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ICPR2000, Invited paper
Automated Generation of Representations in Vision Hans Knutsson
Mats Andersson Magnus Borga
Johan Wiklund
Computer Vision Laboratory Link¨oping University Sweden
[email protected]
is in the range 105 108. The situation is well exemplified by the medical field where new imaging techniques are consistently producing increasing amounts of data, e.g. high resolution CT and MR volumes and volume sequences. It is important here to note that the number of relevant examples at hand puts a hard limit on the attainable performance of a learning system and the higher the dimension of the input the more examples will be needed even for the best system possible. It is, for this reason, essential that adaptability is restricted and that learning is only attempted for that which can not be reasonably well modeled a priori. In the present context this implies that a competitive system will necessarily include state of the art, data independent, multidimensional signal processing techniques. The present work will typically include techniques presented in, for example, [1, 2, 3, 4]. For reasons mentioned above the research of the group has evolved to incorporate learning. This work has quickly received the attention of the research community, [5, 6]. A condensed account of the main ideas is given below. Our general approach for introducing learning is founded on the belief that adaptive model synthesis is the key to success. An in depth discussion of the fundamental topics involved can be found in [7] and [8]. In accordance with the above discussion the group has adopted a simple yet very powerful approach. Combining task specific preprocessing and canonical correlation analysis (CCA) has resulted in a number of novel and efficient methods, [7, 9, 10]. A related line of action is taken by other research groups and the interest in these ideas can be expected to increase in pace with reports of success, [11, 12, 13]. A good example is given by [14] where it is that methods based on CCA are far superior to traditional PCA for analyzing and visualizing differential multispectral satellite data. All this being said it should be apparent that, even if employing CCA certainly helps, the task of designing systems having an appropriate level of adaptability is still a major challenge.
Abstract This paper presents a general strategy for automated generation of efficient representations in vision. The approach is highly task oriented and what constitutes the relevant information is defined by a set of examples. The examples are pairs of situations that are dependent through the chosen feature but are otherwise independent. Particularly important concepts in the work are mutual information and canonical correlation. How visual operators and representations can be generated from examples are presented for a number of features, e.g. local orientation, disparity and motion. Interesting similarities to biological vision functions are observed. The results clearly demonstrates the potential of combining advanced filtering techniques and learning strategies based on canonical correlation analysis (CCA).
1 Introduction For high dimensional data relevant feature extractors are in practice impossible to design by hand due to the overpowering amount off possible signal combinations. It is common practice to handle high dimensional problems by reducing the dimensionality to typically < 10 by throwing away almost all available information in a, perhaps well-informed but, basically ad hoc manner. This approach is however likely to be unsuccessful if the mechanisms by which the necessary information can be extracted are not well understood and this is almost always the case for non-trivial spatio-temporal problems. For this reason designing adaptive systems, comprising mechanisms for learning, appears to be the only possible route to attain a major improvement in system performance. The need for a generally applicable method for learning relations in high dimensional signal spaces is particularly evident in problems involving images, image sequences and volumes where the dimensionality of the input data typically 1
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ICPR2000, Invited paper
2 Basic Considerations
obvious (and the correlation equals one) according to the second matrix below.
Model complexity A primary question in signal processing in general and image analysis in particular is: What is the appropriate model complexity for the analysis? A naive approach would indicate a need for an extreme complexity. Consider, for example, a visual input as small as a 5x5 window of 4 bit pixels giving 2100 1030 possible different situations. The following observation is, however, enlightening and somewhat reassuring. Suppose new neighborhoods can be attained at a rate of 10 MHz. It will take 1020 seconds, i.e. more than 3000 billion years, to see them all! This shows that the distribution of real world event samples will necessarily, even for relatively low dimensional systems, be very sparse indeed. There is simply not time enough for everything to happen. The number of actual events will leave the space almost empty.
01 00 BB0 00 BB0 00 BB0 00 BB0 25 B@0 25 0 25 : : : : : : :
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Constraint manifolds What then, if anything, can in general be said about the distribution of real world samples? Fortunately it turns out that there is, as a rule, a manifold defined by nonlinear constraints, on which the values are jointly restricted to lie [15]. Our hope is that it is possible to find a model generating structure adaptive enough to model only this manifold.
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Canonical correlation Relations of this kind, where linear combinations of parameters are correlated and a standard correlation analysis gives weak indications of correlation, can be found if we instead look for canonical correlations [16, 17]. Canonical correlation analysis (CCA) is base independent and finds correlations between inner products, i.e. Corr(xT wx ; yT wy ). In cases where the dimensionality prohibits the use of standard techniques for finding the solutions we have developed an efficient algorithm estimating only a few of the largest canonical correlations, [18].
Mutual information The most general measure of the interdependency between input and output is the amount of mutual information. However, all attempts to measure mutual information necessarily implies the use of a model. Assuming distributions that are continuous almost everywhere a system consisting of several local models can approximately find the global mutual information between the input and output spaces if each local model is valid and finds the mutual information in it’s region. For continuous signals locally linear models will suffice and for a linear system the mutual information is maximized when correlation is maximized. It is crucial, however that the local models are base independent. Standard correlation analysis will serve to exemplify this fact.
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Figure 1: A general approach for finding maximum mutual information. Finding suitable function classes and efficient implementations of these functions is a central issue.
3 Adaptive Unfolding
Base independent models Correlation analysis only measures correlations in fixed directions in the parameter space, i.e. the projections of the signal onto the basis vectors in a given coordinate system. Consider the following simple example, where the process is described by eight parameters: Suppose that a linear combination of the first four parameters correlates fully with a linear combination of the last four. In the coordinate system given by the process parameters, the correlation matrix looks like the first matrix below. It is hard to see that the first four parameters have much to do with the last four. However, if we make the analysis in a coordinate system where the two linear combinations define two of the basis vectors, the relation is
A broadly applicable general approach is to maximize mutual information subject to a particular set of models. Using linear models this is equivalent to CCA. If mapping functions are introduced, fx and fy in figure 1, more complex relations can be found by maximizing correlation between the mapped variables. This is equivalent to maximizing mutual information subject to constraints imposed by the chosen functions. The choice of fx and fy is of major importance as it intrinsically defines all the possible ways in which information can be unfolded by the canonical correlation analysis. 2
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ICPR2000, Invited paper
f
CCA
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Figure 4: Projections of Fourier components on canonical correlation vectors 1 to 8. The result shows that the generated representation of orientation information include angular operators of orders 2, 4, 6 and 8.
Figure 2: A symbolic illustration of the method of using CCA for finding feature detectors in images. The desired feature (here orientation: illustrated by a solid line) is varying equally in both image sequences while other features (here illustrated with dotted curves) vary independently.
the product of two output variables (of the same type). This dependence can not be seen by a linear analysis in the input space but will, of course, be obvious in the outer product space. c) The adaptive projection is obtained through CCA.
3.1 The expansion-projection paradigm The type of adaptive systems that will be investigated can be described as a sequence of three different types of operations: a) Dimensionality reduction b) Dimensionality expansion and c) Adaptive projection, not necessarily in that order. a) is typically very problem specific, e.g. chosing a neighborhood size, a set of filter outputs etc. In some cases the choice can be simple, e.g. when neighborhood size is given by the spatial extent of a particular feature. In general, however, making a good choice will require a substantial amount of task related knowledge. b) can also be done in a number of ways: 1. Expanding a scalar into several scalars by introducing channels, i.e. dividing the range of a scalar into parts and using one scalar per part as a new representation. 2. Expanding a number of scalars by producing an outer product. 3. Introducing any other set of basis functions that is believed to be useful in the problem at hand. The purpose of this step is to enable the adaptive linear projection to detect nonlinear relationships. As an example consider a situation where the product of two (uncorrelated, zero mean, symmetricly distributed) input variables is always equal to
4 Generating Representations The basic idea behind the automated representation generation approach is illustrated in figure 2. Two signals where the feature that is to be represented generates the dependency between the signal components are feed into the system and CCA is used to find the most informative subspace.
4.1 Local orientation It is shown in [7] and [19] that if f is an outer product and the image pairs contain sine wave patterns with equal orientations but different phase, the CCA finds linear combinations of the outer products that convey information about local orientation and are invariant to local phase. Figures 3, 4 and 5 show results from a similar experiment this time using image pairs of edges having equal orientation and different, independent positions. Independent white Gaussian noise to a level of 12 dB SNR was added to all images. Figure 3 shows the values of the 20 first canonical correlations. The values appear to come in pairs, the first two values being 0:98 demonstrating that the mutual information mediated through local orientation is high. Figure 4 show the projections of Fourier components on canonical correlation vectors 1 to 8. The result shows that angular operators of orders 2, 4, 6 and 8 has been formed and are important information carriers. The magnitude of the projections are close to shift-invariant having a position dependent variation in the order of 5 %. Comparing to figure 3 it can be seen that the decrease in the canonical correlation values corresponds to an increase in angular order of the operators.
1 Canonical correlation
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Figure 3: Plot of canonical correlation values.
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ICPR2000, Invited paper
Figure 5: Spectra of eigenimages interpreted as complex quadrature filter pairs. The 1:st and 3:rd spectrum is due to canonical correlation vector 1. The 2:nd and 4:th is due to canonical correlation vector 2.
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It is demonstrated that the algorithm can handle traditionally difficult problems such as: 1. Producing multiple disparity estimates for semi-transparent images, see figure 7, 2. Maintain accuracy at disparity edges, and 3. Allowing differently scaled images. Canonical correlation analysis is used to create adaptive linear combinations of quadrature filters. These linear combinations are new quadrature filters that are adapted in frequency response and spatial position maximizing the correlation between the filter outputs from the two images. The disparity estimate is obtained by analyzing the phase of the scalar product of the adapted filters. A result for depth estimates using semi-transparent images is shown in figure 7.
In this 1-dimensional experiment data was made dependent through size and position of a rectangular pulse. Pairs of signals were generated where size and position were similar but not exactly equal. The variation of both was 50% of the pulse width. Also in this case the functions fx and fy where outer products. Analyzing the result revealed that a family of filters similar to quadrature wavelets had been generated. The filters had different spatial AND Fourier domain locations enabling simultaneous representation of position and size.. Figure 6 shows three filter with the same Fourier location but different spatial position. 6
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‘Complex cells’ Performing an eigenvalue decomposition of the canonical correlation vectors the corresponding linear combinations, in the outer product space, can be seen as quadratic combinations of linear filters [7]. The linear filters (eigenimages) obtained display a clear tendency to form pairs of odd and even filters having similar spectra. Such quadrature filter pairs allow for a local shift-invariant feature and are functionally similar to the orientation selective ‘complex cells’ found in biological vision, [20, 21]. Figure 5 shows the spectra of four such filter pairs.
5 Learning Motion The final and most advanced example of automated representation generation will show how a representation of motion can be obtained. As have already been pointed out an appropriate preprocessing of data is in most critical for a good result. The present example involves a substantial amount of preprocessing and consists of feeding the input data through a quadrature filter bank. This type of preprocessing was chosen using the fact that quadrature filters were formed when learning orientation representation and that motion can be seen as orientation in a spatio-temporal space. The implementation was done using a particularly efficient technique that deserves a somewhat more detailed explanation before the learning part is described.
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Figure 6: Three of the quadrature ‘wavelets’ generated by the position/size experiment. Left: Magnitudes of the filters in the spatial domain. Right: Magnitude of the filters in the Fourier domain.
4.3 Local disparity An important problem in computer vision that is suitable to handle with CCA is stereo vision, since data in this case naturally appear in pairs. In [7] a novel stereo vision algorithm that combines CCA and phase analysis is presented.
5.1 Convolver network Convolver networks is a new and efficient approach for optimization and implementation of filter banks. The multi 4
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Figure 9: Partitioning of the Fourier domain.
The figures cover the range [ π=2; π=2] in the FD. Left: Stylized scale/orientation patches. Constructed by a sequence of a 1/2 octave scale change and a coupled rotation of 45 degrees. This has a direct correspondence to the structure of the convolver network used. Right: Plot of 6dB limit of the resulting filters.
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The Fourier weighting function W (u) provide an appropriate Fourier space metric. The choice of W (u) should if possible be done in the light of the expected spectra for the signal and the noise, see [4].
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Figure 8: Convolution network producing 8 quadrature filters spanning over 3 octaves using less than 150 real valued multiplications.
Convolver network optimization It is possible to optimize all filters on one level simultaneously with respect to the output filters and the current state of the network. This procedure is repeated for another layer and so on creating a sequential optimizer loop over the network structure by for each layer, n, computing the partial derivatives
layered structure of a convolver network enable a powerful decomposition of complex filters into simple filter components. Compared to a direct implementation a convolver network uses only a fraction of the coefficients to provide the same result. The individual filters of the network contain very few coefficients. Figure 8 depicts the convolver network used for learning motion. The nodes constitute summation points and the filters are located on the arcs connecting two consecutive layers. In fig. 8 the filters are represented by plain arcs or by a circle containing a number of 3-5 dots. The dots correspond to both the number of nonzero coefficients in the filter and the position (coordinates) for these coefficients. The definition of the filter network can be divided into structure and internal properties. The structure of the network are the properties that can be read out from a sketch like in fig. 8 i.e. the number of levels and the number of nodes on each level. The internal properties comprise the coordinates for the nonzero coefficients, the ideal filter function fm (u) and weight functions Wm (u) for each output node, where u are Fourier domain coordinates. The filter coefficients cn in the network are computed to minimize the weighted difference between the resulting and the ideal filter functions. min ε2 =
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The convergence of this approach cannot be guaranteed but for educated choices of filter structure and coefficient distribution the network converges to a stable solution which in terms of computational complexity outperforms a conventional implementation by orders of magnitude. For more details see [22]. A quadrature channel network The network in fig. 8 corresponds to the the partitioning of the Fourier domain illustrated to the left in fig. 9. The upper right branch in fig. 8 consists of sequential 1D LP-filters. The centered squares in fig. 9 illustrate the effect of this branch. Note that as this branch is traversed the distance between the coefficient are gradually increased i.e. the filters are becoming more sparse towards the lower levels. The eight first output nodes in fig. 8 are broadband quadrature filters. These filters are computed pairwise where the relative direction of the filters differ by π=2. Left in fig. 9 one filter from each such pair are visualized by (somewhat overlapping) shaded areas. (The last output node is a real
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ICPR2000, Invited paper
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Figure 10: Fourier domain magnitude of one of the eight
Figure 13: Estimated amount of mutual information con-
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valued LP-filter which is not used for learning motion but improves optimization convergence.) The network produces 8 quadrature filters using less than 150 real valued multiplications per pixel. Each quadrature filter is consequently obtained by the same effort as a direct implementation of a complex valued 3 3 filter. Figure 9 shows the 6dB limits of the resulting quadrature filters (and the LP-filter). Figure 10 shows the Fourier domain magnitude in detail of one of the eight quadrature filter produced by the convolver network.
5.2 Generation of motion representation The basic idea with this step is to feed filter output to the CCA in such a way that velocity is the only common information in the two inputs. Therefore the input is taken from two windows moving with the same velocity and in the same direction. Each window consists of 9 points i a square grid as illustrated in figure 11. The offset between the windows was chosen randomly every 500 step. The total training was 5000 steps. The steps were taken with a velocity vector that was randomly changed every step with a change of -1, 0 or 1 in each direction. The norm of the velocity vector was limited to 5 (pixels per step). For each step, the filter output from the filter network was multiplied by the conjugate of the corresponding filter output from the previous step:
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Figure 11: Generation of trainingdata.
(3)
where qi j (t ) is the output of filter i in position j at time t. 8 filters are used which means that Qi j (t ) can be represented as 72-dimensional complex vectors. These vectors are then normalized and divided into a real and an imaginary part. The resulting pair of a 144-dimensional vector from each window is then used as input to the CCA. The resulting canonical correlations are plotted in figure 12. The solid line shows the random correlation estimates for white Gaussian noise of the same dimensionality and the same number of samples. the correlations can be expressed as mutual information according to:
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To compensate for the random correlations the difference between the total estimated mutual information and the “virtual” mutual information due to the limited set of samples was calculated. This difference is plotted in figure 13. As can be seen, there are 9 canonical correlations contributing
Figure 12: Plot of the canonical correlations obtained in the motion experiment.
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15 we see that the CCA has generated a three-dimensional representation of the velocity approximately on the surface of a sphere. In the first two dimensions, the direction is represented, while the magnitude of the velocity also uses the third dimension. A stylized version of the generated 3D representation is shown in figure 16.
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Figure 16: A stylized version of the generated 3D representation of motion.
Figure 14: Projections of motion induced signals onto the first two canonical vectors. The true direction of motion is indicated with an arrow.
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to the mutual information conveyed by motion. The total mutual information conveyed is approximately 6.5 bits. To visualize the resulting velocity representation, the filter response vectors for different velocities were projected onto the three most significant CCA-vectors (corresponding to 4.3 bits of mutual information). In figure 14, the projections onto the first two vectors are plotted and at each point, the true direction of motion is indicated with an arrow. The clusters in the middle of the figure are caused by the quantization of the velocity in whole pixels per time unit. Figure 15 shows the projections onto the first and the third CCA-vector. Here, also the magnitude of the true velocity is indicated as the length of the arrow. From figures 14 and
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proach for finding mutual information.
6 Conclusion The concept of mutual information provides a solid and general basis for the study of a broad spectrum of problems. Using a simple strategy of nonlinear preprocessing followed by canonical correlation analysis has been shown to be a very powerful and efficient way for automatic generation of operators and representations in vision. It is likely that adopting an even more general approach, illustrated in figure 17, will prove to be useful. The added generality is here obtained by introducing parametric functions fx and fy and performing a simultaneous CCA and search in the parameter space. Finding suitable function classes and efficient parameterization/implementations for these functions will then be the central issue and an important theme in our continued investigations. The most important outcome of the present automated representation generation approach is perhaps not the solutions to a number of different problems in vision but a better understanding of how, and on what grounds, the above choices should be made.
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Figure 17: A more powerful, but also more complex, ap-
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Figure 15: Projections of motion induced signal onto the first and the third CCA-vector. The true velocity is indicated as arrows.
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Acknowledgment
[10] M. Hemmendorff. Single and Multiple Motion Field Estimation. Lic. Thesis LiU-Tek-Lic-1999:22, Dept. EE, Link¨oping University, SE-581 83 Link¨oping, Sweden, April 1999. Thesis No. 764, ISBN 91-7219-4782.
We like to thank The Swedish National Board for Industrial and Technical Development (NUTEK), The Swedish Research Council for Engineering Sciences (TFR), The Swedish National Science Research Council (NFR) and The Swedish Foundation for Strategic Research (SSF).
[11] W. M. Wells, P. Viola, H. Atsumi, S. Nakajima, and R. Kikinis. Multi-modal volume registration by maximization of mutual information. Medical Image Analysis, (1):35–51, 1996.
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