A Probabilistic Representation of LiDAR Range Data for Efficient 3D ObjectDetection

Theodore C. Yapo1∗, Charles V. Stewart2, and Richard J. Radke1

1Department of Electrical, Computer, and Systems Engineering2Department of Computer Science

Rensselaer Polytechnic Institute, Troy, New York 12180yapot@rpi.edu, stewart@cs.rpi.edu, rjradke@ecse.rpi.edu

Abstract

We present a novel approach to 3D object detection inscenes scanned by LiDAR sensors, based on a probabilisticrepresentation of free, occupied, and hidden space that ex-tends the concept of occupancy grids from robot mappingalgorithms. This scene representation naturally handles Li-DAR sampling issues, can be used to fuse multiple LiDARdata sets, and captures the inherent uncertainty of the datadue to occlusions and clutter. Using this model, we for-mulate a hypothesis testing methodology to determine theprobability that given 3D objects are present in the scene.By propagating uncertainty in the original sample points,we are able to measure confidence in the detection resultsin a principled way. We demonstrate the approach in ex-amples of detecting objects that are partially occluded byscene clutter such as camouflage netting.

1. IntroductionLight Detection and Ranging (LiDAR) scanners use

time-of-flight measurements of narrow beams of laser lightto produce estimates of the locations of 3D points in a scene.The resolution of commercially available LiDAR scannerscan be very good, achieving an accuracy of a few mm at100m range [13]. However, unlike digital image sensorsthat use an optical low-pass filter to prevent the aliasing ofhigh spatial frequencies in the scene, LiDAR sensors arevery susceptible to sampling artifacts, as illustrated in Fig-ure 1. For example, if the samples are too far apart, a Li-DAR scan of a picket fence might be interpreted as a solidwall. Conversely, if a solid wall is sampled at a shallowgrazing angle by nearly parallel LiDAR rays, it can be dif-ficult to connect the distant sample points into a single sur-

∗This work was supported in part by the US Army Intelligence and Se-curity Command under the award W9124Q-04-F-2159, and by the DARPAComputer Science Study Group under the award HR0011-07-1-0016.

face. Hence, even though each range point is measured withhigh accuracy, there can still be quite a bit of uncertaintyabout the scene in each LiDAR scan.

Occlusions in the scene introduce a second source of un-certainty into LiDAR range data. Objects may be whollyor partially hidden from the point of view of the scanner,resulting in uncertainty about their presence or position inthe scene. To deal with this issue effectively, a 3D objectdetection algorithm must allow fusion of data taken fromdifferent viewpoints, and model occlusion explicitly, notingwhat parts of the scene are visible from each viewpoint.

Much previous research on analyzing LiDAR data isbased on generating a 3D model of the scene, either reduc-ing the data to a polygonal model, or in some cases, pro-ducing an implicit function representation of the scene sur-faces. Instead of irrevocably collapsing information aboutthe scene into a likely “crisp” estimate, we propose to pre-serve the inherent uncertainty of the original data whentesting hypotheses against the scene using a probabilisticframework.

We propose a discrete scene data structure to maintain aprobabilistic model of the 3D scene, and provide a naturaland tractable means to update this model that properly han-dles LiDAR sampling issues. The scene data structure isfundamentally a site occupancy probability model, extend-ing the concept of occupancy grids from robotics [15]. Weapproximate the scene by a set of random fields that de-scribe the probabilities that any single site (3D voxel) is inone of three states: free space, occupied, or hidden. This ap-proach provides a sound basis for fusing data from disparatesensors that observe the scene from different viewpoints.

While we believe that the precision of available LiDARsensors far surpasses that required for reliable object detec-tion (since most objects of interest in outdoor scenes arevery large relative to the uncertainty of a single LiDAR re-turn), we cannot scan the scene with fewer LiDAR pointswithout exacerbating the undersampling and aliasing prob-

(a)

(b)

(c)

Figure 1. Illustrating sources of uncertainty in LiDAR data. The black dot represents the LiDAR sensor, the black lines LiDAR rays, thegray lines object surfaces, and the white dots LiDAR returns. LiDAR is susceptible to (a) undersampling of objects with holes, (b) poorsampling of surfaces due to shallow grazing angle, and (c) unsampled space resulting from occluding surfaces.

lems. To address these issues, we propose a downsamplingtechnique to reduce data storage requirements and allow ef-ficient hypothesis testing, while not glossing over importantLiDAR sampling issues.

Our initial results are based on range data alone, but wenote that our proposed data structure allows for the fusionof data from sensors of different modalities, such as digitalimages. Using our probabilistic model, 3D inferences fromstereo pairs or single images could be used to “fill in” re-gions that are ambiguous or occluded from the viewpointof the LiDAR scanner. Since image sensors typically havemuch higher resolution than their LiDAR counterparts, thiscombination could also be used to reduce uncertainty aris-ing from LiDAR undersampling in the scene.

Finally, we describe how 3D hypothesis tests relatedto object detection can be easily performed on the datastructure, using an efficient method based on the 3D FastFourier Transform. We demonstrate our results on an ex-ample where objects are hidden behind a camouflage netthat makes the detection problem in raw data difficult.

2. Related Work

Some of the earliest models for representing range dataused an octree with each leaf representing one of three un-ambiguous states: empty, occupied, or unseen [4]. Matthiesand Elfes [15] initially proposed maintaining probabilitiesin a 2D structure called an occupancy grid for mapping thelocal environment of an autonomous robot. In the orig-inal approach, each cell in the grid was assigned a sin-gle value, P occ , representing the probability that the sitewas occupied; initial values for each cell were set to 1

2and the probabilities were updated using a Bayesian rule[16]. Subsequent methods [8, 19] assigned two evidencemasses, one for empty (me) and one for full (mf ) such thatme + mf ≤ 1 to capture some notion of uncertainty, andapplied Dempster-Shafer theory [21] to update the grid.

Most approaches have treated the collection of cells asa Markov Random Field (MRF) of order 0 [5], i.e., assum-ing that the occupancy probabilities of neighboring cells areindependent, greatly simplifying the occupancy probability

estimation procedure. In contrast, Thrun [22] used a for-ward model of the sensor and an expectation-maximizationalgorithm to explicitly model the conditional dependenciesof cells within a sensor sweep area. Jang et al. [10] clus-tered cells into sets using neural networks and modeled theconditional probabilities within these clusters. Ribo andPinz [20] compared three types of models for updating thegrid: probabilistic (using the Bayes update rule), Dempster-Shafer (using the Dempster rule of combination), and “pos-sibilistic” (using fuzzy logic). Gorodnichy and Armstrong[7, 8] formulated the update rule as a regression. Matthiesand Elfes [15] illustrated the importance of an update rulethat is both commutative and associative, allowing data tobe fused in any order.

Although much of the previous work on occupancy gridshas been limited to 2D representations, several researchershave extended the notion to 3D scenes. One of the mostimportant issues is avoiding a computationally intractableincrease in the number of cells. Payeur et al. [19] describedan octree representation in which spherical occupancy gridsare computed around each sensor to accurately model thesensor characteristics; these are then merged into a commonCartesian grid. Several schemes for storing functional rep-resentations of occupancy fields have also been proposed.Payeur [18] described one such method in which a con-tinuous spatial potential force is computed from the occu-pancy probabilities of each cell, allowing a continuous es-timation of the field. Gorodnichy and Armstrong [8] pro-posed a parametric representation using a min-max tree ofpiecewise linear functions. Yguel et al. [24] proposed a 2Dwavelet-based occupancy field, exploiting the continuousand natural multi-resolution properties of the wavelet de-composition. Paskin and Thrun [17] proposed using polyg-onal random fields for probabilistic mapping of 2D scenes,avoiding discretization errors and independence assump-tions associated with occupancy grids. However, we notethat most of these representations do not easily lend them-selves to object recognition.

Once the data has been fused into a single model, hy-potheses (e.g., related to the presence of objects) can betested against it. One class of basic hypotheses concerns the

detection or recognition of known 3D object models in thescene. Campbell and Flynn [3] give a good overview of thebasic approaches to 3D object recognition. Among these,the most relevant to scenes containing heavy clutter and in-tentional occlusions (camouflage) are “point-feature” basedmethods. In these algorithms, a vector descriptor is calcu-lated for a subset of points in the scene. These descriptorsare then matched with those calculated from object modelsto detect and recognize objects in the scene. These methodscan be robust to scene clutter and occlusions provided thatinitial local surface normals can be estimated.

Johnson and Hebert [11, 12] devised a point-featurebased method known as “spin images”. In this approach,a local surface normal is first estimated for each point in thescene to be tested. A 2D histogram is then rotated aroundthis normal, creating a “spin image” representative of thelocal surface features of the object. The resulting imageserves as a point descriptor which can then be matched withdescriptors from object models.

Frome et al. [6] proposed two point-feature techniquesand compared their performance to spin images. The firstmethod extended the concept of shape contexts (Belongie etal. [1]) into 3D. In this method, a local surface normal is firstestimated for each feature point. A spherical histogram ofneighboring points is then calculated as the feature descrip-tor. A spherical harmonic transformation is then applied tothis histogram to create a “harmonic shape context”. Theirresults indicate better recognition rates than spin images.

Several researchers have applied these techniques tomore real-world recognition problems. Vasile and Marino[23] developed an automatic target recognition algorithmfor twelve vehicle classes in relatively-unoccluded LiDARdata based on spin images [12]. Huber et al. [9] described aparts-based classification algorithm for vehicles, applied toisolated recognition in the absence of clutter.

In these point-feature approaches, a local surface rep-resentation (either polygonal meshes or oriented points) isfirst estimated from the range points, discarding much of theuncertainty in the original data. For example, to generateand match spin images, one must accurately estimate sur-face normals at each 3D scene point to define spin axes. Incontrast, our method assumes no a priori model of the sceneor object structure, and was designed for datasets wheremillions of range points acquired from sensors at many dif-ferent viewpoints are fused into a common frame. Real out-door examples of this type of data include “fuzzy” regionslike trees, large hidden regions, and sparsely-sampled areasarising from rays nearly parallel to object surfaces. Gen-erating reliable surface and normal estimates in such caseswould be difficult. However, as we describe below, thesecases are naturally handled by occupancy modeling.

Our proposed object detection method uses FFT-basedcorrelation to efficiently compute joint probabilities of in-

dependent voxels. In this way, it is similar to FFT-basedregistration methods such as that presented by Luccheseet al. [14], which uses a correlation technique to estimatetranslational parameters in range data registration once ro-tational parameters have been estimated.

3. Probabilistic Occupancy ModelWe assume the viewpoint for each LiDAR return is

known; this is used to generate a model representing freeand hidden space as well as regions occupied by solid ob-jects. However, we do not require that the range points havea particular structure, such as lying on an angular grid; infact, each range point can be taken from a different (known)viewpoint. Since each LiDAR point is considered indepen-dently, we treat estimation of occupancy probabilities as asensor fusion problem even for data points obtained froma single sensor at a fixed position. This approach naturallyscales to handle multiple LiDAR data sets that can be regis-tered into a common world coordinate system.

Figure 2 illustrates our representation of both the scenespace and objects that may be located within it. The scenespace ideally comprises three disjoint sets: (1) the occupiedpoints, which have been seen by one or more LiDAR sen-sors, (2) the hidden points, which have been obscured fromall sensors, and (3) the free-space points, which have been“seen through” by at least one sensor. Also shown in Fig-ure 2 is our representation of objects to be detected in thescene, as will be described in Section 4. Again, we considerthree disjoint sets of points: (1) points on the object sur-face, which could possibly be seen by a LiDAR sensor, (2)points on the object’s topological interior, which are alwaysself-occluded by the object, and hence can never be seen,and (3) points exterior to the object, which may either beunoccupied, or contain part of a different object.

Using these models of the scene and object spaces, wecan list the constraints that must be satisfied for an object toexist in the scene, as summarized in Table 1. The object’ssurface points must coincide with occupied scene points,and its interior points must coincide with hidden points.Conversely, points in an object’s surface or interior cannotcoincide with scene space known to be empty. Hidden spacepoints may or may not be consistent with an object’s surfacebeing present; we address this issue in Section 4. We finallynote that scene points exterior to the object have no bearingon detection.

Our model for spatial occupancy of the scene is a set ofrandom fields, in which we assign probabilities of being ineach state independently to points in the scene, extendingthe ideas of [5]. For any point x in the scene space, weconsider the probabilities that x is in one of the three states:occupied, hidden, or free, as illustrated in Figure 2:

P occ(x) + P hid(x) + P free(x) = 1. (1)

FreeSpa eHiddenSpa eO upiedPointsviewpoint

Obje tSurfa e Obje tInterior Obje tExterior(b)(a)Figure 2. Six spaces of interest for constructing probability density functions of scene occupancy. (a) Representation of the scene. Eachpoint in space is assumed to be in one of three states: occupied, hidden, or free. (b) Representation of an object. The object model isrepresented by three sets of points: surface, topological interior, and exterior.

nhid++nhid++nhid++nocc++nfree++nfree++nfree++nfree++nfree++

Figure 3. A single LiDAR return provides information about scene structure at every point along the view ray.

Scene SpaceObject Space occupied hidden free

surface√

? ×interior ×

√×

exterior - - -Table 1. Constraints imposed by scene and object space states.“√

” indicates a required coincidence, “×” indicates mutual ex-clusivity, “?” indicates possible consistency, and “-” indicates noinformation gain.

We note the distinction between this model and previ-ous work, which has either modeled site occupancy witha single probability [15] such that P occ + P free = 1, orwith two evidence masses such that mocc + mfree ≤ 1[8]. In our model, we consider the “hidden” state to be adistinct third possibility, not simply a lack of knowledgeabout the occupied or free state of the point, thereby ac-counting for points that are inside solid objects. In doingso, we allow a resolution to the problem of disambiguatingunknown states from conflicting evidence [8]. For exam-ple, in single-probability models [15], unseen areas of spaceare assigned the noninformative prior P occ = P free = 1

2 ,but there is no way to distinguish this from a completelyobserved site where equal, but conflicting, evidence existsfor either of the two states. Although Dempster-Shaferapproaches [16] resolve this ambiguity by setting initialmasses to mocc = mfree = 0, they do not capture theimportant representation of hidden or occluded space. Bymaking the third state explicit, we maintain true probabili-ties at each point, and can test hypotheses that the randomfield model is in a state consistent with the presence of anobject in a principled way. Moreover, by maintaining a no-tion of hidden space, we enable more complex hypothesesinvolving the interior space of objects, as well as their sur-

faces, to be tested. These tests are described in Section 4.For each voxel in the grid, we wish to estimate the three

probabilities, P occ , P free , and P hid . Since we assume theaccuracy of each LiDAR point is much greater than the res-olution of the voxel grid, we consider only the number ofLiDAR returns that fall into each voxel for occupancy prob-ability calculation. Likewise, for estimating P free and P hid ,we consider the number of view rays that pass through thevoxel, or terminate in a LiDAR return before reaching it,respectively.

To efficiently accumulate these statistics, view rays aretracked through the grid using a variation of Bresenham’sline drawing algorithm [2]. Each voxel is augmented withthree counters: nfree , nocc , and nhid . Starting from thescanner position, each voxel in the path of the ray encoun-tered before the LiDAR return has nfree incremented. Forthe voxel containing the LiDAR return, nocc is incremented.The ray path is then continued until it leaves the grid, witheach subsequent voxel having nhid incremented. This pro-cess is illustrated in Figure 3.

Once all view rays have been traced through the grid, thethree probabilities are estimated for each voxel. In doing so,we want completely unexplored voxels to be assigned thenoninformative prior probabilities P free = P occ = P hid =13 . As the number of points accumulated in the voxel in-creases, the collected statistics should begin to dominate thea priori estimates. This behavior would naturally be pro-vided by a Bayesian update rule. However, in our case wehave neither an accurate noise model for the scanner charac-teristics nor any knowledge about the scene structure withina voxel, so estimating the conditional probabilities requiredfor a Bayesian rule is problematic. Instead, we estimate thevoxel occupancy probability by:

P occ =13αn + (1− αn)

(nocc

n

), (2)

for the given voxel, where n = nfree + nocc + nhid . Theparameter α causes the influence of the a priori estimatesto diminish exponentially with each additional ray travers-ing the voxel. The remaining two probabilities for eachvoxel, P hid and P free , are estimated similarly using nhid

and nfree , respectively.

4. Hypothesis TestingOnce the spatial probabilities at each voxel have been

estimated, we can test the hypothesis that an object modelas illustrated in Figure 2 is present in the scene, centered ona specific voxel. Instead of testing this hypothesis directly,we test the surrogate hypothesis that the random field modelof the scene is in a state consistent with the presence of theobject. The difference is subtle, as described below.

The object model comprises two sets of points, the set ofpoints on the object surface:

S = {xSi , i = 1, . . . , N}, (3)

and the set of points topologically interior to the object:

I = {xIj , j = 1, . . . ,M}, (4)

where x represents a point location in the object’s coordi-nate system. We denote Sx0 as the set of surface pointstranslated so that their center is at point x0:

Sx0 = {xSi + x0, i = 1, . . . , N}. (5)

Ix0 is defined similarly.To determine the probability that a single voxel in the

scene space is consistent with the presence of an object’ssurface, we could evaluate the probability that the point isoccupied, P occ(x), alone. However, doing so neglects thepossible states of hidden points, which may either be hiddenbecause they are truly part of an object’s topological inte-rior, or may simply have been occluded from all viewpointsin the LiDAR dataset. To account for this possibility, weapply a noninformative prior, assuming that hidden pointscan be either part of a surface or not with equal probabil-ity, since detection of intentionally hidden objects is par-ticularly important in our application. We define the newprobabilities that the voxel is consistent with the surface ofan object or not:

P surf (x) = P occ(x) +P hid(x)

2(6)

P¬surf (x) = P free(x) +P hid(x)

2(7)

We note that these adjusted probabilities still obeyP surf (x) + P¬surf (x) = 1. We take P surf (x) to repre-sent the probability that an object surface point is present atx.

We now consider the probability P (Sx0) that the entireobject surface S exists in the scene with the object origin atlocation x0. To do so, we calculate the probability that therandom field model of the scene space is in a state consistentwith this hypothesis:

P (Sx0) =N∏

i=1

P surf (xSi + x0). (8)

We similarly define P (Ix0), the probability that the ob-ject interior I is present in the scene centered at x0. Sincethe salient characteristic of object interior points is that theyare always hidden from view (occluded by the object itself),we obtain:

P (Ix0) =M∏

j=1

P hid(xIj + x0) (9)

for the probability that the random field model is in a stateconsistent with the interior of the object being located at x0.

Our ultimate goal is to express the probability P obj(x0)that the object is present in the scene at location x0. For thisto be the case, the random field must be in a state consistentwith both the surface and the interior of the object beingpresent centered at x0:

P obj(x0) = P (Sx0)P (Ix0)

=N∏

i=1

P surf (xSi + x0)

M∏j=1

P hid(xIj + x0).

(10)

Since we have modeled spatial occupancy with a ran-dom field for which occupancy probabilities are indepen-dent, calculation of these probabilities is straightforward.To facilitate efficient calculation and avoid numerical is-sues, we transform the product of (10) to a sum by con-sidering the log likelihood:

lnP obj(x0) =N∑

i=1

lnP surf (xSi + x0)

+M∑

j=1

lnP hid(xIj + x0). (11)

The sums of (11) can more easily be interpreted by achange of notation, in which we explicitly index voxels andobject models by a triple index:

P obji,j,k = exp

(N1∑l=1

N2∑m=1

N3∑n=1

Sl,m,n lnP surfi+l,j+m,k+n

+M1∑l=1

M2∑m=1

M3∑n=1

Il,m,n lnP hidi+l,j+m,k+n

)= exp

(S ? lnP surf + I ? lnP hid

), (12)

where P obji,j,k is the probability that the random field model is

in a state consistent with the object being centered at voxel(i, j, k), Si,j,k is the discretized object surface model, Ii,j,k

is the discretized object interior model, Pi,j,k is the appro-priate discretized spatial probability, and ? denotes cross-correlation. In this form, (12) is easily recognized as thesum of two 3D cross correlations of the object surface andinterior models with the log probabilities of surface and hid-den states of the random field model, respectively.

Calculation of the full cross correlations of (12) can beachieved in O(N3 lg N) time for a 3D grid of N voxels perside using a Fast Fourier Transform (FFT) algorithm. Sincethis computation simultaneously calculates the probabilityof the object existing in the scene centered at each of the N3

voxel sites, the cost per hypothesis test is O(lg N). In prac-tice, further savings can be realized when testing multipleobjects against the same scene, since the computationallyintensive step of transforming the log spatial probabilitiesinto the Fourier domain only needs to be done once.

Moreover, the computation required for hypothesis test-ing is not a function of the number of LiDAR returns orscans used; this only affects the pre-processing time, whichis linear in the number of points. Once the scene proba-bilities are constructed, the object detection time is not afunction of the number of points used to estimate the spatialprobabilities, nor is it a function of object complexity; theonly parameter affecting the detection time is the number ofvoxels in the grid.

5. Results and DiscussionThe proposed method was applied to an example scene

consisting of a cylinder of 0.26m radius and a box of 0.41mper side obscured behind camouflage netting, as shown inFigure 4a. The scene was scanned with a Leica HDS Li-DAR scanner that collected 130,299 range points with anapproximate depth accuracy of 2mm. We present the pro-cessing of two horizontal slices of the full 3D scan to fa-cilitate visualization. As shown in Figure 4b, one slice in-tersects the cylinder, while the second slice intersects thebox; the cross-sections of the objects become a circle and asquare, respectively. A slice consists of 400 × 400 voxels,each measuring 12mm on a side.

The parameter α used in estimating the voxel probabili-ties of (2) controls the rate at which the a priori probabilitiesdecay as LiDAR rays are added to the scene. In practice,we found a value of α = 0.9 to work well, which was usedin obtaining the results of Figure 4. The mean number ofLiDAR points in each non-empty voxel was 3.2, while themaximum number in any voxel was 26.

Figure 4c illustrates a map of the random field model ofthe first scene slice, with P free , P occ , and P hid mappedto the red, green, and blue channels, respectively. Hence,completely unseen voxels which have equal a priori prob-

abilities appear as a mid gray. Figure 4d illustrates similarmaps for the second slice containing the box. As can beseen in the figures, the camouflage netting occludes mostof each object from the LiDAR viewpoint (i.e., virtually allpoints behind the net have P hid ≈ 1).

We formed object models consisting of the cylinder at itstrue size and two boxes smaller than the true size, as illus-trated in Figure 4e. Testing the cylinder model against thevoxel slice intersecting the cylinder produces the detectionprobability map of Figure 4f. In this figure, the true cen-ter of the cylinder is clearly identified, with much smallerprobability that the cylinder is centered on nearby voxels,and nearly zero chance that a cylinder is present elsewherein the scene.

In Figure 4g, the same test is repeated for the case of the0.38m box model. In this case, we see that since the boxmodel is slightly smaller than the actual box in the scene,there is some correctly-estimated ambiguity about wherethe box could exist in the scene. The smaller box model“slides” within the boundaries of the scanned box’s frontedge, since the sides of the box are almost entirely occludedfrom the single LiDAR viewpoint. These occlusions pro-duce a natural uncertainty in the result, which could be re-solved by scanning the scene from an additional viewpoint.This effect is even more pronounced when the smaller boxmodel (0.19m on a side) is tested against the same scene(Figure 4h). In this case, the random field model is con-sistent with placement of the smaller box anywhere along ashort line of positions, with the greatest probability near themiddle of the front box surface.

We note that it would be difficult to directly comparethese results against spin images or 3D shape contexts sincewe do not assume we know or can find local surface nor-mals, which are required for either method. In many scenesof interest, where objects are intentionally hidden, for ex-ample, by camouflage netting, estimating reliable surfacenormals will be problematic. Additionally, our data struc-ture has been designed so that images can be fused with Li-DAR range scans to fill in occluded regions or compensatefor LiDAR undersampling. Our detection algorithm couldbe directly applied to such fused data. However, spin im-ages and 3D shape contexts are not naturally extended toleverage additional information from images.

6. Conclusions and Future WorkOur framework enables probabilistic 3D object detection

using a spatial occupancy model derived from LiDAR rangedata. Within this framework, range points from multiple Li-DAR scans can be fused into a common representation thatcaptures uncertainty in the scene caused by occlusion andsampling artifacts. This uncertainty can then be meaning-fully propagated to the object detection results.

We believe there are a number of areas for further re-

(a) (b)Free O HidCylinderNetWallFree O HidNetWallBox Cylinder (r=0.26m)Box (s=0.38m)Box (s=0.19m)

(c) (d) (e)

-95.39−∞

-62.89−∞

-18.16−∞

(f) (g) (h)Figure 4. Probabilistic detection of objects partially occluded by camouflage netting. (a) LiDAR data of the scene. (b) LiDAR data showingmanually-inserted true location of cylinder (red) and box (blue). (c) Probabilistic occupancy map of green horizontal slice of (b). Freeprobabilities are mapped to the red channel, hidden probabilities to the blue channel, and occupied probabilities to the green channel. (d)Probabilistic occupancy map of purple horizontal slice of (b). (e) Object models (horizontal slices). (f) The calculated log probability ofthe presence of the cylindrical model at each voxel in the scene. (g) Calculated probability of presence of large box model. (h) Calculatedprobability of presence of small box model.

search extending our scene representation and object de-tection framework. While the current formulation of thetechnique provides no explicit rotational invariance, manyobject models are invariant to some degree. This natural in-variance can be exploited in a naive search over pose param-eters, made feasible using the fast FFT-based algorithm. Weare currently investigating more elegant and efficient meth-ods for imparting full rotational invariance to the method.

Although the independent random field model of spa-tial occupancy facilitates efficient computation, it impliessome counter-intuitive assumptions about the scene struc-ture, e.g., that the probability of an object’s interior andits surface being collocated are independent. We plan toinvestigate higher-order MRF models of spatial occupancythat include conditional probabilities between neighboring

cells. While this approach may yield more accurate modelsand more manageable probabilities, the main challenge willbe maintaining computational efficiency, since this will pre-clude the use of the fast FFT-based algorithm in its currentform. It remains to be seen whether detection results wouldbe improved.

Accurate estimation of spatial occupancy is critical to thesuccess of the proposed technique. The current method usesan implicit spatial “box” filter to downsample the LiDARreturns. While this greatly mitigates sampling issues, it isnot a perfect anti-aliasing filter. Ideally, we should filter theLiDAR data with a “box” filter in the frequency domain thatremoves all frequencies above the voxel Nyquist rate. Sincethe LiDAR points are relatively sparse, this could be effi-ciently accomplished with a straightforward sin(x)/x inter-

polation.Although using a logarithmic representation avoids nu-

merical issues involving the small magnitudes of the detec-tion probabilities, these magnitudes currently depend on thecardinality of the points in the object model. Hence, whilewe can straightforwardly interpret the detection results fora single object, comparing results derived from object mod-els of widely different sizes is problematic. We are investi-gating the normalization of the results relative to the objectmodel size so that different detection maps can be directlycompared.

Finally, we note that if multiple objects are to be testedagainst a single scene, the linearity of the cross correlationoperations can be exploited to improve efficiency. If thecollection of object models can be decomposed into a com-mon set of primitive objects, these primitives can be testedagainst the scene, and the detection results combined in thelogarithmic representation of (11) to produce detection re-sults for the composite objects.

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