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Learning Compact Geometric FeaturesSupplementary Material
Marc KhouryUC Berkeley
Qian-Yi ZhouIntel Labs
Vladlen KoltunIntel Labs
A. Additional Controlled Experiments
There are a number of hyperparameters and designchoices that affect performance. We present a set of con-trolled experiments that validate each choice and provideintuition on how these choices affect the results. These ex-periments are performed on the validation set of the laserscan data (the Dancing Children model).
A.1. Dimensionality
The most important parameter is the dimensionality ofthe feature space. Figure 1 shows the performance of thelearned feature for five different settings of dimensionality.As expected, increasing the dimensionality improves per-formance. However, we observe diminishing returns aftern = 32.
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10D20D32D40D50D
Figure 1. Precision of our learned feature as we increase the di-mensionality of the embedding space.
A.2. Input Parameterization
Recall that our proposed input parameterization isR = 17 subdivisions in the radial direction, E = 11 in theelevation direction, A = 12 in the azimuth direction, and asearch radius of 17% of the diameter of the model. The re-sultant dimensionality of the input histogram is N = 2,244.In each experiment we select one of these parameters and
consider the parameterization resulting from changing thisvalue by ±2.
Figure 2 shows the precision for different values of thenumber of radial subdivisions. The number of radial subdi-visions is the only parameter where precision does not in-crease as we increase the value of the parameter. As Figure2 shows, R = 17 performs slightly better than the other twosettings.
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R = 15R = 17R = 19
Figure 2. Precision of our learned feature as we increase the num-ber of radial subdivisions.
If instead we increase both the number of radial subdivi-sions and the search radius in tandem, as shown in Figure 3,then precision continues to increase. The gain of going fromR = 17 with a search radius of 17% to R = 19 with asearch radius of 19% is smaller than the gain of going fromR = 15 with a search radius of 15% to R = 17 with asearch radius of 17%.
Figure 4 shows the precision for different values of thenumber of elevation subdivisions. There is a gain of 1.7percentage points in going from E = 9 to E = 11, and neg-ligible gain thereafter.
Figure 5 shows the precision for different values of thenumber of azimuth subdivisions. Precision increases as thenumber of azimuth subdivisions increases, but the gains arenegligible.
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R = SR = 15R = SR = 17R = SR = 19
Figure 3. Precision of our learned feature as we increase the num-ber of radial subdivisions and the search radius in tandem.
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E = 9E = 11E = 13
Figure 4. Precision of our learned feature as we increase the num-ber of elevation subdivisions.
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A = 10A = 12A = 14
Figure 5. Precision of our learned feature as we increase the num-ber of azimuth subdivisions.
A.3. Architecture Design
The architecture of our model consists of L = 5 hid-den layers, each with H = 512 hidden units. As Figure 6shows, L = 5 performs slightly better than using a modelarchitecture with L = 4 or L = 6 hidden layers. Further-more there is a gain of 1.5 percentage points in going from
H = 256 to H = 512, with L = 5.
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L = 4L = 5L = 6L = 5, H = 256
Figure 6. Precision of the learned feature as we vary the depth andwidth of our model.
A.4. Other Design Choices
Alternatively, we could consider different loss functionsor different input parameterizations. Figure 7 shows the ef-fect of two such modifications. The first uses SHOT fea-tures as the input to our model, instead of our input pa-rameterization. The second uses the contrastive loss fortraining [1] instead of the triplet loss. Substituting SHOTfeatures for our input parameterization reduces precisionfrom 22.4% to 12%. Substituting the contrastive loss forthe triplet loss reduces precision to 2.75%.
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SHOT raw inputContrastive lossOurs
Figure 7. Effect of the input parameterization and the embeddingobjective. Our input parameterization and loss are compared to us-ing SHOT features as input (everything else held fixed) and usinga contrastive loss instead of the triplet loss (everything else fixed).
B. Additional Baselines
As an additional set of baselines we apply PrincipalComponents Analysis (PCA) to each of the baseline featuredescriptors to embed them into a 32-dimensional space.
Laser scan data. Figure 8 shows the precision of differ-ent feature descriptors after applying PCA to embed eachinto a 32-dimensional space on the laser scan test set. Theprecision values for each feature after applying PCA are27.9% for PCA USC, 26.5% for PCA SI, 21.2% for PCAFPFH, 15.7% for PCA PFH, 14.8% for PCA RoPS, and12% for PCA SHOT. These values can be compared tothe precision achieved by the original descriptors: 31.5%for USC, 32.2% for SI, 21.2% for FPFH, 15.8% for PFH,15.3% for RoPS, and 14.8% for SHOT. Most prior featureslose a few percentage points of precision when projectedinto the lower-dimensional space. Both the original de-scriptors and their lower-dimensional versions are consid-erably less discriminative than our learned low-dimensionaldescriptor.
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CGF-32PCA FPFHPCA PFHPCA RoPSPCA SHOTPCA SpinImagePCA USCPCA of raw input
Figure 8. Precision of prior feature descriptors embedded into a32-dimensional space using PCA, compared to CGF-32. Resultson the laser scan test set.
SceneNN data. Figure 9 shows the precision of differ-ent feature descriptors after applying PCA to embed eachinto a 32-dimensional space on the SceneNN test set. Theprecision values for each feature after applying PCA are22.4% for RoPS, 21% for PCA PFH, 20.7% for PCA FPFH,20.6% for PCA USC, 18.2% for PCA SHOT, and 6.6%for PCA SI. These values can be compared to the preci-sion achieved by the original descriptors: 22.7% for RoPS,21.1% for PFH, 20.7% for FPFH, 29.8% for USC, 20.2%for SHOT, and 8.2% for SI. Most prior features lose afew percentage points of precision when projected into thelower-dimensional space. Both the original descriptors andtheir lower-dimensional versions are considerably less dis-criminative than our learned low-dimensional descriptor.
C. Approximate Nearest Neighbors
Approximate nearest neighbor algorithms acceleratenearest neighbor queries in high-dimensional spaces byloosening the constraint that the exact nearest neighbor
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CGF-32PCA FPFHPCA PFHPCA RoPSPCA SHOTPCA SpinImagePCA USCPCA of raw input
Figure 9. Precision of prior feature descriptors embedded into a32-dimensional space using PCA, compared to CGF-32. Resultson the SceneNN test set.
must be returned. Given a query point q, an approximatenearest neighbor query returns a point p at distance withina factor of K of the nearest-neighbor distance. The tradeoffbetween speed and accuracy is controlled by the parameterK.
We demonstrate that CGF-32 is robust to approximation.Thus, in practice, our reported query times can be sped upeven further using approximate nearest neighbor queries, atalmost no loss in precision. The results for different valuesof K are shown in Figure 10. For K = 20, CGF-32 losesonly 0.8 percentage points of precision at 1% of the modeldiameter.
For the baseline features, at K = 20, FPFH loses 0.8 per-centage points, PFH loses 1 percentage point, SHOT loses1.6 percentage points, SI loses 2.3 percentage points, andUSC loses 9.9 percentage points.
D. Geometric RegistrationFigure 11 (multiple pages) shows 20 randomly sampled
fragment pairs from the SceneNN test set and correspond-ing alignments produced by FGR with CGF-32. This illus-trates the quantitative results presented in Section 7.5 in thepaper.
References[1] R. Hadsell, S. Chopra, and Y. LeCun. Dimensionality reduc-
tion by learning an invariant mapping. In CVPR, 2006. 2
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FPFHFPFH K = 10FPFH K = 20FPFH K = 30FPFH K = 40FPFH K = 50FPFH K = 100
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SHOTSHOT K = 10SHOT K = 20SHOT K = 30SHOT K = 40SHOT K = 50SHOT K = 100
(a) FPFH (b) SHOT
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PFHPFH K = 10PFH K = 20PFH K = 30PFH K = 40PFH K = 50PFH K = 100
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USCUSC K = 10USC K = 20USC K = 30USC K = 40USC K = 50USC K = 100
(c) PFH (d) USC
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SpinImageSpinImage K = 10SpinImage K = 20SpinImage K = 30SpinImage K = 40SpinImage K = 50SpinImage K = 100
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CGF-32CGF-32 K = 10CGF-32 K = 20CGF-32 K = 30CGF-32 K = 40CGF-32 K = 50CGF-32 K = 100
(e) SI (f) CGF-32
Figure 10. Robustness of different feature spaces to approximate nearest neighbor search. As the approximation factor K increases, theprecision of retrieved matches decreases. Some feature spaces are more robust than others and the decline in precision as a factor of Kis smaller. For K = 20, our feature space loses only 0.8 percentage points in precision. USC is the least robust feature space, losing 9.9percentage points for K = 20.
target source alignment
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Figure 11. Randomly sampled fragment pairs from the SceneNN test set (left, middle) and corresponding alignments produced by FGRwith CGF-32 (right).
target source alignment
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Figure 11 (cont.). Randomly sampled fragment pairs from the SceneNN test set (left, middle) and corresponding alignments produced byFGR with CGF-32 (right).
target source alignment
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Figure 11 (cont.). Randomly sampled fragment pairs from the SceneNN test set (left, middle) and corresponding alignments produced byFGR with CGF-32 (right).
target source alignment
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Figure 11 (cont.). Randomly sampled fragment pairs from the SceneNN test set (left, middle) and corresponding alignments produced byFGR with CGF-32 (right).
