# Not Seeing is Also Believing: Combining Object and Metric Spatial Information

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3 object pose) and an occupancy grid (for metric occupancy) to make filtering efficient. Specifically, we propose to maintain the following distributions in two filters: P(x z ) and P(m w ), (2) and only incorporate the other source of information at query time. Computing the posteriors in Eqn. 1 from the filtered distributions in Eqn. 2 is the subject of the next section. IV. THE ONE-DIMENSIONAL, SINGLE-OBJECT CASE To ground our discussion of the solution strategy, in this section we consider a simple instance of the general problem discussed in the previous section. In particular, we focus on the case of estimating the (discrete) location of a single static object and the occupancy of grid cells in a discretized one-dimensional world. The general problem involving more objects and other attributes is addressed in Sec. V. A. Formulation The single-object, 1-D instance is defined as follows: The 1-D world consists of C contiguous, unit-width cells with indices 1 j C. A static object of interest, with known length L, exists in the world. Its location, the lowest cell index it occupies, is the only attribute being estimated. Hence its state x satisfies x [1, C L + 1] {1,..., C L + 1}. We are also interested in estimating the occupancy of each cell cell j. Each cell s state m j is binary, with value 1 if it is occupied and 0 if it is free. Cells may be occupied by the object, occupied by dirt / stuff, or be free. Stuff refers to physically-existing entities that we either cannot yet or choose not to identify. Imagine only seeing the tip of a handle (which makes the object difficult to identify) or, as the name suggests, a ball of dirt (which we choose to ignore except note its presence). We will not explicitly distinguish between the two types of occupancy; the cell s state has value 1 if it is occupied by either the object or stuff, and 0 if it is free. The assumption above, that cells can be occupied by nonobject entities, allows us to ascribe a simple prior model of occupancy: each cell is occupied independently with known probability P(m j = 1) = ψ. This prior model and cell independence assumption are commonly used in the occupancy grid literature (see, e.g., [16]). This may be inaccurate, especially if the object is long and ψ is small. Noisy observations z of the single object s location x and observations w of the cells occupancies m are made. We will be intentionally agnostic to the specific sensor model used, and only assume that appropriate filters are used in light of the noise models. The object and metric filters maintain P(x z ) and P(m w ) respectively. We assume that the former is a discrete distribution over the domain of x, and the latter is an occupancy grid, using the standard log-odds ratio l j = log P(mj =1 w j ) for each cell s occupancy. P(m j =0 w ) j States of distinct cells are assumed to be conditionally independent given the object state x. This is a relaxation of the assumption cells are independent, which is typically assumed in occupancy grids. The current assumption disallows arbitrary dependencies between cells; only dependencies mediated by objects are allowed. For example, two adjacent cells may be occupied by the same object and hence are dependent if the object s location is not known. As mentioned in the previous section, what makes this problem interesting is that x and m are dependent. In this case, the crucial link is that an object that is located at x necessarily occupies cells with indices j J (x) [x, x + L 1], and therefore these cells must have as state m j = 1. This means that states of a subset of cells are strongly dependent on the object state, and we expect this to appear in the metric posterior P(m z, w ). Likewise, occupancy/freeness of a cell also supports/opposes respectively the hypothesis that an object overlaps the cell. However, the latter dependency is weaker than the former one, as an occupied cell can be due to dirt (or other objects, though not in this case), and a free cell typically only eliminates a small portion of the object location hypotheses. B. Cell occupancy posterior We now use this link between x and m to derive the desired posterior distributions from Eqn. 1. We first consider the posterior occupancy m j of a single cell cell j. Intuitively, we expect that if the object likely overlaps cell j, the posterior occupancy should be close to 1, whereas if the object is unlikely to overlap the cell, then the posterior occupancy should be dictated by the stuff prior and relevant occupancy observations (w j ). Since we do not know the exact location of the object, we instead have to consider all possibilities: P(m j z, w ) = x P(m j x, w j ) P(x z, w ). (3) In the first term, because x is now explicitly considered, object observations z are no longer informative and are dropped. Since we assumed that cells are conditionally independent given the object state, all other cells observations are dropped too. The second term is the posterior distribution on the object location, which will be discussed later. The term P(m j x, w j ) can be decomposed further: P(m j x, w j ) P(w j m j ) P(m j x) (4) The second term, P(m j x), serves as the link between cells and objects. By the discussion above, for j J (x), i.e., cells that the object at location x overlaps, m j must be 1. In this case, Eqn. 4 is only non-zero for m j = 1, so P(m j = 1 x, w ) j must also be 1. For j / J (x), the cell is unaffected by the object, hence P(m j x) = P(m j ). Eqn. 4 in this case is, by reverse application of Bayes rule, proportional to P(m j w ), j and since this is in fact a distribution, P(m j x, w ) j = P(m j w ). j This reflects that for j / J (x), the cell s state is independent of the object state. In summary: { P(m j x, w ) j 1 if j J (x), = P(m j w ) j (5) otherwise.

6 VI. APPLICATIONS In this section, we will look at several scenarios where object-based and metric-based information need to be considered together. First, we will introduce additional attributes (besides location and occupancy from Sec. IV). A. Shape-based object identification When detecting and tracking objects in the world, uncertainty typically arises in more attributes than just location/pose. In particular, object recognition algorithms are prone to confusing object types, especially if we only have a limited view of the object of interest. When multiple instances of the same object type are present, we also run into data association issues. Furthermore, we may even be unsure about the number of objects in existence. Sophisticated filters (e.g., [10], [11], [12], [13]) can maintain distributions over hypotheses of the world, where a hypothesis in our context is a assignment to the joint state x. Let us revisit the one-dimensional model of Sec. IV again, this time with uncertainty in the object type. In particular, the single object s length L is unknown, and is treated as an attribute in x (in addition to the object s location). Suppose that after making some observations of the object, we get P(x z ) from the filter, as shown in Fig. 5(a) (top). The filter has identified two possible lengths of the object (L = 3, 5). Here we visualize the two-dimensional distribution as a stacked histogram, where the bottom bars (red) shows the location distribution for L = 3, and the top bars (black) for L = 5. The total height of the bars is the marginal distribution of the object s location. Suppose we have also observed that cell 7 is most likely empty, and cell 8 most likely occupied (the occupancy grid gives probability of occupancy 0.01 and 0.99 for the two cells respectively). The posterior distributions obtained by combining the filters distributions is shown on in Fig. 5(b). In the object-state posterior, the main difference is that the probability mass has shifted away from the L = 5 object type, and towards locations on the left. Both effects are caused by the free space observations of cell 7. Because all locations for L = 5 states cause the object to overlap cell 7, implying that cell 7 is occupied, the observations that suggest otherwise cause the marginal probability of L = 5 to drop from 0.50 to The drop in probability for locations 5 and 6 is due to the same reason. In conclusion, incorporating occupancy information has allowed us to reduce the uncertainty in both object location and object type (length). Interestingly, among the L = 5 states, although location 3 had the highest probability from the object filter, it has the lowest posterior probability mass. This minor effect comes from the likely occupancy of cell 8, which lends more evidence to the other L = 5 states (which overlap cell 8 ) but not for location 3 (which does not overlap). However, the strong evidence of cell 8 s occupancy has much less of an effect compared to the free space evidence of cell 7. This example highlights the fundamental asymmetry in occupancy information. Recall that the prior model allows for unidentified, non-object stuff to exist in the world, stochastically (a) Filter distributions (input) (b) Posterior distributions (output) Fig. 5. A 1-D scenario with a single object, but now the object s length is uncertain as well. The object filter (top left) determines that the object may have length L = 3 or 5, and for either case, may be in one of several locations. Because of a strong free space occupancy observation in cell 7, the uncertainty in object length has decreased significantly in the posterior object distribution (top right), because a L = 5 object must contradict the free space evidence of cell 7. Please see text in Sec. VI-A for more details. with probability ψ. That cell 8 is occupied only suggests it is overlapped by some object, or contains stuff. In particular, this is the interpretation for cell 8 for the two most likely L = 3 states in the posterior. An object overlapping the cell would gain evidence, but the cell s occupancy does not need to be explained by an object. In contrast, cell 7 being free means that none of the objects can overlap it, thereby enforcing a strong constraint on each object s state. In the example shown in Fig. 5, this constraint allowed us to identify that the object is most likely of shorter length. B. Physical non-interpenetration constraints When multiple objects are present in the world, a new physical constraint appears: objects cannot interpenetrate each other ([17]). For example, in the 1-D scenario, this means that for any pair of blocks, the one on the right must have location x r x l + L l, where x l and L l is the location and length of the left block respectively. This is a constraint in the joint state space that couples together all object location/pose variables. One possible solution is to build in the constraint into the domain of x by explicitly disallowing joint states that violate this constraint. Although this is theoretically correct, it forces filtering to be done in the intractable joint space of all object poses, since there is in general no exact way to factor the constraint. We now consider an alternate way to handle object noninterpenetration that is made possible by considering metric cell occupancies. So far, we have only distinguished between cells being occupied or free, but in the former case there is no indication as to what occupies the cell. In particular, the model so far allows interpenetration because two objects can occupy the same cell, and the cell state being occupied is still consistent. To disallow this, we consider expanding the occupancy attribute for grid cells. We propose splitting the previous occupied value (m j = 1) into separate values, one for each object index, and one additional value for stuff /unknown. That is, the cell not only indicates that it is occupied, but also which object is occupying it (if known). Then in the object-metric link P(m j x ), if, for example, obj 2 overlaps cell j, m j is enforced to have value 2. The noninterpenetration constraint naturally emerges, since if obj 1 and obj 2 interpenetrate, they must overlap in some cell cell j,

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