Supplementary Material: Piecewise Planar and Compact Floorplan Reconstruction from Images

Transcription

1 Supplementary Material: Piecewise Planar and Compact Floorplan Reconstruction from Images Ricardo Cabral Carnegie Mellon University Yasutaka Furukawa Washington University in St. Louis 1. Algorithm Details We provide details of several steps in our algorithm, which are not the contributions of the work and ommited in the main paper, but are necessary to reproduce the work Structure classification initialization As in Sect. 6 of the main paper, we use 3D points and free-space to initialize structure classification labels. From 3D points We first detect 3D points that are on the floor, ceiling or walls, based on their positions and normals. A 3D point is detected to be on the floor if the point is at the floor height and its normal is along the up direction, allowing an error of 0.3m and 15 degrees, respectively. Points on the ceiling are detected in the same way. A 3D point belongs to a wall if its height is between the floor and the ceiling, and the point normal is near horizontal, allowing an error of 15 degrees. Detected 3D points are projected onto visible panoramas. A superpixel is assigned a structure label if the number of projected points with the label is at least 5 and more than the numbers for the other labels. From core free-space A wall should not exist in the free-space. Given a panorama, which is usually in the middle of the core free-space region, if its surrounding core free-space region is large, the distance between the panorama and the closest wall must be at least that far. Therefore, given a panorama and a (2D) direction, which corresponds to an image column, the distance from the panorama to the first pixel that is outside the core-free space along the direction, gives a lower bound on the distance between the panorama and a wall. As in Sect. 6 of the main paper, this lower-bound yields a set of pixels that must belong to the floor or ceiling in the corresponding image column Dynamic programming construction In Section 5.4 of the main paper, dynamic programming is used to compute the optimal path with the specified number of edges in two phases: 1) between every pair of adjacent anchor points; and 2) for an entire floorplan. In both phases, dynamic programming construction is simple. In Dijkstra s algorithm, each node remembers the cost of the optimal path that has been found so far together with the previous node in that path. In the first phase of our algorithm, suppose we are interested in a shortest path that has at most β edges between consecutive anchor points. Each node remembers β different optimal paths with k(= 1, 2, β) edges from the source, again together with the previous node each. We simply scan the entire graph β times, with a simple modification in the recurrunce formula, then the sink node contains the costs of the β optimal paths that consist of k(= 1, 2, β) edges, where each exact path can be obtained by a standard back-tracking. In the second phase, the first anchor point stores optimal paths with k(= 1, 2, β) edges from the start. We also know the set of optimal paths from the first anchor point to the second one, and hence, can easily construct the optimal paths with k(= 1, 2, 2β} edges from the start to the second anchor point. This step is repeated through all the anchor points until the end by the dynamic programming, and can find optimal paths with different numbers of edges from the start to the end, forming an entire floorplan. 2. Complete results and evaluations We did not have enough space to provide comprehensive experimental results and evaluations in the main paper, which are included in this supplementary material together with those in the main paper for being self-contained. Please see the caption for the explanation of each figure. References [1] Y. Furukawa, B. Curless, S. M. Seitz, and R. Szeliski. Reconstructing building interiors from images. In ICCV, [2] D. C. Lee, M. Hebert, and T. Kanade. Geometric reasoning for single image structure recovery. In CVPR,

2 Italian Figure 1. Sample panorama images illustrating the challenges in our datasets.

3 Figure 2. The final texture mapped mesh model rendered from two different viewpoints for each dataset.

4 Italian Figure 3. Continued.

5 Italian Figure 4. Multi-view stereo reconstructions are incomplete and miss many major structures. Figure 5. The left two columns show the extracted line segments (color represents the corresponding vanishing direction) and the structure classification result based on the line feature [2]. The right three columns show our structure classification results after initialization by MVS points, after initialization by free-space information, and the final result.

6 MVS points Camera centers Start/end-line Core free-space Floorplan-path Anchor points Italian Wine Store Points from structure classification Figure 6. The floorplan reconstruction is based on two kinds of 3D evidence: Wall evidence from 3D points (first column) and free-space evidence from visible rays associated with 3D points (second column). Red, green and blue illustrates high, medium and low confidence, respectively. After identifying a region with high free-space evidence as core free-space (grey), a shortest path problem is formulated to reconstruct a floorplan that goes around it (third column). To overcome shrinkage bias, we solve the problem again but with additional anchor points as constraints (fourth column). Ground is obtained manually by clicking room corners in images for comparison (fifth column).

7 Italian Figure 7. Comparative evaluations against the 2D version of the Manhattan volumetric graph-cuts [1]. See the right column of Fig. 6 for our results. From left to right, smoothness penalties are scaled by a factor of four each time.

8 Italian Figure 8. Comparative evaluations against the 3D volumetric graph-cuts technique [1] (left) against our method (right) for each dataset.