Viewpoint Selection by Navigation through Entropy Maps. Tal Arbel and Frank P. Ferrie. McGill University,. Center for Intelligent Machines,. Montréal, Québec ...
Viewpoint Selection by Navigation through Entropy Maps Tal Arbel and Frank P. Ferrie McGill University, Center for Intelligent Machines, Montr´eal, Qu´ebec, CANADA H3A 2A7 tel. (514) 398 2185 fax:(514) 398 7348
Abstract In this paper, we show how entropy maps can be used to guide an active observer along an optimal trajectory, by which the identity and pose of objects in the world can be inferred with confidence, while minimizing the amount of data that must be gathered. Specifically, we consider the case of active object recognition where entropy maps are used to encode prior knowledge about the discriminability of objects as a function of viewing position. The paper describes how these maps are computed using optical flow signatures as a case study, and how a gaze-planning strategy can be formulated by using entropy minimization as a basis for choosing a next best view. Experimental results are presented which show the strategy’s effectiveness for active object recognition using a single monochrome television camera.
1. Introduction Consider the problem of an active observer moving through a familiar environment with the task of identifying and localizing known objects. Due to limited resources, it cannot spend a lot of time in one place so the computational overhead must be low. Furthermore, it needs to minimize the effort expended in gathering data so it must be economical in its movement. These constraints typify many applications of active vision, particularly in the context of mobile robotics. At another level this problem raises the challenge of how relatively simple percepts can be integrated into solid inferences about the visual world. Our approach to this problem is based on two observations: i) that strong assertions can be made by accumulating evidence that might appear to be weak instantaneously, and ii) knowing how to explore an environment (i.e. where to look) can be learned from local interactions with the objects that populate it. This leads to the computational framework which is at the heart of this paper.
Evidence for different assertions can be expressed in the language of probability theory and the accumulation of evidence defined formally in terms of Bayes accumulation. Since the observer is free to interact with each of the objects prior to exploration (i.e. off-line), then object probabilities associated with image measures can be learned from training. We will refer to these probabilities as belief distributions. This idea can be taken one step further. If the parameters associated with each measurement are also recorded then one can also build a model of how these parameters influence the confidence for the resulting assertions. In Section 2 we use the concept of Shannon entropy to define a measure of ambiguity for the resulting belief distributions called an entropy map. Here the entropy map is used to relate ambiguity to camera (viewing) position and serves as the basis for our active vision system. By choosing viewpoints that minimize ambiguity, the system seeks out locations that are maximally informative. Hence fewer observations are required to arrive at a confident assertion. Other strategies use entropy on-line in order to maximize information gain during navigation [6, 5]. The key difference in this work is that we maximize the a priori information available, by building the entropy maps off-line and using them to guide the on-line navigation. In the present context our mobile observer will consist of a monochrome television camera mounted on the end effector of a gantry robot (Figure 3(a)). The camera is free to move about the workspace of the gantry in which the different test objects are placed (i.e. stationary environment). As the camera moves relative to an object, an optical flow pattern is induced on the retina which results in a discrete image sequence. The task that the system must perform is to generate an optimal trajectory (the shortest sequence) that will result in the correct assertion. What makes this task particularly challenging (and an excellent demonstration of the theory) is that the problem is fundamentally ill-posed. Because optical flow is a function of object shape, camera geometry, and the relative motion between camera and object, factoring out the different
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components can be exceedingly difficult (if not impossible). However, as has been demonstrated elsewhere [1, 12, 3] the factoring problem can be dealt with where suitable a priori constraints are available. In the present case the expected motion of the camera is learned using appearancebased techniques [10, 11] and distances are controlled so as to minimize the effects of camera geometry. We have determined empirically that it is indeed possible to extract a signature that is largely correlated with object shape from the instantaneous optical flow for the expected range of motions. A single such measure is not sufficiently robust to make a confident assertion, but it does have the advantages of being inexpensive to compute and can cope with variations in illumination (relative to training conditions. Illumination is assumed to be constant during measurements). Rather than make an assertion based on a single measurement, we use Bayesian techniques to accumulate evidence for the different assertions over a sequence of measurements. Our hypothesis, which is supported by the experiments presented in Section 4, is that the correct assertion will become apparent over time. Gaze planning using the entropy map further ensures that the viewpoints selected are maximally informative relative to the most probable assertions. The net effect of this feedback is to increase overall robustness (i.e. the probability of making correct assertions) by avoiding viewpoints that are inherently ambiguous. The remainder of this paper is as follows: We begin in Section 2, with a description of the structure of the entropy map and a navigation strategy based on it that is used for gathering new data. Section 3 then describes the application of the theory to an active vision strategy for recognizing objects from their optical flow fields. Experimental results are presented in Section 4 which compares the entropy-driven approach to naive exploration. Finally Section 5 with a brief discussion and pointers to future work.
2. Entropy Maps 2.1. Entropy Map Structure Consider an appearance-based training strategy that associates an image measure, , with some particular object Oi . We define recognition as the posterior probability distribution over the set of n object hypotheses, fOi gi=1::n , given the measurement vector . We denote this distribution as P Oj . We now wish to obtain a measure that predicts the likelihood of ambiguous recognition results as a function of viewing position. A suitable measure is defined in terms of the Shannon entropy [7],
x
x
( x)
X P (O jx) log H (P (Ojx)) = i
i
1 ( jx) ;
P Oi
(1)
which is a measure of the ambiguity of the posterior distribution produced by a recognition experiment. Higher entropies reflect greater ambiguity. For the problem at hand the entropy map is parameterized on a tessellated viewsphere with the object at origin. Off-line during training, image measures, , are sampled at each coordinate of the viewsphere for each object in the database. Stored with each data set are the coordinates of acquisition. A Bayesian learning strategy (see Section 3) is applied to the cumulative set of measures to derive a function for P Oj . Then using Equation (1), a map is constructed for each object in the database by evaluating H P Oj at each coordinate of its respective viewsphere. In practice some additional conditioning is required before the entropy map can be used for gaze planning purposes, namely in the enforcement of particular smoothness constraints which are essential for stability. For example, a gaze planner using the map to determine minimal entropy viewing positions would seek to avoid locations corresponding to singularities or discontinuities in the entropy field. Slight errors in positioning (or equivalently in determining the pose of the entropy map relative to the data acquired) could result in sampling at precisely the wrong locations. These constraints are made explicit by applying the following non-linear smoothing operator,
x
( ( x))
( x)
P cos( ) H (P (Ojx )) P cos( ) H (P (Ojx )) = ; j
ij
j
i
j
x
(2)
ij
where i is the data vector gathered at viewsphere location i of the operator, j are the points in the local neighbourhood indexed by j , and ij the angle subtended at the center between i and j . The minimal entropy location on this map will correspond to an optimal location which is stable with respect to localization errors. The resulting entropy map can be very informative in the context of planning gaze for object recognition. It provides a quantitative prediction of the level of difficulty of recognizing each object in an on-line experiment. In contrast to human-generated aspect graphs, e.g., [8, 9, 4]), by linking location and discriminability using entropy maps, a set of such characteristic views can be automatically generated off-line.
x
x x
2.2. Using Entropy Maps to Plan Gaze Two problems must be solved prior to planning gaze: 1) a particular map must be selected and 2) the pose of that map must be determined relative to the data acquired. As measurements are made on-line, the maximum a posteriori (MAP) solution corresponding to P Oj is used to determine the most likely object hypothesis for the measured data . This estimate is subsequently used to select the entropy map to be used for planning the next best view. Of
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( x)
course the particular gaze planning strategy must be carefully structured to operate stably in these circumstances. More will be said about this shortly. Pose can be estimated at minimal expense by retaining the location information along with the image measures acquired during training. For example, appearance-based methods can be used to index these measures using the data acquired on-line [10]. In fact, the implementation described in Section 3 already uses appearance-based techniques in the process of determining the likelihoods for the different object hypotheses. As such, the computational overhead of determining pose is minimal. Once the camera pose is established in the coordinates of the training viewsphere (Figure 1, Steps 1–2), it is straightforward to determine the relative transform taking the camera to the desired position within this frame (Figure 1, Steps 2–3). By applying this same transform to the current camera frame, the camera is positioned accordingly (Figure 1, Step 4). The gaze planning strategy itself must be sufficiently robust to accommodate errors in pose determination and entropy map selection. Errors in the former are accommodated in part by the smoothing applied to the entropy map and a strategy that avoids placement in the vicinity of singularities and discontinuities. A partial solution to the selection problem is effected by choosing a next best view that minimizes the entropy on the most likely object hypothesis map. Over time the expectation is that confidence in an incorrectly chosen hypothesis will decrease as further evidence is uncovered. 2
1
stationary environment with the task of identifying objects within it. Motion induces a sequence of optical flow images on the camera retina which will serve as the basic input to the recognition system. Our strategy will be to accumulate evidence for the different object hypotheses over time until a clear winner emerges. To minimize the chances of arriving at an incorrect assertion, the entropy map will be used to plan a trajectory that safely leads to the correct minimum entropy solution in a minimal number of steps. In this section, we briefly describe how to compute the probabilities in the various hypotheses using flow images as inputs, and use a Bayesian framework to update the beliefs over time. We also show how this is used to compute the entropy maps off-line and how the entropy map structure can be exploited during the on-line recognition phase. An appearance-based approach is used to represent the optical flow patterns that result from the relative motions between the moving observer and each of the objects in the database. Of course the recognition task is complicated by the fact that these patterns confound motion, structure and imaging geometry not to mention the difficulty of estimating the optical flow field in the first place. However by using an active vision system the sensor trajectories and camera positions can be controlled to the point where the flow patterns associated with different objects can be learned by the training procedure off-line. Robustness to confounding and flow estimation errors are further enhanced by the facts that evidence is accumulated and viewpoints chosen specifically to minimize ambiguity. Use of optical flow also has some advantages with respect to minimizing sensitivity to illumination variations and background conditions.
3.1. Object Recognition Strategy
4
x ( x)
3 Current Viewsphere
Now we describe a Bayesian recognition strategy that represents the posterior beliefs over the entire set of ob::n, given the current ject hypotheses, fOi g where i data vector, , by a posterior probability distribution of the form P Oj , with discrete (conditional) probability density function p Oi j ji=1:::n . The question becomes: how do we compute these probabilities given a sequence of optical flow images gathered over time? In the case study in question, at each time interval, the input to the system is a flow image gathered by the sensor, represented by the vector . We wish to establish the belief in each object, given each of the images gathered, and accumulate evidence in each hypothesis over time. During the off-line (training) phase the camera is moved about the objects of interest along trajectories that reflect the expected on-line motions. Each coordinate of a tessellated viewsphere surrounding an object is sampled with a sequence of short, curvilinear sweeps along different directions. The resulting set of flow images acquired from all
Training Viewsphere LEGEND: nearest neighbour motion transform
Figure 1. Navigation Strategy.
3. Case Study: Recognizing Objects Based on Optical Flow Images The preceding framework is now applied to the problem of planning the gaze of an active observer moving through a
=1
( x)
x
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viewpoints for each object is used to determine a basis for representation using Principal Components Analysis (PCA) [11]. A representation for each object is then constructed by projecting its corresponding flow image vectors f j g onto this lower dimensional basis. The projected vectors f j g are subsequently parameterized by a multivariate normal jOi . This represents the physical thedistribution, P ory predicting possible variations in parameters given each object in the database. On-line, an image, , corresponding to the unknown object is projected onto the basis determined during training, resulting in the parametric description x. Using standard Bayesian techniques to determine the data support for each object hypothesis gives:
x
m
(m )
x
m
( jx) = K1 p(m jO ) p(O ); 1 : : : n (3) where p(O ) defines the prior probability for each object hypothesis, O , p(m jO ) is the multivariate normal distrip Oi
x
i
i
i
i
x
i
bution derived during training evaluated at the location in space defined by x, and K is the normalization constant such that:
m
K
=
X p(m jO ) p(O ): n
j
x
=1
j
(4)
j
The result is a discrete conditional probability distribution describing the belief in each of the models in the database, given the flow data. Because of confounding and errors in flow estimation, recognition from a single viewpoint alone will not necessarily be stable. We therefore hypothesize that by accumulating evidence over time, a more robust solution is possible. Here, as each of the flow images is gathered by the sensor, evidence can be accumulated on the level of the probabilities over time, by using a Bayesian chaining strategy that assigns the posterior probability at time t, P Oi j t , as the . This can be expressed as: prior at time t
( x)
+1 1 p(O jx ) p(m p(O jx +1 ) = K i
t
x
i
t
+1 jO );
xt
i
1:::n
where t is defined as the data set at time t, and is the parametric description of the measured flow, . time t
+1
(5)
m +1 m , at xt x
3.2. Navigation Based on Entropy Maps With the system defined in this Bayesian framework, the entropy of the probability distribution can be computed exactly as in Equation (1). Using flow images is different from the general case in that, at each position on the viewsphere, the system stores a series of entropy values, each associated with a different movement of the camera relative to the object at that location. The system then computes the average
entropy at that location and proceeds with the non-linear smoothing as in Equation (2). The best location to go to is chosen as before, and the movement to induce is the one that generates the lowest entropy among the set of possible movements at that location. Figure 2 illustrates two viewpoints of the entropy map resulting from gathering optical flow images densely about a viewsphere surrounding a particular database object. Each tile corresponds to a camera view of the object at the origin (superimposed onto a shaded sphere). For illustration purposes, we show the best entropy value among all the movements trained on at that location. The tiles are shaded in accordance with their raw entropy value in Figure 2(a), and to their smoothed value in Figure 2(b). The shading ranges from low entropy (black) to high entropy (white). Large areas with no tiles indicate viewpoints that lead to a false recognition result in terms of the MAP solution. Examining the maps, one can see that their structure is such that areas that result in inter-class confusion are concentrated into relatively isolated patches on the viewsphere. In addition, the ambiguity increases in the areas surrounding the “worst” locations. The structure of the map lends itself to our navigation strategy as generally large patches are comprised of optimal viewpoints, in terms of high confidence in the correct model. The smoothed versions of the entropy map at the same location, as seen in Figure 2(b), illustrate both the best and worst locations (in terms of location and movement) for discrimination. Notice that the best location is maximally far from the worst areas of high confusion or strong belief in the wrong object. This leads to the hypothesis that moving towards this location will lead to a correct solution with relative safety. The navigation based on these entropy maps works exactly as described in Section 2.2, with the system now using the location and direction of movement to build a camera frame at its current position. The difference here is that the sensor is directed to the optimal location and optimal direction of motion, where new data are gathered. Over time, the expectation is that confidence in an incorrect hypothesis will decrease as further evidence is uncovered.
4. Experimental Results The theories presented in this paper were tested through a sequence of real experiments. The goals of the experiments were: (1) to build an entropy map off-line for each of a series of objects in a database, and to examine its structure when using optical flow images as signatures for object shape, (2) to test the ability of using these maps to guide a sensor towards optimal locations in on-line recognition experiments. We will show that the structure of the entropy maps are conducive for navigation, and that active recognition experiments based on these maps indicate their supe-
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(a)
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Figure 3. (a) Experimental Setup, (b) Database of Objects. (b) Entropy
0
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Figure 2. (a) Two Views of an Entropy Map, (b) Corresponding Smoothed Maps. riority to random sequential recognition methods in terms of recognition accuracy and speed of convergence. Further, we will illustrate that despite the difficulties in recognizing objects based on single flow images, the system is able to quickly converge to a correct solution by using Bayesian updating techniques.
Entropy maps were built for each object in the database off-line during training. The hypothesis was that the structure of the maps lend themselves to on-line navigation based on them. Figure 4 shows images of the toothpaste tube, and its corresponding smoothed entropy map, taken from two different camera viewpoints. The system was successful at identifying which viewpoint was informative for recognition (in dark), and which was not (in light). The structure of the maps were such that the good locations were found in large patches, with the resulting map varying continuously. The optimal locations were chosen to be those at the center of the patches, so that navigation towards them should lead to correct recognition results, even if slight errors in positioning occur. Notice that the entropy maps match an intuitive notion of viewpoint ambiguity.
4.1. Building Entropy Maps 4.2. Navigation Experiments Off-line, training images of 15 household products (see Figure 3(b)) were gathered at equally spaced locations around a coarsely tessellated viewsphere. Specifically, each object was placed on a rotary table. At each position on the viewsphere, a gantry robot arm moved along a horizontal and along a vertical arc at fixed distances from the object. A CCD camera, mounted on its end-effector (see Figure 3(a)), gathered three images in sequence along each trajectory from which optical flow was computed (using a strategy as in [2]). This served to create a local basis for flow. The expectation was that other on-line motions could be inferred from this basis. Speed normalization was achieved by normalizing the optical flow magnitudes to lie between 0 and 255. Flow was used to localize the object of interest within the images. A low dimensional parametric basis for flow was determined by using standard PCA techniques [11]. Empirically, it was found that 20 eigenvectors were sufficient to represent the images. Low dimensional parametric flow descriptors were established by projecting each flow image onto this basis.
Having computed the entropy maps off-line, a series of experiments were devised to test the navigation strategy that uses the maps on-line for recognition. The hypothesis is that the maps would guide the sensor towards the optimal locations for recognition, and that convergence to the correct hypothesis would be more accurate than if one were to use a random sequential navigation strategy. For these experiments, we used the same set-up as in training. Starting from a random location, the gantry arm was moved along a local, curvilinear trajectory on the viewsphere according to the proposed navigation strategy. At each coordinate sampled along this path, a local flow measurement was made by sweeping the arm along two short curvilinear arcs. Recognition was then performed using the corresponding optical flow generated by this local motion. As a precautionary measure, an initialization procedure was performed to ensure that the system did not start out from a local minimum in the entropy map. Empirically, it was found that this lead to an improvement in the results. The system iterated until the entropy reached an arbitrarily small
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Figure 5. Percentage Correct Recognition Results at Convergence. (b) Images of the toothpaste tube and the corresponding smoothed entropy map are seen at two locations. The system chose the right view as the most informative (seen with darker shading on the map), and the left view as a relatively bad one (lighter tiles).
model was present (lighter tiles).
Figure 4. Images of a Toothpaste Tube, (b) Smoothed Entropy Maps at Corresponding Locations.
convergence value (e.g. 0.01 was chosen). Five hundred such tests were performed on all the objects in the database in order to examine average performances. The percentage of correct MAP recognition results at convergence using the described navigation strategy can be found in Figure 5, where the results are plotted for each of the objects of the database. Here, one can see that the system converges to the correct solution in 80–100 of the cases. It does so in less than 3 iterations on average. A similar experiment to the one above was performed, this time using a random navigation strategy. The results indicate that both strategies performed quite well, in terms of recognition results and quick convergence. This is mostly due to the strength of the Bayesian chaining algorithm at eliminating false hypotheses quickly. However, it was found that navigating based on entropy maps outperformed the random approach in cases where the system started in high entropy locations. Figure 6 shows two examples comparing the two strategies when started from the same high entropy location. On the right of the figure, one can see the results of an entropy-based navigation sequence, superimposed onto the entropy map of the object of interest. Here, convergence to the correct solution occurs in three iterations. Notice that the system leads the sensor near to the entropy map minimum of the correct hypothesis (darker tiles). On the left of the figure, one can see the result of a random sequence, where convergence to the wrong model, reached in five iterations, resulted from the effect of several images taken at locations where strong belief in the wrong
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Above we can see the results of a random walk experiment (seen on the left), and one using our navigation strategy (on the right), both initialized from the same location. Each iteration (seen in black) is superimposed onto the entropy map of the bread roll. Using our approach, the system converged to the correct solution in the desired neighbourhood (darker tiles). Using a random strategy, the system converged to the wrong solution, as many random views were chosen in a “bad” neighbourhood (lighter tiles).
Figure 6. Comparison of Entropy Map and Random Navigation Strategies. Figure 7 illustrates a comparison of navigation results (using both strategies) starting at high entropy locations in the cases of the duck and the bread roll (chala). One can see that the proposed strategy converges quicker than the random strategy in both cases. Notice that, in Figure 7(a), the random strategy caused the sensor to move to a “bad” local minimum (low entropy, wrong model case) at iteration 2. A formal comparison of the performance of the two approaches when started from ambiguous viewpoints can be seen in Figure 8. Only the results where a difference exists are plotted. For the most part, our strategy wins over the random one. Empirical evidence indicates the benefits of using an offline entropy minimization strategy, over on-line methods, in leading the system towards the global entropy minimum. In cases where the sensor began with a high confidence in the wrong model, the entropy may increase with each step as
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Navigation Results Over Time
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Figure 7. Navigation Results Over Time for (a) duck, (b) bread roll.
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estimating pose is quite effective, a more precise pose estimate would ensure even faster convergence to the correct solution. Further, we are currently developing a strategy that navigates based on the entropy maps of several competing hypotheses, rather than only using the map of the MAP solution. This will ensure greater stability during traversal. Finally, the on-line entropy results were not used during experimentation, except to establish convergence. We are currently developing a strategy that will use the on-line entropy as a measure of ambiguity in the recognition assertions made at the current location, permitting us to assess the validity of using the MAP assertion in navigation.
References
Figure 8. Comparison of Two Strategies from Ambiguous Viewpoints.
it leaves the local minimum, before converging to a global low entropy state. On-line entropy minimization strategies converge to a local entropy minimum, even if it belongs to a false assertion.
5. Discussion and Conclusions In this paper, we have introduced an active recognition strategy whereby entropy maps, encoding object discriminability as a function of viewing position, are computed off-line. During on-line recognition experiments, these maps serve to guide a sensor towards areas that minimize the inter-class confusion between competing object hypotheses. We have applied the strategy to the problem of recognizing objects based on signatures in their optical flow images. We have developed a Bayesian recognition framework that computes the degree of confidence in each of the objects in the database given a single flow field image, and shown how robustness can be achieved by using Bayesian chaining to accumulate evidence as sequences of such images are introduced. Experimental results indicate that the strategy works in practice, as convergence occurs in relatively few iterations, and MAP solutions indicate near perfect results for all the database objects. Various improvements to the navigation strategy can be employed. First, although the nearest neighbour strategy for
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