Supplementary Material:Discrete Optimization for Optical Flow
Moritz Menze1 , Christian Heipke1 , and Andreas Geiger212

Leibniz Universität Hannover, Germany
Max Planck Institute for Intelligent Systems, Tübingen, Germany

Abstract. In this supplementary material we first provide the parameters we used in our discrete optical flow model. Next, we present histograms of the statistics of optical flow for MPI Sintel and KITTI which
we use to select the upper bounds for the magnitude of flow vectors
in our model. Finally, more detailed qualitative results on the processed
datasets are shown, including comparisons to state-of-the-art approaches.

1Parameters

The parameters in our model were set using block coordinate descent on therespective training data. In this section we report the values we used for theexperimental evaluation on MPI Sintel. For reproducibility we also report theparameters we used for the feature descriptor.
Parameter
Symbol Value
Relative weight of unary and pairwise terms
γ0.95
τφ2.5
Truncation threshold of the data term
Truncation threshold of the smoothness term τψ15α10Edge weight parameter
Size of the label setL
500
Number of neighbors to be sampled
N
200
Stride for discrete CRF in pixels
4Standard deviation for sampling neighbors
5
Similarity threshold (fw-/bw-check)
10
Minimum segment size (outlier removal)
100
10Consistency threshold (outlier removal)
Table 1. Parameters of the proposed model.

In our implementation we use the relative weight γ to balance the data andsmoothness terms
XXψp,q (lp , lq )E(l) = (1 − γ)
ϕp (lp ) + γ| {z }|{z}p∼qp∈Pdatasmoothness

In Equation (1) of the paper we encode this weight as λ in for the sake ofsimplicity.

2Moritz Menze, Christian Heipke, and Andreas Geiger

DAISY Parameters
We leverage the DAISY descriptor [4] due to its computational efficiency and its
robustness against changes in illumination. The parameter setting is adapted toour application as stated in Table 2. Note that due to the global optimization
performed by our model, a small radius of 5 pixels led to the best results.
Symbol Value
Parameter
RadiusR
5
Radius Quantization No.
Q
4
Angular Quantization No.
T
4Histogram Quantization No.
H
4
Grid Point No.
S1DS68Descriptor Size
Table 2. Parameters of the DAISY-Descriptor.

2Ground Truth Statistics

To avoid grossly wrong flow proposals, we limit the admissible flow range to±250 pixels. As evidenced by the histograms of the respective ground truth flow
vectors in Fig. 1, this flow range is appropriate for the datasets considered inthis paper. For the sake of simplicity, we employ a single (loose) upper boundwhich is valid for all scenarios under consideration.
7070-87.3

73.3

60

50

Frequency (%)

Frequency (%)

60

4030
20
100-2000u-flow (Px)

30
20-50

0v-flow (Px)

50

100

70-114.8

133.5

40
30
20
10-5.060
Frequency (%)

Frequency (%)

40

0-100

200

50050.6

1070
60

-50.3

5049.350
40
30
20
10-2000
u-flow (Px)

200

0-100

-50

0v-flow (Px)50

100Fig. 1. This figure shows histograms of the optical flow u-component (red) and theoptical flow v-component (blue) for the training sets of Sintel [1] (top) and KITTI [2](bottom). Vertical lines indicate the 0.01 and 0.99 percentile, respectively.

Supplementary Material: Discrete Optimization for Optical Flow

33

Qualitative Results

In this section, we provide additional qualitative results. In addition to our final
result, we include visualizations of the individual stages of our algorithm. “OursForward” refers to the flow map from block coordinate descent as described inSection 3.3 in the paper and “Ours Clean” comprises our outlier rejection stepfrom Section 3.4. The following figures show comparisons of our post-processed
results (“Ours+DeepFlow” and “Ours+EpicFlow”) to the full DeepFlow [5] andEpicFlow [3] pipelines which use DeepMatches [5] as features.
At the top of each error map in the right column we specify the percentageof outliers and the endpoint error (EPE). The color coding visualizes outliers(> 3 px EPE) in red and inliers (< 3 px EPE) in blue on a logarithmic color
scale. The reference images are the same as shown in the paper, the sixth resultof each dataset shows a failure case (Fig. 7 and Fig. 13).

4

Moritz Menze, Christian Heipke, and Andreas Geiger

Fig. 2. Qualitiative Results. From top-to-bottom: Reference data (topmost row),
optical flow result and error map of the oracle solution, our result without refinement,
our result with outlier removal, our method with DeepFlow refinement, the result ofDeepFlow [5], our method with EpicFlow refinement, the result of EpicFlow [3].

Supplementary Material: Discrete Optimization for Optical Flow

5Fig. 3. Qualitiative Results. From top-to-bottom: Reference data (topmost row),
optical flow result and error map of the oracle solution, our result without refinement,
our result with outlier removal, our method with DeepFlow refinement, the result ofDeepFlow [5], our method with EpicFlow refinement, the result of EpicFlow [3].

6Moritz Menze, Christian Heipke, and Andreas Geiger

Fig. 4. Qualitiative Results. From top-to-bottom: Reference data (topmost row),
optical flow result and error map of the oracle solution, our result without refinement,
our result with outlier removal, our method with DeepFlow refinement, the result of
DeepFlow [5], our method with EpicFlow refinement, the result of EpicFlow [3].

Supplementary Material: Discrete Optimization for Optical Flow

7Fig. 5. Qualitiative Results. From top-to-bottom: Reference data (topmost row),
optical flow result and error map of the oracle solution, our result without refinement,
our result with outlier removal, our method with DeepFlow refinement, the result ofDeepFlow [5], our method with EpicFlow refinement, the result of EpicFlow [3].

8Moritz Menze, Christian Heipke, and Andreas Geiger

Fig. 6. Qualitiative Results. From top-to-bottom: Reference data (topmost row),
optical flow result and error map of the oracle solution, our result without refinement,
our result with outlier removal, our method with DeepFlow refinement, the result ofDeepFlow [5], our method with EpicFlow refinement, the result of EpicFlow [3].

Supplementary Material: Discrete Optimization for Optical Flow

9Fig. 7. Qualitiative Results. From top-to-bottom: Reference data (topmost row),
optical flow result and error map of the oracle solution, our result without refinement,
our result with outlier removal, our method with DeepFlow refinement, the result of
DeepFlow [5], our method with EpicFlow refinement, the result of EpicFlow [3].

10Moritz Menze, Christian Heipke, and Andreas Geiger

Fig. 8. Qualitiative Results. From top-to-bottom: Reference data (topmost row),
optical flow result and error map of the oracle solution, our result without refinement,
our result with outlier removal, our method with DeepFlow refinement, the result ofDeepFlow [5], our method with EpicFlow refinement, the result of EpicFlow [3].

Supplementary Material: Discrete Optimization for Optical Flow

11Fig. 9. Qualitiative Results. From top-to-bottom: Reference data (topmost row),
optical flow result and error map of the oracle solution, our result without refinement,
our result with outlier removal, our method with DeepFlow refinement, the result ofDeepFlow [5], our method with EpicFlow refinement, the result of EpicFlow [3].

12Moritz Menze, Christian Heipke, and Andreas Geiger

Fig. 10. Qualitiative Results. From top-to-bottom: Reference data (topmost row),
optical flow result and error map of the oracle solution, our result without refinement,
our result with outlier removal, our method with DeepFlow refinement, the result ofDeepFlow [5], our method with EpicFlow refinement, the result of EpicFlow [3].

Supplementary Material: Discrete Optimization for Optical Flow

13Fig. 11. Qualitiative Results. From top-to-bottom: Reference data (topmost row),
optical flow result and error map of the oracle solution, our result without refinement,
our result with outlier removal, our method with DeepFlow refinement, the result ofDeepFlow [5], our method with EpicFlow refinement, the result of EpicFlow [3].

14Moritz Menze, Christian Heipke, and Andreas Geiger

Fig. 12. Qualitiative Results. From top-to-bottom: Reference data (topmost row),
optical flow result and error map of the oracle solution, our result without refinement,
our result with outlier removal, our method with DeepFlow refinement, the result ofDeepFlow [5], our method with EpicFlow refinement, the result of EpicFlow [3].

Supplementary Material: Discrete Optimization for Optical Flow

15Fig. 13. Qualitiative Results. From top-to-bottom: Reference data (topmost row),
optical flow result and error map of the oracle solution, our result without refinement,
our result with outlier removal, our method with DeepFlow refinement, the result ofDeepFlow [5], our method with EpicFlow refinement, the result of EpicFlow [3].

16Moritz Menze, Christian Heipke, and Andreas Geiger

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4. Tola, E., Lepetit, V., Fua, P.: Daisy: An efficient dense descriptor applied to widebaseline stereo. PAMI 32(5), 815–830 (May 2010)
5. Weinzaepfel, P., Revaud, J., Harchaoui, Z., Schmid, C.: DeepFlow: Large displacement optical flow with deep matching. In: ICCV (2013)