal Cartesian coordinate x.
Note that horizontal angle @ is not a function of the ver- tical Cartesian coordinate y and that the vertical angle ¢ is not a function of the horizontal Cartesian coordinate x.
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Information SystemsRegion growingRange ImageElectrical and Electronic Engineering
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