# An Iterative Image Registration Technique with an Application to Stereo Vision

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2 possible values of h is of size MxM, then this method requires 0(M 2 N 2 ) time to compute. SpeeduD at the risk of Dossible failure can be achieved by using a hill climbing technique This technique begins with an initial estimate h 0 of the disparity To obtain the next guess from the current guess h k, one evaluates the difference function at all points in a small (say, 3x3) neighborhood of h k and takes as the next guess h k+1 that point that minimizes the difference function. Another technique, known as the sequential similarity detection algorithm (SSDA) [2], only estimates the error for each disparity vector h. In SSDA, the error function must be a cumulative one such as the L 1 or L 2 norm. One stops accumulating the error for the current h under investigation when it becomes apparent that the current h is not likely to give the best match. Some registration algorithms use a coarse fine search strategy. See 6] for an example. One of the techniques discussed above is used to find the best registration for the images at low resolution, and the low resolution match is then used to constrain the region of possible matches examined at higher resolution. The algorithm we present specifies the order in which to search the space of possible h's In particular, our technique starts with an initial estimate of h, and it uses the spatial intensity gradient at each point of the image to modify the current estimate of h to obtain an h that yields a better match This process is repeated in a kind of NewtonRaphson iteration. If the iteration converges, it will do so in 0(M 2 log N) steps on the average. This registration technique can be combined with a coarse fine strategy, since it requires an initial estimate of the approximate disparity h. 4. The registration algorithm In this section we first derive an intuitive solution to the one dimensional registration problem, and then we derive an alternative solution which we generalize to multiple dimensions. We then show how our technique generalizes to other kinds of registration. We also discuss implementation and performance of the algorithm One dimensional case In the one dimensional registration problem, we wish to find the horizontal disparity h between two curves F(x) and G(x) = F{x + h). This is illustrated in figure 2. Our solution to this problem depends on a linear approximation to the behavior of F{x) in the neighborhood of x, as do all subsequent solutions in this paper. In particular, for small h, The success of our algorithm requires h tn be small enouqh that this approximation is adequate. In section 4.3 we will show how to extend the range of h's over which this approximation is adequate by smoothing the images. The approximation to h given in (2) depends on x. A natural method tor combining the various estimates of h at various values of x would be to simply average them: We can improve this average by realizing that the linear approximation in (1) is good where F{x) is nearly linear, and conversely is worse where F"(x) is large. Thus we could weight the contribution of each term to the average in (3) in inverse proportion to an estimate of F"(x). One such estimate is Since our estimate is to be used as a weight in an average, we can drop the constant factor of 1/h in (4), and use as our weighting function This appeals to our intuition; for example, in figure 2, where the two curves cross, the estimate of h provided by (2) is 0, which is bad; fortunately, the weight given to this estimate in the average is small, since the difference between F'(x) and G'(x) at this point is large. The average with weighting is (3) (4) (5) (6) Having obtained this estimate, we can then move F(x) by our estimate of h, and repeat this procedure, yielding a type of Newton Raphson iteration. Ideally, our sequence of estimates of h will converge to the best h. This iteration is expressed by (7) 675

3 4.2. An alternative derivation The derivation given above does not generalize well to two dimensions because the two-dimensional linear approximation occurs in a different form. Moreover, (2) is undefined where F'(x) 0, i.e. where the curve is level. Both of these problems can be corrected by using the linear approximation of equation (1) in the form to find the h that minimizes the L 2 norm measure of the difference between the curves (8) which can be accomplished by smoothing the image. The tradeoff is that smoothing suppresses small details, and thus makes the match less accurate. If the smoothing window is much larger than the size of the object that we are trying to match, the object may be suppressed entirely, and so no match will be possible. Since lowpass-filtered images can be sampled at lower resolution with no loss of information, the above observation suggests that we adopt a coarse-fine strategy. We can use a low resolution smoothed version of the image to obtain an approximate match. Applying the algorithm to higher resolution images will refine the match obtained at lower resolution. While the effect of smoothing is to extend the range of convergence, the weighting function serves to improve the accuracy of the approximation, and thus to speed up the convergence Without weighting, i.e. with w(x) = 1. the calculated disparity h 1 of the first iteration of (10) with F{x) = sin x falls off to zero as the disparity approaches one half wavelength. However, with w(x) as in (5), the calculation of disparity is much more accurate, and only falls off to zero at a disparity very near one half wavelength Thus with w(x) as in (5) convergence is faster for large disparities 4.4. Implementation (9) Implementing (10) requires calculating the weighted sums of the quantities F'G, F'F, and (f') 2 over the region of interest R. We cannot calculate F'{x) exactly, but lor the purposes of this algorithm, we can estimate it by This is essentially the same solution that we derived in (6), but with the weighting function w(x) = F'(x) 2. As we will see, the form of the linear approximation we have used here generalizes to two or more dimensions. Moreover, we have avoided the problem of dividing by 0, since in (9) we will divide by 0 only if F'(*) = 0 everywhere (in which case h really is undefined), whereas in (3) we will divide by 0 if F'(x) = 0 anywhere. The iterative form with weighting corresponding to (7) is F'(x) ~ F{x + Ax)- F(x) and similarly for G'(x), where we choose Ax appropriately small (e.g. one pixel) Some more sophisticated technique could be used for estimating the first derivatives, but in general such techniques are equivalent to first smoothing the function, which we have proposed doing for other reasons, and then taking the difference Generalization to multiple dimensions The one-dimensional registration algorithm given above can be generalized to two or more dimensions We wish to minimize the 676

4 views, due to the difference of the viewpoints of the cameras or to differences in the processing of the two images. If we model this difference as a linear transformation, we have (ignoring the registration problem for the moment) where may be thought of as a contrast adjustment and as a brightness adjustment. Combining this with the general linear transformation registration problem, we obtain The discussions above of iteration, weighting, smoothing, and the coarse fine technique with respect to the one dimensional case apply to the n dimensional case as well. Calculating our estimate of h in the two dimensional case requires accumulating the weighted sum of five products ((G - F)F K, (G - F)F y, F 2 x, F 2 r and F XF ) over the region R, as opposed to accumulating one product for correlation or the L 2 norm. However, this is more than compensated for, especially in high resolution images, by evaluating these sums at fewer values of h Further generalizations Our technique can be extended to registration between two images related not by a simple translation, but by an arbitrary linear transformation, such as rotation, scaling, and shearing. Such a relationship is expressed by where A is a matrix expressing the linear spatial transformation between f(x) and G(x). The quantity to be minimized here is To determine the amount AA to adjust A and the amount Ah to adjust h, we use the linear approximation (11) When we use this approximation the error expression again becomes quadratic in the quantities to be minimized with respect to Differentiating with respect to these quantities and setting the results equal to zero yields a set of linear equations to be solved simultaneously. This generalization is useful in applications such as stereo vision, where the two different views of the object will be different because of the difference in the viewpoints of the cameras. Linear transformations account for the distortion of planar objects assuming orthographic projection; this is often an adequate approximation to the local distortions of arbitrary objects under the actual perspective projection. A further generalization is possible to account for the fact that the brightness values of the pixels may be different in the two as the quantity to minimize with respect to and h. The minimization of this quantity, using the linear approximation in equation (11), is straightforward. This is the general form promised in section 2. If we ignore A, minimizing this quantity is equivalent to maximizing the correlation coefficient (see, for example, [3]); if we ignore a and B as well, minimizing this form is equivalent to minimizing the L 2 norm. 5. Application to stereo vision In this section we show how the generalized registration algorithm described above can be applied to extracting depth information from stereo images The stereo problem The problem of extracting depth information from a stereo pair has in principle four components: finding objects in the pictures, matching the objects in the two views, determining the camera parameters, and determining the distances from the camera to the objects. Our approach is to combine object matching with solving for the camera parameters and the distances of the objects by. using a form of the fast registration technique described above. Techniques for locating objects include an interest operator [6], zero crossings in bandpass-filtered images [5], and linear features [1]. One might also use regions found by an image segmentation program as objects. Stereo vision systems that work with features at the pixel level can use one of the registration techniques discussed above. Systems whose objects are higher level features must use some difference measure and some search technique suited to the particular feature being used. Our registration algorithm provides a stereo vision system with a fast method of doing pixel-level matching. Many stereo vision systems concern themselves only with calculating the distances to the matched objects. One must also be aware that in any real application of stereo vision the relative positions of the cameras will not be known with perfect accuracy. Gennery [4] has shown how to simultaneously solve for the camera parameters and the distances of objects A mathematical characterization The notation we use is illustrated in figure 3. Let c be the vector of camera parameters that describe the orientation and position of 677

5 camera 2 with respect to camera 1s coordinate system. These parameters are azimuth, elevation, pan, tilt, and roll, as defined in [4]. Let x denote the position of an image in the camera 1 film plane of an object. Suppose the object is at a distance z from camera 1. Given the position in picture 1 x and distance z of the object, we could directly calculate the position p(x,z) that it must have occupied in three space. We express p with respect to camera 1 's coordinate system so that p does not depend on the orientation of camera 1. The object would appear on camera 2's film plane at a position q(p,c) that is dependent on the object's position in three-space p and on the camera parameters c. Let G(x) be the intensity value of pixel x in picture 1, and let F(q) the intensity value of pixel q in picture 2. The goal of a stereo vision system is to invert the relationship described above and solve for c and z, given x, F and G Applying the registration algorithm First consider the case where we know the exact camera parameters c, and we wish to discover the distance z of an object. Suppose we have an estimate of the distance z. We wish to see what happens to the quality of our match between F and G as we vary z by an amount Az. The linear approximation that we use here is (12) This equation is due to the chain rule of the gradient operator; 3q/3p is a matrix of partial derivatives of the components of q with respect to the components of p, and df/dq is the spatial intensity gradient of the image F(q). To update our estimate of z, we want to find the Az that satisfies We have implemented the technique described above in a system that functions well under human supervision. Our program is capable of solving for the distnaces to the objects, the five camera paprameters described above, and a brightness and contrast parameter for the entire scene, or any subset of these parameters. As one would expect from the discussion in section 4.3, the algorithm will converge to the correct distances and camera parameters when the initial estimates of the z,'s and c are sufficiently accurate that we know the position in the camera 2 film plane of each object to within a distance on the order of the size of the object. A session with this program is illustrated in figures 4 through 8 The original stereo pair is presented in figure 4 (Readers who can view stereo pairs cross eyed will want to hold the pictures upside down so that each eye receives the correct view) The camera parameters were determined independently by hand-selecting matching points and solving for the parameters using the program described in [4]. Figures 5 and 6 are bandpass filtered versions of the pictures in figure 4. Bandpass filtered images are preferred to lowpass filtered images in finding matches because very low spatial frequencies tend to be a result of shading differences and carry no (or misleading) depth information. The two regions enclosed in rectangles in the left view of figure 5 have been hand selected and assigned an initial depth of 7.0 in units of the distance between cameras. If these were the actual depths, the corresponding objects would be found in the right view at the positions indicated figure 5. After seven depth adjustment iterations, the program found the matches shown in figure 6. The distances are 6.05 for object 1 and 5 86 for object

6 Figures 7 and 8 are band pass-filtered with a band one octave higher than figures 5 and 6. Five new points have been handselected in the left view, reflecting the different features that have become visible in this spatial frequency range. Each has been assigned an initial depth equal to that found for the corresponding larger region in figure 6. The predicted position corresponding to these depths is shown in the right view of figure 7. After five depth-adjustment iterations, the matches shown in figure 8 were found. The corresponding depths are 5.96 for object 1, 5.98 for object 2,5.77 for object 3, 5.76 for object 4, and 6.09 for object Future research The system that we have implemented at present requires considerable hand-guidance. The following are the issues we intend to investigate toward the goal of automating the process. Providing initial depth estimates for objects: one should be able to use approximate depths obtained from low-resolution images to provide initial depth estimates for nearby objects visible only at higher resolutions. Constructing a death map: one could build a depth map from depth measurements by some interpolation method. Selecting points of interest: the various techniques mentioned in section 3 should be explored. Tracking sudden depth changes: the sudden depth changes found at the edges of objects require some set of higher-level heuristics to keep the matching algorithm on track at object boundaries. Compensating for the different appearances of objects in the two views: the general form of the matching algorithm that allows for arbitrary linear transformations should be useful here. References 1. Baker, H. Harlyn. Edge Based Stereo Correlation. Proceedings: Image Understanding Workshop, DARPA, April, 1980, pp Barnea, Daniel I. and Silverman, Harvey F. "A Class of Algorithms for Fast Digital Image Registration.** IEEE Transactions on Computers C-21,2 (February 1972), Dudewicz, Edward J.. Introduction to Statistics and Probability. Holt, Rinehart and Winston, New York, Gennery, Donald B. Stereo-Camera Calibration. Proceedings: Image Understanding Workshop, DARPA, November, 1979, pp Marr, D. and Pogglo, T. "A Computational Theory of Human Stereo Vision." Proceedings of the Royal Society of London (1979), Moravec, Hans P. Obstacle Avoidance and Navigation in the Real World by a Seeing Robot Rover. Tech. Rept. CMURI-TR-3, Carnegie-Mellon University Robotics Institute, September,

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