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# 2 Theory

Our method has a learning and an image sequence synthesis stage. In the learning stage the problem we solve is: Given a (time) sequence of intensity images and the corresponding sequence of motor states of an articulated agent, find an approximate invertible mapping between image space and motor space, so that (for the training set): To make the problem tractable we divide into two parts , where is in a space of appearance vectors. We will show how to estimate (learn) the function from watching a moving articulated agent. For images of size , , and an agent with n freedoms ( ) we choose the appearance vector space so that a significant dimensionality reduction is achieved. For example in the two joint robot arm simulation in section 3.1, , m = 24, n = 2.

Figure 1 outlines our method. During training, sequences of intensity images are transformed into sequences of appearance vectors by either a disparity or a subspace method described in the next section. The appearance vectors and the corresponding motor vectors are used to estimate the visual-motor model either on-line while the training images are acquired (section 2.2) or off-line (section 2.3.2) from a batch set of images.

Figure 1: Overview of appearance based view synthesis.

## 2.1 Appearance based representations

In this section we briefly review the principles of the two visual front ends we have experimented with. We require visual representations to be: (1) Approximately invertible so reasonable quality images can be reconstructed. (2) Well behaved (smooth) w.r.t. motor space so that the visual motor model can be estimated. We also prefer representations not restricted to a particular class of agents, or requiring a-priori models.

### 2.1.1 Subspace eigen-image methods

The idea in the subspace eigen image method is to project the raw intensity values onto a basis of m eigen images. Representations based on this idea have been used for recognition problem ``what''[1, 3] and location determination problem ``where''[2]. There are several ways to choose the eigen images. In our case we will be looking at the same agent, in different poses, and all the images we want to represent are fairly similar. In this case it is advantageous to use a basis specifically designed for the agent. In summary (see also [1, 3]) this can be done by acquiring a (large) number p of size images of the agent in different poses. Let the mean image , and for each image in the data set form the difference image . Form a measurement matrix , and calculate the covariance matrix . The principal components of this data are the eigenvectors to the matrix C. The eigenvectors form an orthogonal basis for the original image set, accounting for the variation in the data in decreasing order, according to the corresponding eigenvalues. A dimensionality reduction is achieved by using instead of all eigenvectors only a subspace of say the first eigenvectors. For practical reasons usually , and the covariance matrix C will be rank deficient. We can then save computational effort by instead computing and using the p eigenvectors of L to form the m first eigenvectors of A by , where .

After a basis has been acquired (which for a particular agent typically only needs to be done once), any new image can be represented in this basis as an appearance vector

(1)

and a given can be transformed (with some quality loss) into a corresponding image by the inverse formula

(2)

### 2.1.2 Disparity based methods

Previous image based view synthesis methods have often been based on dense disparity maps. The objective has been to synthesize different viewpoints on a 1D viewing circle, or possibly a 2D viewing sphere [5, 6, 7]. An advantage of using disparity is that considerable research effort has been spent on developing good ``stereo'' vision algorithms, in which finding the disparity between the images in the two cameras is usually a crucial step. We base our method on an algorithm by Cox et al [4], which uses a Bayesian ML approach to do an image intensity based matching of image features searching along the epipolar lines in two images.

We cannot directly apply these disparity based methods. While in previous view synthesis work, image motion caused by a change in viewpoint is along an epipolar line, we study a multi DOF articulated agent, where image motion can occur in many directions. We have instead tried an approximate method, based on image rectification[6], aligning the scan lines in the rectified images with the major direction of motion. Disparity is then measured along this line only with the 1D disparity algorithm. This approach works relatively well for simulating our PUMA robot, where the rigid links are attached to each other, and often the major motion is caused by the two joints in the base, and the other joint motions can be seen as small corrections to this big motion.

Given the disparity map represented in a column vector, between the view at time k-1 and the view at time k, for the 1D case we experiment with, the appearance vector is chosen so that . This determines the set of up to a constant vector. The constant vector is set with reference to a special view, which for the on-line case is the current view, and for the off-line case is a small set of reference views. The inverse problem of getting the image back from the appearance vector, assuming no occlusion, is solvable given reference image (in the on-line simulation case discussed later, this is just the current rectified real image) and a desired change in visual appearance as

(3)

where the vectors are indexed in the corresponding 2-D image coordinate locations (i,j). The synthesized image is then ``unrectified'' into the original camera frame, and can be displayed.

## 2.2 On-Line visual-motor model estimation

The visual appearance vector and the agent's actions are related by an initially unknown visual motor model f, satisfying . At any time k we want to estimate a first order model , valid around the current system configuration , and described by the ``image''[12] or visual-motor Jacobian defined as

(4)

The image Jacobian not only relates visual changes to motor changes, as has been previously exploited in visual feedback control[10], but also highly constrains the possible visual changes to the set of possible solutions . Thus the Jacobian J is also a visual model, parameterized in exactly the same degrees of freedom our system can change in. (Typically so the solution set is only a small subspace of .)

In observing the moving agent we obtain the changes in visual appearance corresponding to a particular controller command . This is essentially a secant approximation of the derivative of f along the direction . We want to update the Jacobian in such a way so as to satisfy our most recent observation (secant condition):

(5)

The above condition is under determined, and a family of updating formulas, called the Broyden hierarchy, is defined as follows:

(6)

Where are different rank 1 matrices so the rank of the correction term is equal to the number of non-zero We choose an unsymmetric correction term:.

(7)

This is a rank 1 updating formula in the Broyden hierarchy. For a set of orthogonal movements about a point (7) is identical to a finite difference approximation , in coordinate frame rotated so that . Note however that our estimation accepts movements along arbitrary directions and thus is applicable also when the agent movement cannot be specifically and precisely tailored for the training, such as when learning the model of a biological (human) agent, or when the real agent can only be observed, not controlled.

Over a course of time our estimation method will generate a piecewise linear estimate of the visual-motor model. This is illustrated in fig.2. The size of the mesh element on which a particular Jacobian is used is determined using a trust region method [18]. Let be the trajectory length ( ) between Jacobian updates, and define a model agreement The trust region method adjusts adjust according to ( and are predefined bounds):

(8)

Figure 2: On-line piecewise linear function approximation.

## 2.3 View synthesis

We describe view synthesis in the off-line and on-line case. The off-line case is: Given a motor command sequence ( ) and a previously learned visual motor model of an agent, generate a movie of the agent performing the task. The on-line case is to generate arbitrary simulated views, representing (reasonably small) deviations from the current state of the real physical agent, while that same agent is executing some task and while learning and refining the model used to generate the synthesized views.

### 2.3.1 On-line view synthesis

In the on-line case we are currently observing, and possibly controlling the physical agent we want to be able to simulate. Assume observation of the visual motor system has been going on for at least n linearly independent moves , where n is the number of controlled DOF's of the agent. Then through observation of the process by the method in section 2.2 we have an estimate of the current visual motor Jacobian . Consider one step in an on-line algorithm. At time k we have current image , appearance vector , estimated visual motor Jacobian , and current agent state in motor space. Around this state the visual appearance change caused by a hypothetical movement can be predicted by:

(9)

The description of a system is easily cast in a telemanipulation application, where the tele-operator controls the, but for instance long delays, or limited bandwidth between the tele-operation site and the agent prevents immediate and/or full frame rate visual feedback to the operator. Instead we use the view synthesis method to generate the immediate visual feedback, and use the real visual feedback to calibrate the model used for the view synthesis. Note that for simplicity the synthesis and estimation processes are described synchronized here, but they can be asynchronous.

After some delay d, and possibly at a lower rate than full frame rate, the real image arrives. From it the real measured feature vector is extracted, The on-line method thus estimates, and uses successive piecewise linear models of the visual motor function, each estimated model valid around a particular physical motor state . How long a delay d we can tolerate depends on the validity range for our linear model represented in (8), which in turn depends on the visual-motor model of our system, and on the visual measures we choose.

### 2.3.2 Off-line view synthesis

In the off-line case we have a data set of images and a set of the corresponding agent control vectors . If sampled relatively densely ordering the set so allow us to use the on-line method to estimate the visual-motor model f. However, we assume more processing time is available in the off-line case, and one of several well known function approximation techniques can be used. We show experiments with (cubic) spline interpolation and weighted inverse distance interpolation. The latter is interesting in that manipulating the weight vector we can vary the ``smoothness'' of the estimate, and as shown in section 3.2 trade positioning accuracy for image sharpness.

(10)

The batch part of the off-line view synthesis is as follows:

During use desired action sequences are supplied, for instance by a tele-operator interacting with the ``virtual'' agent. They are made into images in a straightforward way:

Note that the task in the training data set and the motor space description of the desired simulated actions need not be related in any way, other than that the desired simulated actions should be from (a subset of) the workspace explored in the training. That is, the robot could have been doing another task in the training stage one being simulated, and the training task does not need to contain the poses of the simulation stage, only reasonably close ones.

Next: 3 Experiments Up: Model Free View Synthesis Previous: 1 Introduction

Martin Jägersand