菜鳥看論文——Stochastic Occupancy Grids and Dynamic Programming（一）

參考論文：

07.Free Space Computation Using Stochastic Occupancy Grids and Dynamic Programming

08.Stereo-based Free Space Computation in Complex Traffic Scenarios

08.Dynamic Stereo Vision for Intersection Assistance

08.Towards optimal stereo analysis of image sequences

An occupancy grid M is a two-dimensional array or grid which models occupancy evidence of the environment

There are two main types of occupancy grids: deterministic and stochastic grids. Deterministic grids are basically bidimensional histograms counting 3D points. They are usually obtained by projecting the 3D view of measurements onto the road and counting the number of points falling onto the same cell area . The noise properties of the 3D measurements are not explicitly modeled. Grid cells with a large amount of hits are more likely to be occupied as those with few or no points. The choice of the discretization of cells of a determistic grid requires the usual compromise of any sampling process. Small discretization values might avoid the accumulation of points. Large discretization values loses spatial resolution, and therefore, estimation accuracy.

On the other hand, the cells of the stochastic occupancy grids maintain a likelihood or probability of occupancy. Instead of counting points, the stochastic grids define an update function which specifies the operation to perform on every cell, based on the measurement and its noise properties. Stochastic occupancy grids are more expensive to compute but are more informative and do not suffer from discretization effects, as compared to deterministic grids

However, the error of the triangulated 3D points is modelled as a Gaussian distribution. This leads to biased estimates of 3D position [11], and therefore to the wrong estimation of free space.

As mentioned in the introduction, it is common to accumulate all 3D points above ground in a stochastic occupancy grid. Figure 2 shows such a grid obtained for the urban situation considered in the sequel. The origin of the coordinate system is centered in our own vehicle. Our standard stereo camera system has a base line of 30cm and an angle of view of 42 deg. The imagers have VGA resolution.The Occupancy Grid Integration block performs a low-pass filter with the predicted occupancy grid based on the ego-motion of the camera. Finally, the free space is computed using dynamic programming on the polar grid.

 It becomes obvious that the uncertainty of stereo depth measurements increases quadratically

with distance. Therefore, the bicyclist (zoomed out in the left image) at around 60m is highly blurred in the occupancy grid. Free space analysis of those occupancy grids is not very reliable, thus we are looking for strategies to reduce the uncertainty.

One way to reduce the disparity noise is the tracking of features in the images over multiple frames. If the disparity measurements are uncorrelated, the variance decreases with 1/N, if N is the number of images. The 6D-Vision algorithm described below exploits this fact. 

Fortunately, tracking becomes redundant in static scenes when the ego-motion of the camera is known a priori. This is beneficial since it allows working with dense stereo disparity maps despite the real-time constraint. Disparity measurements which are consistent over time are considered as belonging to the same world point, and therefore, disparity variance is reduced accordingly. 

Figure 2 shows an example of the improvement achieved. The occupancy grid shown at the right was computed with an integrated disparity image. Note the significantly reduced uncertainties of the registered 3D points. A bicyclist at approximately 60 meters away is marked in the images.

Figure 5(a) shows the procedure so far. By observing Figure 5(a) carefully, it can be seen that the solution forms a path from left to right segmenting transversely the polar grid in two regions.Instead of thresholding each column as usually done [10] [8], dynamic programming is used. The new method based on dynamic programming has the following properties:

– Global optimization: every row is not considered independently, but as part of a global optimization problem that is optimally solved.

– Spatial and temporal smoothness of the solution: the spatial smoothness is imposed by the use of a cost that penalizes jumps in depth while temporal smoothness is imposed by a cost that penalizes the deviation of the current solution from a prediction.

– Preservation of spatial and temporal discontinuities: the saturation of the spatial and temporal costs allows the preservation of discontinuities.

Dynamic programming is applied to the grid in order to segment the image into two regions. For computation of the optimal path, a graph G(V,E) is generated.V is the set of vertices, and contains one vertex for every cell in the grid. E is the set of edges which connect every vertex of one column, with every vertex of the following column. Every edge has an associated value which defines the cost of segmenting the image through the connected vertices. The objective is to find the minimal path using dynamic programming. The cost of each edge is composed of a data and a smoothness term, i.e.;

    (10)

 is the cost of the edge connecting the vertices Vi j and Vk l where;

     (11)

 is the data term defined by the inverse likelihood of the cell and If D(i, j) = 0, then a very large value is assigned to Ed(i, j) and;

    (12)

is a smoothness term containing a spatial and a temporal part. The spatial term penalizes jumps in depth and is defined as:

    (13)

The function d(j, l) returns the distance in meters between cells in rows j and l of the grid. The constant Cs is a cost parameter penalizing jumps in depth, and the threshold Ts (also measured in meters) saturates the cost function, allowing the preservation of depth discontinuities.

The temporal term of Equation 12 has the same form, i.e.;

    (14)

 where Ct is the cost parameter, Tt is the maximal distance for the saturation, and j0 is the prediction obtained by applying ego-motion to the segmentation result of the previous cycle.

 

 07.A Global Optimization Algorithm for Real-Time On-Board Stereo Obstacle Detection Systems

The DP calculation is done from the right most column to the left most column using the following recursive equations,

 where mi(di) is the matching score for node (i, di) (i.e. the node that represent i th column with disparity value di) ,ci(di, di.1) is the cost of the path from node (i.1, di.1) to node (i, di), and Mi(di) is the best score up to node (i, di).Then, the best path is given as a sequence of disparity values d1, · · · , d W using the following recursive equations,

  Any obstacle area in the left image has a half-occluded area on its left and points inside the half-occluded area are invisible in the right image (see area A and A’ in Fig.9). This poses the following two requirements on the optimization procedure.

—The road-obstacle boundary must not go inside halfoccluded areas (which can be interpreted as the ordering constraint[3]), since the road-obstacle boundary should be visible in the both images.

—Half-occluded areas have to be excluded from the matching score calculations, since they do not have corresponding areas. 

The above requirements restrict paths on DSI: no partial path can have slope any steeper than 45 degree toward upward-left and the partial path whose slope is 45 degree is considered to be a boundary between road area and half-occluded area. This constraint can be realized by the following matching score mi(di) and path cost ci(di, di-1):

 The optimum path on the DSI is converted to the road obstacle boundary in the left image using (2). The partial paths whose disparity values are smaller than a threshold or those which are considered to be boundary between road area and half-occluded areas are eliminated from the result.Fig.10 shows an example of a DSI and the estimated roadobstacle boundary.

The algorithm described in the paper is based on gradient directions in the left and right images of a rectified stereo image pair.