Figure 1

Figure 2

Figures through show the results of applying various estimation schemes to fill the holes in the acquisition. Starting from upper-left and moving clockwise, each of the four figures shows a) the ``answer,'' i.e., the 1024x1024 topographical surface, subsampled by a factor of four, b) the estimated model, c) The error in the estimate clipped to a common value and overlain by the the 2250 known data locations, and d) a crossplot of the estimated model and answer at 4000 randomly chosen spatial locations.

Figure shows 100 iterations with Laplacian regularization. Obviously, convergence has not been achieved. Figure shows 1000 iterations with Laplacian regularization. The crossplot is very tight, making this result tough to beat. Figure shows 100 iterations of Laplacian regularization with preconditioning. This result is disappointing: Convergence to this result occured in only 10-20 iterations, but the result itself is not desirable. Although not the subject of this paper, the ``ice-cream-cone'' nature of this result is alarming and merits further investigation. Figure shows 100 iterations of multiscale Laplacian regularization. This result is quite similar to the 1000 iterations Lapacian result, but the crossplot is not as tight. Also, we expect this result to be a bit smoother than the pure Laplacian, but it is difficult to see if it is. Figure shows the explicit Quadtree Pyramid interpolation. The quadtree structure of the result is quite apparent. Although the interpolated map is not smooth spatially, the general structure of the topography are reproduced quite convincingly. Figure shows 100 iterations of Laplacian regularization, with the explicit Quadtree Pyramid interpolation (Figure ) used as a starting guess. This result is quite similar to the 1000 iterations Laplacian result; also the 100 iterations with the multiscale Laplacian.

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

As the ``answer'' is known in this problem, we can quantitatively check the accuracy of an estimated model. The statistics below speak for themselves. Taking the standard deviation of the error to be a measure of goodness, we can rank the results from best to worst:

- 1.
- Laplacian regularization, 1000 iterations.
- 2.
- Multiscale Laplacian regularization, 100 iterations.
- 3.
- Laplacian regularization, 100 iterations, quadtree pyramid interpolation used as starting guess.
- 4.
- Quadtree pyramid direct interpolation.
- 5.
- Preconditioned Laplacian regularization, 100 iterations.
- 6.
- Laplacian regularization, 100 iterations.

| Min. Val. | Max. Val. | Mean Val. | Median | Std. Dev. | ---------------------------------------- True values: -42.10075 253.25171 33.82695 12.50521 48.19902 Estimated values: 0.00000 246.00000 44.96923 18.00000 54.02688 Errors: -70.55768 235.33752 11.14226 0.87845 29.06191 Absolute errors: 0.00000 235.33752 12.89230 2.21318 28.32898 Relative errors: 0.00000 1206.58191 3.02346 0.06742 33.63662

| Min. Val. | Max. Val. | Mean Val. | Median | Std. Dev. | ---------------------------------------- True values: 3.19786 232.39223 45.01379 20.00496 53.70287 Estimated values: 0.00000 246.00000 44.96923 18.00000 54.02688 Errors: -68.31208 76.44979 -0.04465 -0.00007 5.77580 Absolute errors: 0.00000 76.44979 2.51392 0.80804 5.20023 Relative errors: 0.00000 1.00000 0.06820 0.02891 0.10538

| Min. Val. | Max. Val. | Mean Val. | Median | Std. Dev. | ---------------------------------------- True values: 2.55815 220.00000 38.08480 18.64038 45.06718 Estimated values: 0.00000 246.00000 44.96923 18.00000 54.02688 Errors: -61.61859 174.82333 6.88441 0.54753 19.72276 Absolute errors: 0.00000 174.82333 8.05753 1.45995 19.27327 Relative errors: 0.00000 43.94756 0.26195 0.07687 1.28934

| Min. Val. | Max. Val. | Mean Val. | Median | Std. Dev. | ---------------------------------------- True values: -1.53884 229.31610 44.82654 20.00943 53.62170 Estimated values: 0.00000 246.00000 44.96923 18.00000 54.02688 Errors: -62.51950 107.75475 0.14272 -0.00067 8.23634 Absolute errors: 0.00000 107.75475 3.38367 1.06108 7.51047 Relative errors: 0.00000 7.89127 0.09558 0.03850 0.23027

| Min. Val. | Max. Val. | Mean Val. | Median | Std. Dev. | ---------------------------------------- True values: 0.00000 235.00000 42.54507 18.00000 52.90804 Estimated values: 0.00000 246.00000 44.96923 18.00000 54.02688 Errors: -77.00000 102.00000 2.42414 0.00000 12.79912 Absolute errors: 0.00000 102.00000 6.17795 1.00000 11.46838 Relative errors: 0.00000 5.82759 0.18037 0.04881 0.40353

| Min. Val. | Max. Val. | Mean Val. | Median | Std. Dev. | ---------------------------------------- True values: -3.58206 237.56772 43.53872 18.33733 53.00369 Estimated values: 0.00000 246.00000 44.96923 18.00000 54.02688 Errors: -68.67423 71.94943 1.43058 0.00038 8.61643 Absolute errors: 0.00000 71.94943 3.64835 0.85361 7.93589 Relative errors: 0.00000 9.47873 0.10503 0.03159 0.31080

9/5/2000