# Accurate Dense Stereo Matching Based on Image Segmentation Using an Adaptive Multi-Cost Approach

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

## 3. Stereo Matching Algorithm

#### 3.1. Image Segmentation

#### 3.2. Initial Disparity Map Estimation

#### 3.2.1. The Multi-Cost Function

_{x}(x directions) and I

_{y}(y directions) are shown as:

_{horizonal}and vertical vector V

_{vertical}. Define V(P) as the INV of point P.

_{horizonal}and vertical vector V

_{vertical}are defined as follows:

#### 3.2.2. Cost Aggregation

_{q}) = −(T

_{max}/L

_{max}) × L

_{q}+ T

_{max}. This linear threshold T(L

_{q}) in color similarity involves the maximum semi-dimension L

_{max}of the support window size, the maximum color dissimilarity T

_{max}between pixels p and q, and the spatial closeness L

_{q}. According to [7], the values of T

_{max}and L

_{max}are 20 and 35, respectively. The final support window U(p) for p is formulated as a union of horizontal segment H(q), in which q traverses the vertical segment V(p). A symmetric support window is also adopted to avoid distortion by the outliers in the reference image [7]. This is shown in Figure 7b.

_{x}

_{,y}at coordinates (x,y) is estimated by using the WTA strategy where the lowest matching cost is selected:

#### 3.3. Disparity Plane Fitting

_{reference}be the disparity map from reference image to target image, and D

_{target}be the disparity map from target image to reference image. The mutual consistency check is formulated as:

_{consistency}is a constant threshold (typically 1). If the pixels of the reference image satisfy Equation (19), these pixels are marked as non-occluded pixels; otherwise these pixels are marked as occluded pixels, which should be filtered out as outliers.

_{confidence}is a threshold to adjust the confidence level. If the cost score of the pixels in the reference image satisfies Equation (20), these pixels are considered reliable. If the ratio between the number of the reliable pixels and the total number of the pixels in arbitrary segment region is equal to or greater than 0.5, this segment region is considered a reliable segment region. Otherwise segment regions are marked as unreliable regions, which lack sufficient data to provide reliable plane estimations. The disparity plane of the unreliable region is stuffed through its nearest reliable segment region.

_{outlier}is a constant threshold (typically 1). If the pixel does not satisfy Equation (21), it would be an outlier. Then we can exclude the outliers, update the reliable pixels of the segment region, and re-estimate the disparity plane parameters of the segment region.

_{convergence}is the convergence threshold of the iterative and is usually set as (typically 10

^{−6}).

#### 3.4. Disparity Plane Optimization by Belief Propagation

_{N}is a set of all adjacent segment regions, S

_{i}, S

_{j}are neighboring segment regions, and ${\mathsf{\lambda}}_{\mathrm{disc}}(x,y)$ is a discontinuity penalty function.

## 4. Experimental Results

**Runtime**. The algorithm implementation is written in VS2010 and uses the OpenCV core library for basic matrix operations. The runtime is measured on a desktop with Core i7-6700HQ 2.60 GHz CPU and 16 GB RAM, and no parallelism technique is utilized. All operations are carried out with floating point precision. Our algorithm require 0.59 s/megapixels (s/mp) for image segmentation, 3.4 s/mp for initial disparity estimation, 15.8 s/mp for disparity plane fitting, and 7.6 s/mp for disparity plane optimization.

## 5. Discussion and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**The image segmentation results. (

**a**) Jade plant image and corresponding segmentation results; (

**b**) motorcycle image and corresponding segmentation results; (

**c**) playroom image and corresponding segmentation results; (

**d**) play table image and corresponding segmentation results; and (

**e**) shelves image and corresponding segmentation results.

**Figure 4.**The similarity cost functions and the shape of the adaptive support window. (

**a**) Jade plant; (

**b**) motorcycle; (

**c**) playroom; (

**d**) play table; and (

**e**) shelves. From top to bottom: the reference images; the modulus images of the illumination normal vector; the gradient maps along horizontal direction; the gradient maps along vertical direction; and the examples of the adaptive support window.

**Figure 5.**Comparison of different ways of similarity functions combination for Middlebury stereo datasets. (

**a**) Jade plant; (

**b**) motorcycle; (

**c**) playroom; (

**d**) play table; and (

**e**) shelves. Disparity maps are estimated by different combinations of similarity functions after the aggregation step. From top to bottom: the reference images; the ground truth; the disparity maps estimated by ICT; ICT+TADc; ICT+TADc+TADg; ICT+TADc+TADg+INV; the corresponding bad 2.0 error maps for ICT; ICT+TADc; ICT+TADc+TADg; and ICT+TADc+TADg+INV. The same region of the error maps is marked by red rectangles.

**Figure 6.**The visualized quantitative performance of similarity functions (in % of erroneous disparities at 2 error threshold) by comparing different combinations of similarity functions against the ground truth. From left to right, the charts correspond to the error matching rate of (

**a**) non-occluded pixels and (

**b**) all image pixels. On the horizontal axis, A: ICT; B: ICT+TADc; C: ICT+TADc+TADg; and D: ICT+TADc+TADg+INV.

**Figure 7.**The illustration of the adaptive cross-based aggregation algorithm. (

**a**) The upright cross skeleton. The upright cross consists of a horizontal segment $H(p)$ = $\{(x,y)|x\in [{x}_{p}-{h}_{p}^{-},{x}_{p}+{h}_{p}^{+}],y={y}_{p}\}$ and a vertical segment $V(p)$ = $\{(x,y)|x={x}_{p},y\in [{y}_{p}-{v}_{p}^{-},{y}_{p}+{v}_{p}^{+}]\}$; (

**b**) the support region U(P) is a combination of each horizontal segment H(q), where q traverses the vertical segment V(p) of p; (

**c**) a schematic of a 1D integral image technique.

**Figure 8.**The flow chart of the iterative outlier suppression and disparity plane parameters fitting algorithm.

**Figure 9.**Diagrams presenting the response of the algorithm to the tuning parameters with the rest of the parameter set remaining constant. The average error rate of the stereo pairs for each evaluation area (all, nonocc) is displayed. (

**a**) Spatial bandwidth and (

**b**) spectral bandwidth used for image segmentation in Step 1 of the algorithm; (

**c**) Gamma INV(Illumination Normal Vector); (

**d**) Gamma ICT(Improved Census Transform); (

**e**) Gamma TADC(Truncated Absolute Difference on Color); and (

**f**) Gamma TADG(Truncated Absolute Difference on Gradient) used for matching cost computation in Step (2) of the algorithm. (

**g**) Threshold consistency; (

**h**) threshold confidence; (

**i**) threshold outlier; and (

**j**) threshold convergence used for outliers filter and disparity plane parameters fitting in Step (3) of the algorithm; (

**k**) smoothness penalty and (

**l**) occlusion penalty used for smoothness and occlusion penalty in Step (4) of the algorithm.

**Figure 10.**Results of Middlebury stereo datasets “Jade plant”, “Motorcycle”, “Playroom”, “Play table” and “Shelves” (from top to bottom). (

**a**) Reference images; (

**b**) ground truth images; (

**c**) results of the proposed method; and (

**d**) error maps (bad estimates with absolute disparity error >2.0 are marked in black).

**Figure 11.**The statistical results of different combinations of similarity functions and different fitting algorithms. (

**a**) The average error rates of the non-occlusion areas (nonocc) and (

**b**) of the complete image (all). A: initial disparity; B: disparity plane fitting by RANSAC; C: disparity plane optimization of B; D: disparity plane fitting by our iterative outlier suppression and disparity plane parameters fitting algorithm; E: disparity plane optimization of D.

**Figure 12.**Performance of each step of the algorithm regarding disparity map accuracy. (

**a**) The average error rates of the complete image (all) and non-occlusion areas (nonocc) for the 0.75 pixel threshold; (

**b**) the average error rates of the complete image (all) and non-occlusion areas (nonocc) for the 1.0 pixel threshold; (

**c**) the average error rates of the complete image (all) and non-occlusion areas (nonocc) for the 2.0 pixel threshold; and (

**d**) the average error rates of the complete image (all) and non-occlusion areas (nonocc) for the 4.0 pixel threshold. A: initial disparity estimation, B: disparity plane fitting, and C: disparity plane optimization.

**Figure 13.**Results of representative data on the Middlebury website. From top to bottom: Moebius, Laundry, Bowling 2 and Plastic. (

**a**) Reference image; (

**b**) ground truth images; (

**c**) results of the proposed method; and (

**d**) error maps (bad estimates with absolute disparity error >1.0 are marked in black).

**Figure 14.**Results of synthesized stereo pairs. From top to bottom: Tanks, Temple, and Street. (

**a**) Reference image; (

**b**) ground truth images; (

**c**) results of the proposed method; and (

**d**) error maps (bad estimates with absolute disparity error >1.0 are marked in black).

**Figure 15.**Results of real-world stereo data. (

**a**) Frames of ”Book Arrival” stereo video sequence; (

**b**) estimated disparity maps of ”Book Arrival”; (

**c**) frames of ”Ilkay” stereo video sequence; and (

**d**) estimated disparity maps of ”Ilkay”.

Parameter Name | Purpose | Algorithm Steps | Parameter Value |
---|---|---|---|

Spatial bandwidth hs | Image segmentation | Step (1) | 10 |

Spectral bandwidth hr | 7 | ||

Gamma ${\mathsf{\gamma}}_{\mathrm{INV}}$ | Matching cost computation | Step (2) | 40 |

Gamma ${\mathsf{\gamma}}_{\mathrm{ICT}}$ | 20 | ||

Gamma ${\mathsf{\gamma}}_{\mathrm{TADC}}$ | 40 | ||

Gamma ${\mathsf{\gamma}}_{\mathrm{TADG}}$ | 20 | ||

Threshold ${t}_{consistency}$ | Outliers filter and disparity plane parameters fitting | Step (3) | 1 |

Threshold ${t}_{confidence}$ | 0.04 | ||

Threshold ${t}_{outlier}$ | 1 | ||

Threshold ${t}_{convergence}$ | 10^{−6} | ||

Smoothness penalty ${\mathsf{\lambda}}_{\mathrm{disc}}$ | Smoothness and occlusion penalty | Step (4) | 5 |

Occlusion penalty ${\mathsf{\omega}}_{\mathrm{occ}}$ | 5 |

**Table 2.**Quantitative evaluation based on the training set of the 2014 Middlebury stereo datasets at 2 Error Threshold. The best results for each test column are highlighted in bold. Res and Avg represent resolution scale and average error respectively. Adiron, ArtL, Jadepl, Motor, MotorE, Piano, PianoL, Pipes, Playrm, Playt, PlaytP, Recyc, Shelvs, Teddy, and Vintge are the names of experimental data in the training set.

Name | Res | Avg | Adiron | ArtL | Jadepl | Motor | MotorE | Piano | PianoL | Pipes | Playrm | Playt | PlaytP | Recyc | Shelvs | Teddy | Vintge |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

APAP-Stereo [32] | H | 7.78 | 3.04 | 7.22 | 13.5 | 4.39 | 4.68 | 10.7 | 16.1 | 5.35 | 10.1 | 8.60 | 8.11 | 7.70 | 12.2 | 5.16 | 7.97 |

PMSC [33] | H | 8.35 | 1.46 | 4.44 | 11.2 | 3.68 | 4.07 | 11.9 | 18.2 | 5.25 | 12.6 | 8.03 | 6.89 | 7.58 | 31.6 | 3.77 | 17.9 |

MeshStereoExt [34] | H | 9.51 | 3.53 | 6.76 | 18.1 | 5.30 | 5.88 | 8.80 | 13.8 | 8.10 | 11.1 | 8.87 | 8.33 | 10.5 | 31.2 | 4.96 | 12.2 |

MCCNN_Layout | H | 9.54 | 3.49 | 7.97 | 14.0 | 3.91 | 4.23 | 12.6 | 15.6 | 4.56 | 12.3 | 14.9 | 12.9 | 7.79 | 24.9 | 5.20 | 17.6 |

NTDE [35] | H | 10.1 | 4.54 | 7.00 | 15.7 | 3.97 | 4.37 | 13.3 | 19.3 | 5.12 | 14.4 | 12.1 | 11.7 | 8.35 | 33.5 | 3.75 | 17.8 |

MC-CNN-acrt [36] | H | 10.3 | 3.33 | 8.04 | 16.1 | 3.66 | 3.76 | 12.5 | 18.5 | 4.22 | 14.6 | 15.1 | 13.3 | 6.92 | 30.5 | 4.65 | 24.8 |

LPU | H | 10.4 | 3.17 | 6.83 | 11.5 | 5.8 | 6.35 | 13.5 | 26 | 7.4 | 15.3 | 9.63 | 6.48 | 10.7 | 35.9 | 4.19 | 21.6 |

MC-CNN+RBS [37] | H | 10.9 | 3.85 | 10 | 18.6 | 4.17 | 4.31 | 12.6 | 17.6 | 7.33 | 14.8 | 15.6 | 13.3 | 7.32 | 30.1 | 5.02 | 22.2 |

SGM [26] | F | 22.1 | 28.4 | 6.52 | 20.1 | 13.9 | 11.7 | 19.7 | 33.2 | 15.5 | 30 | 58.3 | 18.5 | 23.8 | 49.5 | 7.38 | 49.9 |

TSGO [38] | F | 31.3 | 27.3 | 12.3 | 53.1 | 23.5 | 25.7 | 33.4 | 54.5 | 22.5 | 49.6 | 45 | 27 | 24.2 | 52.2 | 13.3 | 57.5 |

Our method | Q | 33.9 | 36.5 | 20.6 | 35.7 | 27.6 | 30.5 | 38.8 | 59 | 26.6 | 46.8 | 56.9 | 31.8 | 29.6 | 53.3 | 12.2 | 52.8 |

**Table 3.**Quantitative evaluation based on the test set of the 2014 Middlebury stereo datasets at 2 Error Threshold. The best results for each test column are highlighted in bold. Austr, AustrP, Bicyc2, Class, ClassE, Compu, Crusa, CrusaP, Djemb, DjembL, Hoops, Livgrm, Nkuba, Plants and Stairs are the names of experimental data in the test set.

Name | Res | Avg | Austr | AustrP | Bicyc2 | Class | ClassE | Compu | Crusa | CrusaP | Djemb | DjembL | Hoops | Livgrm | Nkuba | Plants | Stairs |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

PMSC [33] | H | 6.87 | 3.46 | 2.68 | 6.19 | 2.54 | 6.92 | 6.54 | 3.96 | 4.04 | 2.37 | 13.1 | 12.3 | 12.2 | 16.2 | 5.88 | 10.8 |

MeshStereoExt [34] | H | 7.29 | 4.41 | 3.98 | 5.4 | 3.17 | 10 | 8.89 | 4.62 | 4.77 | 3.49 | 12.7 | 12.4 | 10.4 | 14.5 | 7.8 | 8.85 |

APAP-Stereo [32] | H | 7.46 | 5.43 | 4.91 | 5.11 | 5.17 | 21.6 | 9.5 | 4.31 | 4.23 | 3.24 | 14.3 | 9.78 | 7.32 | 13.4 | 6.3 | 8.46 |

NTDE [35] | H | 7.62 | 5.72 | 4.36 | 5.92 | 2.83 | 10.4 | 8.02 | 5.3 | 5.54 | 2.4 | 13.5 | 14.1 | 12.6 | 13.9 | 6.39 | 12.2 |

MC-CNN-acrt [36] | H | 8.29 | 5.59 | 4.55 | 5.96 | 2.83 | 11.4 | 8.44 | 8.32 | 8.89 | 2.71 | 16.3 | 14.1 | 13.2 | 13 | 6.4 | 11.1 |

MC-CNN+RBS [37] | H | 8.62 | 6.05 | 5.16 | 6.24 | 3.27 | 11.1 | 8.91 | 8.87 | 9.83 | 3.21 | 15.1 | 15.9 | 12.8 | 13.5 | 7.04 | 9.99 |

MCCNN_Layout | H | 9.16 | 5.53 | 5.63 | 5.06 | 3.59 | 12.6 | 9.97 | 7.53 | 8.86 | 5.79 | 23 | 13.6 | 15 | 14.7 | 5.85 | 10.4 |

LPU | H | 10.5 | 11.4 | 3.18 | 8.1 | 6.08 | 20.9 | 9.84 | 6.94 | 4 | 4.04 | 33.9 | 16.9 | 15.2 | 17.8 | 9.12 | 11.6 |

SGM [26] | F | 25.3 | 45.1 | 4.33 | 6.87 | 32.2 | 50 | 13 | 48.1 | 18.3 | 7.66 | 29.6 | 36.1 | 31.2 | 24.2 | 24.5 | 50.2 |

Our method | Q | 38.7 | 40.4 | 20.3 | 27.3 | 35.1 | 55.9 | 22.3 | 56.1 | 50.9 | 24.2 | 58 | 56.3 | 36.5 | 32.1 | 38.7 | 69.7 |

TSGO [38] | F | 39.1 | 34.1 | 16.9 | 20 | 43.3 | 55.4 | 14.3 | 54.1 | 49.2 | 33.9 | 66.2 | 45.9 | 39.8 | 42.6 | 47.2 | 52.6 |

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**MDPI and ACS Style**

Ma, N.; Men, Y.; Men, C.; Li, X.
Accurate Dense Stereo Matching Based on Image Segmentation Using an Adaptive Multi-Cost Approach. *Symmetry* **2016**, *8*, 159.
https://doi.org/10.3390/sym8120159

**AMA Style**

Ma N, Men Y, Men C, Li X.
Accurate Dense Stereo Matching Based on Image Segmentation Using an Adaptive Multi-Cost Approach. *Symmetry*. 2016; 8(12):159.
https://doi.org/10.3390/sym8120159

**Chicago/Turabian Style**

Ma, Ning, Yubo Men, Chaoguang Men, and Xiang Li.
2016. "Accurate Dense Stereo Matching Based on Image Segmentation Using an Adaptive Multi-Cost Approach" *Symmetry* 8, no. 12: 159.
https://doi.org/10.3390/sym8120159