15arspc submission 97

8/8/2019 15arspc Submission 97

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Vision-Based Terrain Modelling with Applicationto Change Detection

Tariq S. Abuhashim, Mitch Bryson, Salah SukkariehAustralian Centre for Field Robotics

The University of SydneyNSW 2006, Australia

[email protected]

1 Abstract

Remotely sensed Multi-temporal images collected via satellites, manned aircrafts andUnmanned Aerial Vehicles (UAVs) are important in understanding ecological phe-nomenon and provide an important record to better understanding of ecosystems overspace and time. Since ecology deals with ever changing ecosystems, both time andspace must be taken into account when modelling the terrain surface and land cover.Moreover, sparsity of observations, data association, and change in scale, all make thistask more difficult. In this paper, we tackle the terrain modelling problem by buildinga continuous surface representation that allows formulating a sampling criterion using

Locally Weighted Partial Least Squares (LWPLS). Motivated by number of applica-tions including combining and comparing multi-temporal and multi-resolution datasets,dynamic update, and change detection, the model is able to adapt for changes, andprovides adequate representation of prediction uncertainty. We demonstrate its sim-plicity of implementation and its applicability of terrain modelling on images collectedusing monocular camera mounted on a UAV.

2 Introduction

Map reconstruction plays a major role in many applications of mobile robots and re-cently three-dimensional maps have gained substantial interest in different fields ofrobotics community including terrain [5] and sea floor [12] visualisation, mapping andclassification of vegetation [6], path planning, control and navigation [13]. Raw pointclouds of features extracted from images have been used as the geometric primitive ofchoice for several modelling tasks such as rendering [1], editing [30], and compression[20].

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Autonomous navigation in vegetated terrain is a challenging remains a challengingproblem in robotics due to difficulties in modelling the high variability of out door en-

vironments. In this effort, Unmanned Aerial Vehicles (UAVs) have several advantagesover piloted aircrafts; they are less costly, safer, can be deployed easily and repeatedly,and are easy to fly at low altitude. However, their payload is limited as they cantcarry heavy laser scanners. Vision sensors such as stereo rigs and monocular camerasprovide better and more feasible solution due to their light weight and wider field ofview. However, in applications where UAVs are to fly at higher altitudes wider baselineis required to estimate depth accurately, and therefore only monocular cameras can beused to acquire images of terrain and then to reconstruct its surface.

Terrain surface reconstruction can be understood as the process fitting a functionto a set of noisy observed points features. It can be treated either as a geometricfitting problem or as a regression problem. In the first case, it might require too manyparameters to fit an explicit function (polynomial for instance) and with too complexsurfaces the number of parameters might exceed the number of data points. Methodsbased on classical regression typically assume that the surface geometry can be treatedeither globally, such as Gaussian Process Regression [23], or locally, such as LocallyWeighted Projection Regression (LWPR) [29]. Main drawback of global model lies inthe assumption that surface properties, such as surface smoothness or observationsnoise, can be treated globally. An assumption that can be relaxed, either by using non-stationary covariance functions [17] or assuming heteroscedastic noise models [15]. Onthe other hand, the only drawback of local models is in the way they cluster the space.

In this paper, we present a scheme for approximating terrain surface from irregularsampled points that allows formulating a sampling criteria. Motivated by number ofapplications including combining and comparing multi-temporal and multi-resolutiondatasets, dynamic update, and change detection, we build a continuous representationof terrain surface using LWPR. The model is able to adapt for changes, and providesadequate representation of prediction uncertainty. Unlike other batch machine learningtechniques, the system accomplishes online learning without the need to store thetraining data or large scale matrix inversion. It also assumes that the input and outputdistributions of the data are unknown, and in which these distributions may change overtime. The system utilises a nonparametric regression approach to build a receptive field-

based learning system for incremental surface approximation. A local model is fittedincrementally within each receptive field such that local area surface approximationis accomplished in the spirit of a Taylor series expansion. Incremental learning isaccomplished by incrementally minimising a weighted local cross validation error. Forthe application of terrain reconstruction, our approach offers number of advantagesover other machine learning algorithms:

1. Most of terrain representations change over time, and the terrain model requires

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adaptation. One application that suits LWPR very well cause of the embeddedtime scale representing the rate at which changes occur.

2. For applications requiring terrain models to be constructed incrementally andonline, our use of LWPR offers online learning of terrain model.

3. For applications requiring to navigate or to model large terrain, this involvescollecting large amount of training data and for some regression methods such asSVM or GP, computational complexity scales with size of training set.

4. Terrain surface properties, such as smoothness and noise level, vary globally overthe area of interest and hence have to be treated locally. Therefore, multiple localmodels offer better non-linear surface approximation over the local scale.

The rest of the paper is organised as follows, in section 4 we briefly review non-parametric learning using LWPR. Section 5 review surface reconstruction algorithm.Experimental setup is presented in section 6 with results of the algorithms demon-strated on images collected via monocular camera mounted on an Unmanned ArialVehicle (UAV).

3 Related Work

The computer vision literature is rich with approaches to 3D surface reconstruction.With the advent of computers with more memory and computational power, the contin-uous approximations, such as B-splines [10] and Bezier curves [9], gave way to discreteapproximations, such as polygon meshes [14] and voxel grids [8]. In [12] a completereconstruction and visualisation pipeline that generates 3D meshes and voxels with dif-ferent levels of details using sparse features extracted from seafloor is proposed. In [5]a framework for integrating IMU, GPS and monocular vision camera information usinga batch smoothing approach which allows for the construction of dense terrain mapswas presented. The method integrates all the sensor information using a statisticallyoptimally non-linear least squares smoothing algorithm to estimate vehicle poses simul-taneously to a dense point feature map of the terrain, a method thats been known formany years in computer vision community as Bundle Adjustment (BA) [27]. However,in both approaches the reconstructed surfaces are irregular and sparse, and dont allowfor dynamic updates or sampling criterion.

Terrain surface reconstruction can be also understood as the process of fitting afunction to a noisy observed points features. Gaussian Process Regression (GPR) [23]is a powerful tool for accurate function approximation in high-dimensional space. Thehyperparameters of a GP model can be adjusted by maximising the marginal likelihoodof test data given train data. It requires little effort, easy and flexible to be used andprovide an elegant function approximation in probabilistic framework. However, themain limitations of GPR for large data sets are that the computational complexity

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scales cubically with the number of training samples, requires a large memory to storelarge training data, and the assumption that surface properties such as smoothnessand noise can be treated globally.

Many researchers attempt to tackle GPs problems using different approaches. In[28] a GP was used in large-scale terrain modelling in combine with Kd-trees [25] tospeed up the learning and prediction process. Sparse Gaussian Process (SGP) [21, 7]approximates the training data by a smaller set of so-called inducing inputs. However,the selection of such inducing inputs replacing the whole training set is a problem. Incontrast to SGP, an alternative approach is to divide the state space into number oflocal regions where classical PGR can be applied. Mixture of Experts (ME) [26, 22]divide the input space in smaller subspaces by a gating network and the computationalcomplexity is then significantly reduced. However, the performance of ME dependslargely on the way the state space is partitioned and the choice of the optimal numberof local models. In [18] a method of sparsification by incrementally building a dictio-nary of the most informative data points is proposed. An information measure that isbased on linear independence is used to select the most informative points given thecurrent dictionary. A drawback of this method is that new candidates to be added tothe dictionary are selected based on how close they are to the most recent query pointusing nearest neighbour search and the linear independence measure is determined byminimising the spatial distribution between the current points in the dictionary and thenew query point. While this method might work well in applications where the statesof interest are the most recent ones, it wont be beneficial for terrain reconstruction.

Inspired by local learning, number approaches were induced for real-time modellearning for robotics using the receptive field concept, such as the Locally WeightedProjection Regression (LWPR) [29] and Local Gaussian Process Regression (LGPR)[19]. In local regression, the state space (in the case of terrain surface learning thestate space is defined by the 3D positions of points features in a reference coordinates)is partitioned into local regions within which the local model is approximated. Theallocation of these partitions is essential, and therefore an appropriate online clusteringof the state space becomes a central problem in local approaches. In [19], a LGPRmodel was proposed to combine the high accuracy of the GPR and the fast speed ofthe LWPR. The applicability of the algorithm was demonstrated for real-time onlinelearning of the inverse dynamics for robot model-based control. The training data was

partitioned into local regions, for each an individual GP model is trained. The predic-tion for a query point is performed by weighted estimation using nearby local models.A distance-based measure is used for partitioning the data and weighted predictions.However, storing the local training data for inference is still required and therefore thesize of the memory required will scale with the size and dimensions of training data.Another drawback of LGPR is that the covariance function need to be updated everytime new data point arrives, and that the deletion and insertion of new data points in

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order to escape the curse of dimensionality of standard GPs is not trivial. The LWPRalgorithm removes this requirement as the true function is approximated within eachlocal region a linear function and only minimum number of parameters are stored. The

global true function is then approximated with local linear functions covering the wholestate space and learning become computationally feasible due to low computationaldemands of the Local Projection Regression (LPR) which can be performed online.

4 Locally Weighted Learning of Terrain Surface

LWPR is based on a modification of the Partial Least Squares (PLS) regression toperform incremental locally weighted updates. The model assumes that the regressionfunction E{yk|x} is linear in the inputs within local regions (or receptive field RFk).

Where x is input position in north-east coordinates and yk is the predicted height atx within the k-th RF. Given a query point x, the weighted prediction of height y at xis built from the individual predictions yk of all receptive fields as

y =

Kk=1

wkyk

Kk=1

wk

(1)

where K is the number of receptive fields and wk is a weight determining whether adata point x falls into the region of validity of RFk (in machine learning literature,this weight is usually referred to as similarity measure between the query point andthe local model) and is characterised by a kernel (kernel functions represent the dotproduct between two vectors in the feature space). For simplification, we assumeGaussian kernel;

wk = exp

1

2(x ck)

TDk(x ck)

(2)

The kernel function parametrizes the k-th linear model of the k-th RF by its locationin the space, ck, and a positive definite distance metric Dk, determining the size andshape of the receptive field (it is the inverse of the length scale). Hence, the weightedprediction y can be estimated as The individual prediction of RFs, yk, is simply givenby classical linear regression expression

yk = xTk k (3)

xk = [(x ck)T, 1]T (4)

where k are the estimated parameters of the k-th linear model. During the learning(update) process, both the shape of the RFs Dk and the parameters k of the localmodels are adjusted such that the error between predicted values and observed targetsare minimal. Given some sparse observations of the terrain surface, the regression

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parameters k can be computed incrementally and online using Partial Least Squares(PLS) [29, 11], and the distance metric Dk, which is equivalent to the inverse of thesurface length scale determining the smoothness of the predicted terrain surface, can

be updated incrementally using leave-one-out cross validation [29], and will be dis-cussed more detailed in the following section. A pseudocode of the model learning isoutlined in algorithm 1, where K is the total number of clusters, is a threshold thatdetermines if a point belongs to a cluster and hence determines when to create newcluster and Dint is the initial distance metric. The significance of the threshold willbe discussed in details in the results in section 5.

Algorithm 1 Pseudocode for learning local models

Require: K = 0, , Dint1: for every new training sample (x, y) do

2: for k = 1 : K do3: Calculate wk4: if wk > then5: Estimate regression parameter k using PLS

6: Update distance metric Dk using gradient descent

7: end if8: end for9: end for

10: if no RF was activated then11: Create new FR with ck = x and Dk = Dint12: end if

4.1 Updating the Distance Metric

The distance metric specifying the size and shape of the local area in which a localsurface model is valid can be learned and modified for each local area individually bystochastic gradient descent in a penalised leave-one-out cross validation cost function[24]

J=1

N

i=1wi

Ni=1

wi(yi yi)2

(1 wixTi P1xi)2

+

M

Mi,j=1

D2ij (5)

where N denotes the number of data points in the training set, M denotes numberof dimensions and is a tradeoff parameter that can be determined empirically or fromthe assessment of the maximal local curvature of the function to be approximated. Thefirst term of the cost function is the mean leave-one-out cross validation error of thelocal model, and the second term is a penalty term which ensures that the receptivefield wont shrink indefinitely in case of large amounts of training data. The matrixP corresponds to the weighted covariance matrix of the input data. The inversion of

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such matrix can be sometimes expensive, specially with increasing number of pointsor dimensions

To contribute more into the computational efficiency of the algorithm, the optimi-sation process is carried into the projection space were projected inputs are orthogonal.Equation 5 can be exactly formulated in term of the projected inputs zi = [zi,1, zi,R]T

as

J=1

Ni=1

wi

Ni=1

wi(yi yi)2

(1 wizTi P

1z zi)

2+

M

Mi,j=1

D2ij (6)

In this case, P1z is proved to be diagonal and corresponds to the covariance matrixcomputed from the projected inputs [29]. The distance metric is learned locally using

gradient descent

Mn+1 = Mn J

M(7)

where M is an upper triangular matrix resulting from the Cholesky decompositionofD. A stochastic approximation of the gradient J

Mis derived in [29].

5 Results

5.1 Experimental SetupIn this section we the experimental set up of the platform used to collect the trainingterrain data. Data was collected over a farmland location in Queensland Australia usingthe J3 Cub platform shown in figure 1a. The vehicle is capable of carrying a payload of15kg with an endurance of one hour. The vehicle flew at a fixed hight of 100m abovethe ground. The payload, shown in figure 1b, included a low-cost IMU running at 100Hz, a GPS receiver providing updates at 5Hz, a Colour camera acquiring frames at3.75Hz at a resolution of 1024 768 pixels, and a PC104 computer used to log thesensor data.

5.2 Methodology

Inertial measurements were integrated with GPS updates using a navigation ExtendedKalman Filter (EKF) [2] to estimate the vehicle position and attitude. Images wereprocesses as pairs, as shown in figure 2, to form a stereo rig and the displacementof the vehicle between the two positions, where every images pairs were taken, wasused to estimate the baseline between the two frames, and multiple-view geometry

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(a) (b)

Figure 1: The J3 Cup UAV used to collect data and the payload system onboard.

principles were applied to triangulate features extracted from both images using GoodFeatures To Track from OpenCV library [4]. These features were tracked betweenmultiple frames using pyramid implementation of Locas-Kanade optical flow [3].

Tracked features were then used to estimate the fundamental matrix. In this work weuse the fundamental matrix with RANSAC and known rotations proposed by [16]. Theestimated fundamental matrix is them used to remove outliers violating the epipolarconstraints imposed by the fundamental matrix. Matched features pairs were thentriangulated to estimate features locations in world coordinates. Triangulated features

are then fed into the LWPR algorithm learning the smoothness of the surface byclustering the data into local areas. The learning step includes learning the distancemetric and the local regression parameter within every cluster. The learned models arethen used to predict heights give set of random query points.

6 Results

This section we present results from both the terrain reconstruction algorithm and theterrain learning algorithm applied to the data collected by the fixed-wing UAV. Figure 3shows point cloud of triangulated features from pair of images. The extracted features

are noisy and sparse, specially in areas around trees where most of the extracted fea-tures candidates were rejected due to the branches configuration changing with cameraprospective.

In order to build a continuous surface from the sparse observations shown in fig-ure 3, the data has to be clustered into number of local regions where local linearapproximations apply. This is a critical step is the LWPR algorithm since local ap-

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8 meters

p1

p2

Figure 2: Pair of images acquired using the down looking camera on the UAV. The frequencyby which images were captured allowed the UAV to travel average distance of 8 metersbetween frames creating baseline required to estimate depth of features

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Figure 3: Sparse features extracted and triangulated features in the 3D world coordinates

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proximations within every region will heavily correlate to the observations included andpoor result of the clustering stage will result into deterioration of the overall algorithmperformance. The main factor controlling the clustering process is the threshold that

determines if a point belongs to a cluster and hence determines when to create newcluster. Figure 4 shows the results of varying the cluster threshold while learning thedistance metric from the data. In the figure, as the value of the threshold changes fromsmall values to larger values the space changes from being under-clustered into over-clustered and hence the the degree of overlap between individual models increases. Asresult, figure 5 shows the effect of degree of overlap variation. The smoothness ofthe predicted surface is tightly related to the weight threshold value. As the degree ofoverlap increases the surface preserves more details. On the other hand, as the degreeof overlap increases more local approximations become involved in predicting heightat given query point and hence we expect higher accuracy in predictions. This canbe easily noticed by looking at predicted variance variations. One more observation isthat as the threshold value increases the algorithm can extend predictions at regionswith no observations, while at low values of threshold the predicted mean goes to zeroas result of the zero mean assumption. Finally, it worths mentioning that as the valueof the threshold approached 1, the LWPR model approached the accuracy of the GPR.

Finally, for the purpose of texturing, rendering and 3D modelling, the continuoussurface representation can be resampled at regular steps and regular meshes can beefficiently generated. Figure 6 shows the irregular Delaunay triangulation and theregular mesh generated from the constructed terrain model.

7 Conclusions and Future Work

The 3D surface reconstruction process can be understood as the process of fittinga surface to a set of noisy observed points. Methods based on classical regressiontypically assume that the geometry can be treated either globally, such as in the caseof GPR, or locally, such as in LWPR. In large terrain modelling, the computationalcomplexity of GPs becomes a burden. LWPR is an alternative that provides a fasterand incremental solution by partitioning the state space and doesnt need to store thetraining set. However, dividing the space into number of regions using a distance-based measure tends to uncorrelate points inside and outside a local region makingour predictions less accurate. The next step in this research will aim to find a way

of dividing the space based on the geometry of the surface. We believe that inte-grating geometric information extracted from point clouds such as surface normals orDelaunay triangulations will constraint the regression problem and therefore increasethe accuracy of predictions. To estimate the accuracy of surface reconstruction, densestereo approaches are to be used to generate ground truth model of the terrain andprovide dense point clouds that will constraint the regression and improve the overallperformance.

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60 65 70 75 80 85 90 95 100

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North (meters)

East(meters)

(a)

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(b)

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(c)

Figure 4: State space is divided into local regions (receptive field) where the shape and sizeof local areas is learned individually from the data using gradient descent for the cases of(a) = 0.001, (b) = 0.1, and (c) = 0.5

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Figure 5: Predicted mean of terrain surface (left column) and predicted standard deviation(right column) for different values of in the same order as in figure 4

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(a)

(b)

Figure 6: Comparison between sparse Delaunay triangulated mesh and regular mesh of terrainsurface. In figure (a) the Delaunay triangulated mesh of observed features is sparse andirregular. Specially at areas around trees due to the limited field of view of the down lookingcamera. In figure (b) the reconstructed surface was resampled at regular intervals and theDelaunay triangulation will result into a regular and smooth mesh. As shown in the figure,

the surface is too smooth due to sparsity of observations. It is expected that by increasingthe density of observed features, the algorithm will preserve more details.

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