Collisions on inland waters frequently occur due to the absence of clearly defined navigation guidance. To prevent such accidents, optical sensors such as stereo cameras can be employed to detect obstacles at an early stage, enabling timely warnings to vessel operators or the planning of collision avoidance trajectories. In this context, the paper presents a neural network for object detection on inland waters, designed to address the specific challenges of waterborne vehicles, including strong ego-motion and contextual cues like the shoreline appearing in the background. The proposed network leverages a plane sweep approach to integrate multiple views of a scene and predict object locations in bird’s-eye view (BEV) coordinates. Its ability to incorporate more than two camera perspectives is demonstrated using the KITTI dataset. On real-world inland water data, the network is evaluated against both a traditional maritime stereo-based method and a learning-based stereo detector from autonomous driving, showing consistently higher mean average precision. The code is available at https://github.com/dionysos4/IWOD.

Autonomous navigation on inland waters requires an accurate understanding of the environment in order to react to possible obstacles. Deep learning is a promising technique to detect obstacles robustly. However, supervised deep learning models require large data-sets to adjust their weights and to generalize to unseen data. Therefore, we equipped our research vessel with a laser scanner and a stereo camera to record a novel obstacle detection data-set for inland waters. We annotated 1974 stereo images and lidar point clouds with 3d bounding boxes. Furthermore, we provide an initial approach and a suitable metric2to compare the results on the test data-set. The data-set is publicly available3and seeks to make a contribution towards increasing the safety on inland waters.

One-class incremental learning is a special case of class-incremental learning, where only a single novel class is incrementally added to an existing classifier instead of multiple classes. This case is relevant in industrial defect detection scenarios, where novel defects usually appear during operation. Existing rolled-out classifiers must be updated incrementally in this scenario with only a few novel examples. In addition, it is often required that the base classifier must not be altered due to approval and warranty restrictions. While simple finetuning often gives the best performance across old and new classes, it comes with the drawback of potentially losing performance on the base classes (catastrophic forgetting [1]). Simple prototype approaches [2] work without changing existing weights and perform very well when the classes are well separated but fail dramatically when not. In theory, null-space training (NSCL) [3] should retain the basis classifier entirely, as parameter updates are restricted to the null space of the network with respect to existing classes. However, as we show, this technique promotes overfitting in the case of one-class incremental learning. In our experiments, we found that unconstrained weight growth in null space is the underlying issue, leading us to propose a regularization term (R-NSCL) that penalizes the magnitude of amplification. The regularization term is added to the standard classification loss and stabilizes null-space training in the one-class scenario by counteracting overfitting. We test the method’s capabilities on two industrial datasets, namely AITEX and MVTec, and compare the performance to state-of-the-art algorithms for class-incremental learning.

Motion estimation is an essential element for autonomous vessels. It is used e.g. for lidar motion compensation as well as mapping and detection tasks in a maritime environment. Because the use of gyroscopes is not reliable and a high performance inertial measurement unit is quite expensive, we present an approach for visual pitch and roll estimation that utilizes a convolutional neural network for water segmentation, a stereo system for reconstruction and simple geometry to estimate pitch and roll. The algorithm is validated on a novel, publicly available dataset22https://git.ios.htwg-konstanz.de/dgriesse/constance_orientation_dataset/archive/main/constance_orientation_dataset-main.zip recorded at Lake Constance. Our experiments show that the pitch and roll estimator provides accurate results in comparison to an Xsens IMU sensor. We can further improve the pitch and roll estimation by sensor fusion with a gyroscope. The algorithm is available in its implementation as a ROS node33https://github.com/dionysos4/water_surface_detector.

We are interested in computing a mini-batch-capable end-to-end algorithm to identify statistically independent components (ICA) in large scale and high-dimensional datasets. Current algorithms typically rely on pre-whitened data and do not integrate the two procedures of whitening and ICA estimation. Our online approach estimates a whitening and a rotation matrix with stochastic gradient descent on centered or uncentered data. We show that this can be done efficiently by combining Batch Karhunen-Löwe-Transformation [1] with Lie group techniques. Our algorithm is recursion-free and can be organized as feed-forward neural network which makes the use of GPU acceleration straight-forward. Because of the very fast convergence of Batch KLT, the gradient descent in the Lie group of orthogonal matrices stabilizes quickly. The optimization is further enhanced by integrating ADAM [2], an improved stochastic gradient descent (SGD) technique from the field of deep learning. We test the scaling capabilities by computing the independent components of the well-known ImageNet challenge (144 GB). Due to its robustness with respect to batch and step size, our approach can be used as a drop-in replacement for standard ICA algorithms where memory is a limiting factor.

Multi-Dimensional Connectionist Classification is a method for weakly supervised training of Deep Neural Networks for segmentation-free multi-line offline handwriting recognition. MDCC applies Conditional Random Fields as an alignment function for this task. We discuss the structure and patterns of handwritten text that can be used for building a CRF. Since CRFs are cyclic graphical models, we have to resort to approximate inference when calculating the alignment of multi-line text during training, here in the form of Loopy Belief Propagation. This work concludes with experimental results for transcribing small multiline samples from the IAM Offline Handwriting DB which show that MDCC is a competitive methodology.

In this paper we present a method using deep learning to compute parametrizations for B-spline curve approximation. Existing methods consider the computation of parametric values and a knot vector as separate problems. We propose to train interdependent deep neural networks to predict parametric values and knots. We show that it is possible to include B-spline curve approximation directly into the neural network architecture. The resulting parametrizations yield tight approximations and are able to outperform state-of-the-art methods.

Offline handwriting recognition systems often use LSTM networks, trained with line- or word-images. Multi-line text makes it necessary to use segmentation to explicitly obtain these images. Skewed, curved, overlapping, incorrectly written text, or noise can lead to errors during segmentation of multi-line text and reduces the overall recognition capacity of the system. Last year has seen the introduction of deep learning methods capable of segmentation-free recognition of whole paragraphs. Our method uses Conditional Random Fields to represent text and align it with the network output to calculate a loss function for training. Experiments are promising and show that the technique is capable of training a LSTM multi-line text recognition system.

Offline handwriting recognition systems often include a decoding step, that is retrieving the most likely character sequence from the underlying machine learning algorithm. Decoding is sensitive to ranges of weakly predicted characters, caused e.g. by obstructions in the scanned document. We present a new algorithm for robust decoding of handwriting recognizer outputs using character n-grams. Multidimensional hierarchical subsampling artificial neural networks with Long-Short-Term-Memory cells have been successfully applied to offline handwriting recognition. Output activations from such networks, trained with Connectionist Temporal Classification, can be decoded with several different algorithms in order to retrieve the most likely literal string that it represents. We present a new algorithm for decoding the network output while restricting the possible strings to a large lexicon. The index used for this work is an n-gram index with tri-grams used for experimental comparisons. N-grams are extracted from the network output using a backtracking algorithm and each n-gram assigned a mean probability. The decoding result is obtained by intersecting the n-gram hit lists while calculating the total probability for each matched lexicon entry. We conclude with an experimental comparison of different decoding algorithms on a large lexicon.

Echolocating bats use the echoes from their echolocation calls to perceive their surroundings. The ability to use these continuously emitted calls, whose main function is not communication, for recognition of individual conspecifics might facilitate many of the social behaviours observed in bats. Several studies of individual-specific information in echolocation calls found some evidence for its existence but did not quantify or explain it. We used a direct paradigm to show that greater mouse-eared bats (Myotis myotis) can easily discriminate between individuals based on their echolocation calls and that they can generalize their knowledge to discriminate new individuals that they were not trained to recognize. We conclude that, despite their high variability, broadband bat-echolocation calls contain individual-specific information that is sufficient for recognition. An analysis of the call spectra showed that formant-related features are suitable cues for individual recognition. As a model for the bat's decision strategy, we trained nonlinear statistical classifiers to reproduce the behaviour of the bats, namely to repeat correct and incorrect decisions of the bats. The comparison of the bats with the model strongly implies that the bats are using a prototype classification approach: they learn the average call characteristics of individuals and use them as a reference for classification.

A critical step on the way to understanding a sensory system is the analysis of the input it receives. In this work we examine the statistics of natural complex echoes, focusing on vegetation echoes. Vegetation echoes constitute a major part of the sensory world of more than 800 species of echolocating bats and play an important role in several of their daily tasks. Our statistical analysis is based on a large collection of plant echoes acquired by a biomimetic sonar system. We explore the relation between the physical world (the structure of the plant) and the characteristics of its echo. Finally, we complete the story by analyzing the effect of the sensory processing of both the echolocation and the auditory systems on the echoes and interpret them in the light of information maximization. The echoes of all different plant species we examined share a surprisingly robust pattern that was also reproduced by a simple Poisson model of the spatial reflector arrangement. The fine differences observed between the echoes of different plant species can be explained by the spatial characteristics of the plants. The bat's emitted signal enhances the most informative spatial frequency range where the species-specific information is large. The auditory system filtering affects the echoes in a similar way, thus enhancing the most informative spatial frequency range even more. These findings suggest how the bat's sensory system could have evolved to deal with complex natural echoes.

Classification of plants according to their echoes is an elementary component of bat behavior that plays an important role in spatial orientation and food acquisition. Vegetation echoes are, however, highly complex stochastic signals: from an acoustical point of view, a plant can be thought of as a three-dimensional array of leaves reflecting the emitted bat call. The received echo is therefore a superposition of many reflections. In this work we suggest that the classification of these echoes might not be such a troublesome routine for bats as formerly thought. We present a rather simple approach to classifying signals from a large database of plant echoes that were created by ensonifying plants with a frequency-modulated bat-like ultrasonic pulse. Our algorithm uses the spectrogram of a single echo from which it only uses features that are undoubtedly accessible to bats. We used a standard machine learning algorithm (SVM) to automatically extract suitable linear combinations of time and frequency cues from the spectrograms such that classification with high accuracy is enabled. This demonstrates that ultrasonic echoes are highly informative about the species membership of an ensonified plant, and that this information can be extracted with rather simple, biologically plausible analysis. Thus, our findings provide a new explanatory basis for the poorly understood observed abilities of bats in classifying vegetation and other complex objects.

In this paper, we use large neighborhood Markov random fields to learn rich prior models of color images. Our approach extends the monochromatic Fields of Experts model (Roth & Black, 2005a) to color images. In the Fields of Experts model, the curse of dimensionality due to very large clique sizes is circumvented by parameterizing the potential functions according to a product of experts. We introduce simplifications to the original approach by Roth and Black which allow us to cope with the increased clique size (typically 3x3x3 or 5x5x3 pixels) of color images. Experimental results are presented for image denoising which evidence improvements over state-of-the-art monochromatic image priors.

This paper addresses the bottom-up influence of local image information on human eye movements. Most existing computational models use a set of biologically plausible linear filters, e.g., Gabor or Difference-of-Gaussians filters as a front-end, the outputs of which are nonlinearly combined into a real number that indicates visual saliency. Unfortunately, this requires many design parameters such as the number, type, and size of the front-end filters, as well as the choice of nonlinearities, weighting and normalization schemes etc., for which biological plausibility cannot always be justified. As a result, these parameters have to be chosen in a more or less ad hoc way. Here, we propose to learn a visual saliency model directly from human eye movement data. The model is rather simplistic and essentially parameter-free, and therefore contrasts recent developments in the field that usually aim at higher prediction rates at the cost of additional parameters and increasing model complexity. Experimental results show that--despite the lack of any biological prior knowledge--our model performs comparably to existing approaches, and in fact learns image features that resemble findings from several previous studies. In particular, its maximally excitatory stimuli have center-surround structure, similar to receptive fields in the early human visual system.

Volterra and Wiener series are perhaps the best-understood nonlinear system representations in signal processing. Although both approaches have enjoyed a certain popularity in the past, their application has been limited to rather low-dimensional and weakly nonlinear systems due to the exponential growth of the number of terms that have to be estimated. We show that Volterra and Wiener series can be represented implicitly as elements of a reproducing kernel Hilbert space by using polynomial kernels. The estimation complexity of the implicit representation is linear in the input dimensionality and independent of the degree of nonlinearity. Experiments show performance advantages in terms of convergence, interpretability, and system sizes that can be handled.

In recent years, Kernel Principal Component Analysis (KPCA) has been suggested for various image processing tasks requiring an image model such as, e.g., denoising or compression. The original form of KPCA, however, can be only applied to strongly restricted image classes due to the limited number of training examples that can be processed. We therefore propose a new iterative method for performing KPCA, the Kernel Hebbian Algorithm which iteratively estimates the Kernel Principal Components with only linear order memory complexity. In our experiments, we compute models for complex image classes such as faces and natural images which require a large number of training examples. The resulting image models are tested in single-frame super-resolution and denoising applications. The KPCA model is not specifically tailored to these tasks; in fact, the same model can be used in super-resolution with variable input resolution, or denoising with unknown noise characteristics. In spite of this, both super-resolution and denoising performance are comparable to existing methods.

The computation of classical higher-order statistics such as higher-order moments or spectra is difficult for images due to the huge number of terms to be estimated and interpreted. We propose an alternative ap- proach in which multiplicative pixel interactions are described by a se- ries of Wiener functionals. Since the functionals are estimated implicitly via polynomial kernels, the combinatorial explosion associated with the classical higher-order statistics is avoided. First results show that image structures such as lines or corners can be predicted correctly, and that pixel interactions up to the order of five play an important role in natural images. Most of the interesting structure in a natural image is characterized by its higher-order statistics. Arbitrarily oriented lines and edges, for instance, cannot be described by the usual pairwise statistics such as the power spectrum or the autocorrelation function: From knowing the intensity of one point on a line alone, we cannot predict its neighbouring intensities. This would require knowledge of a second point on the line, i.e., we have to consider some third-order statistics which describe the interactions between triplets of points. Analogously, the prediction of a corner neighbourhood needs at least fourth-order statistics, and so on. In terms of Fourier analysis, higher-order image structures such as edges or corners are described by phase alignments, i.e. phase correlations between several Fourier components of the image. Classically, harmonic phase interactions are measured by higher-order spectra [4]. Unfortunately, the estimation of these spectra for high-dimensional signals such as images involves the estimation and interpretation of a huge number of terms. For instance, a sixth-order spectrum of a 1616 sized image contains roughly 1012 coefficients, about 1010 of which would have to be estimated independently if all symmetries in the spectrum are considered. First attempts at estimating the higher-order structure of natural images were therefore restricted to global measures such as skewness or kurtosis [8], or to submanifolds of fourth-order spectra [9]. Here, we propose an alternative approach that models the interactions of image points in a series of Wiener functionals. A Wiener functional of order n captures those image components that can be predicted from the multiplicative interaction of n image points. In contrast to higher-order spectra or moments, the estimation of a Wiener model does not require the estimation of an excessive number of terms since it can be computed implicitly via polynomial kernels. This allows us to decompose an image into components that are characterized by interactions of a given order. In the next section, we introduce the Wiener expansion and discuss its capability of model- ing higher-order pixel interactions. The implicit estimation method is described in Sect. 2, followed by some examples of use in Sect. 3. We conclude in Sect. 4 by briefly discussing the results and possible improvements. 1 Modeling pixel interactions with Wiener functionals For our analysis, we adopt a prediction framework: Given a d d neighbourhood of an image pixel, we want to predict its gray value from the gray values of the neighbours. We are particularly interested to which extent interactions of different orders contribute to the overall prediction. Our basic assumption is that the dependency of the central pixel value y on its neighbours xi, i = 1, . . . , m = d2 - 1 can be modeled as a series y = H0[x] + H1[x] + H2[x] + + Hn[x] + (1) of discrete Volterra functionals m m H0[x] = h0 = const. and Hn[x] = h(n) x . . . x . (2) i i i i 1 ...i 1 n n 1 =1 i =1 n Here, we have stacked the grayvalues of the neighbourhood into the vector x = (x1, . . . , xm) Rm. The discrete nth-order Volterra functional is, accordingly, a linear combination of all ordered nth-order monomials of the elements of x with mn coefficients h(n) . Volterra functionals provide a controlled way of introducing multiplicative inter- i1...in actions of image points since a functional of order n contains all products of the input of order n. In terms of higher-order statistics, this means that we can control the order of the statistics used since an nth-order Volterra series leads to dependencies between maximally n + 1 pixels. Unfortunately, Volterra functionals are not orthogonal to each other, i.e., depending on the input distribution, a functional of order n generally leads to additional lower-order interac- tions. As a result, the output of the functional will contain components that are proportional to that of some lower-order monomials. For instance, the output of a second-order Volterra functional for Gaussian input generally has a mean different from zero [5]. If one wants to estimate the zeroeth-order component of an image (i.e., the constant component created without pixel interactions) the constant component created by the second-order interactions needs to be subtracted. For general Volterra series, this correction can be achieved by de- composing it into a new series y = G0[x] + G1[x] + + Gn[x] + of functionals Gn[x] that are uncorrelated, i.e., orthogonal with respect to the input. The resulting Wiener functionals1 Gn[x] are linear combinations of Volterra functionals up to order n. They are computed from the original Volterra series by a procedure akin to Gram-Schmidt or- thogonalization [5]. It can be shown that any Wiener expansion of finite degree minimizes the mean squared error between the true system output and its Volterra series model [5]. The orthogonality condition ensures that a Wiener functional of order n captures only the component of the image created by the multiplicative interaction of n pixels. In contrast to general Volterra functionals, a Wiener functional is orthogonal to all monomials of lower order [5]. So far, we have not gained anything compared to classical estimation of higher-order mo- ments or spectra: an nth-order Volterra functional contains the same number of terms as 1Strictly speaking, the term Wiener functional is reserved for orthogonal Volterra functionals with respect to Gaussian input. Here, the term will be used for orthogonalized Volterra functionals with arbitrary input distributions. the corresponding n + 1-order spectrum, and a Wiener functional of the same order has an even higher number of coefficients as it consists also of lower-order Volterra functionals. In the next section, we will introduce an implicit representation of the Wiener series using polynomial kernels which allows for an efficient computation of the Wiener functionals. 2 Estimating Wiener series by regression in RKHS Volterra series as linear functionals in RKHS. The nth-order Volterra functional is a weighted sum of all nth-order monomials of the input vector x. We can interpret the evaluation of this functional for a given input x as a map n defined for n = 0, 1, 2, . . . as 0(x) = 1 and n(x) = (xn1, xn-1 1 x2, . . . , x1xn-1 2 , xn 2 , . . . , xn ) m (3) such that n maps the input x Rm into a vector n(x) Fn = Rmn containing all mn ordered monomials of degree n. Using n, we can write the nth-order Volterra functional in Eq. (2) as a scalar product in Fn, Hn[x] = n n(x), (4) with the coefficients stacked into the vector n = (h(n) 1,1,..1, h(n) 1,2,..1, h(n) 1,3,..1, . . . ) Fn. The same idea can be applied to the entire pth-order Volterra series. By stacking the maps n into a single map (p)(x) = (0(x), 1(x), . . . , p(x)) , one obtains a mapping from Rm into F(p) = R Rm Rm2 . . . Rmp = RM with dimensionality M = 1-mp+1 . The 1-m entire pth-order Volterra series can be written as a scalar product in F(p) p Hn[x] = ((p)) (p)(x) (5) n=0 with (p) F(p). Below, we will show how we can express (p) as an expansion in terms of the training points. This will dramatically reduce the number of parameters we have to estimate. This procedure works because the space Fn of nth-order monomials has a very special property: it has the structure of a reproducing kernel Hilbert space (RKHS). As a conse- quence, the dot product in Fn can be computed by evaluating a positive definite kernel function kn(x1, x2). For monomials, one can easily show that (e.g., [6]) n(x1) n(x2) = (x1 x2)n =: kn(x1, x2). (6) Since F(p) is generated as a direct sum of the single spaces Fn, the associated scalar product is simply the sum of the scalar products in the Fn: p (p)(x1) (p)(x2) = (x1 x2)n = k(p)(x1, x2). (7) n=0 Thus, we have shown that the discretized Volterra series can be expressed as a linear func- tional in a RKHS2. Linear regression in RKHS. For our prediction problem (1), the RKHS property of the Volterra series leads to an efficient solution which is in part due to the so called repre- senter theorem (e.g., [6]). It states the following: suppose we are given N observations 2A similar approach has been taken by [1] using the inhomogeneous polynomial kernel k(p) ( inh x1, x2) = (1 + x1 x2)p. This kernel implies a map inh into the same space of monomi- als, but it weights the degrees of the monomials differently as can be seen by applying the binomial theorem. (x1, y1), . . . , (xN , yN ) of the function (1) and an arbitrary cost function c, is a nonde- creasing function on R>0 and . F is the norm of the RKHS associated with the kernel k. If we minimize an objective function c((x1, y1, f (x1)), . . . , (xN , yN , f (xN ))) + ( f F), (8) over all functions in the RKHS, then an optimal solution3 can be expressed as N f (x) = ajk(x, xj), aj R. (9) j=1 In other words, although we optimized over the entire RKHS including functions which are defined for arbitrary input points, it turns out that we can always express the solution in terms of the observations xj only. Hence the optimization problem over the extremely large number of coefficients (p) in Eq. (5) is transformed into one over N variables aj. Let us consider the special case where the cost function is the mean squared error, c(( N x1, y1, f (x1)), . . . , (xN , yN , f (xN ))) = 1 (f (x N j=1 j ) - yj )2, and the regularizer is zero4. The solution for a = (a1, . . . , aN ) is readily computed by setting the derivative of (8) with respect to the vector a equal to zero; it takes the form a = K -1y with the Gram matrix defined as Kij = k(xi, xj), hence5 y = f (x) = a z(x) = y K-1z(x), (10) where z(x) = (k(x, x1), k(x, x2), . . . k(x, xN )) RN . Implicit Wiener series estimation. As we stated above, the pth-degree Wiener expan- sion is the pth-order Volterra series that minimizes the squared error. This can be put into the regression framework: since any finite Volterra series can be represented as a linear functional in the corresponding RKHS, we can find the pth-order Volterra series that min- imizes the squared error by linear regression. This, by definition, must be the pth-degree Wiener series since no other Volterra series has this property6. From Eqn. (10), we obtain the following expressions for the implicit Wiener series 1 p p G0[x] = y 1, G H N n[x] = n[x] = y K-1 p z(p)(x) (11) n=0 n=0 where the Gram matrix Kp and the coefficient vector z(p)(x) are computed using the kernel from Eq. (7) and 1 = (1, 1, . . . ) RN . Note that the Wiener series is represented only implicitly since we are using the RKHS representation as a sum of scalar products with the training points. Thus, we can avoid the "curse of dimensionality", i.e., there is no need to compute the possibly large number of coefficients explicitly. The explicit Volterra and Wiener expansions can be recovered from Eq. (11) by collecting all terms containing monomials of the desired order and summing them up. The individual nth-order Volterra functionals in a Wiener series of degree p > 0 are given implicitly by Hn[x] = y K-1 p zn(x) (12) with zn(x) = ((x1 x)n, (x2 x)n, . . . , (x x)n) . For p = 0 the only term is the N constant zero-order Volterra functional H0[x] = G0[x]. The coefficient vector n = (h(n) 1,1,...1, h(n) 1,2,...1, h(n) 1,3,...1, . . . ) of the explicit Volterra functional is obtained as n = K-1 n p y (13) 3for conditions on uniqueness of the solution, see [6] 4Note that this is different from the regularized approach used by [1]. If is not zero, the resulting Volterra series are different from the Wiener series since they are not orthogonal with respect to the input. 5If K is not invertible, K-1 denotes the pseudo-inverse of K. 6assuming symmetrized Volterra kernels which can be obtained from any Volterra expanson. using the design matrix n = (n(x1) , n(x1) , . . . , n(x1) ) . The individual Wiener functionals can only be recovered by applying the regression procedure twice. If we are interested in the nth-degree Wiener functional, we have to compute the solution for the kernels k(n)(x1, x2) and k(n-1)(x1, x2). The Wiener functional for n > 0 is then obtained from the difference of the two results as n n-1 Gn[x] = Gi[x] - Gi[x] = y K-1 n z(n)(x) - K-1 n-1 z(n-1)(x) . (14) i=0 i=0 The corresponding ith-order Volterra functionals of the nth-degree Wiener functional are computed analogously to Eqns. (12) and (13) [3]. Orthogonality. The resulting Wiener functionals must fulfill the orthogonality condition which in its strictest form states that a pth-degree Wiener functional must be orthogonal to all monomials in the input of lower order. Formally, we will prove the following Theorem 1 The functionals obtained from Eq. (14) fulfill the orthogonality condition E [m(x)Gp[x]] = 0 (15) where E denotes the expectation over the input distribution and m(x) an arbitrary ith- order monomial with i < p. We will show that this a consequence of the least squares fit of any linear expansion in a set of basis functions of the form y = M j=1 j j (x). In the case of the Wiener and Volterra expansions, the basis functions j(x) are monomials of the components of x. We denote the error of the expansion as e(x) = y - M j=1 j j (xi). The minimum of the expected quadratic loss L with respect to the expansion coefficient k is given by L = E e(x) 2 = -2E [ k (x)e(x)] = 0. (16) k k This means that, for an expansion in a set of basis functions minimizing the squared error, the error is orthogonal to all basis functions used in the expansion. Now let us assume we know the Wiener series expansion (which minimizes the mean squared error) of a system up to degree p - 1. The approximation error is given by the sum of the higher-order Wiener functionals e(x) = G n=p n[x], so Gp[x] is part of the error. As a consequence of the linearity of the expectation, Eq. (16) implies E [k(x)Gn[x]] = 0 and E [k(x)Gn[x]] = 0 (17) n=p n=p+1 for any k of order less than p. The difference of both equations yields E [k(x)Gp[x]] = 0, so that Gp[x] must be orthogonal to any of the lower order basis functions, namely to all monomials with order smaller than p. 3 Experiments Toy examples. In our first experiment, we check whether our intuitions about higher-order statistics described in the introduction are captured by the proposed method. In particular, we expect that arbitrarily oriented lines can only be predicted using third-order statistics. As a consequence, we should need at least a second-order Wiener functional to predict lines correctly. Our first test image (size 80 110, upper row in Fig. 1) contains only lines of varying orientations. Choosing a 5 5 neighbourhood, we predicted the central pixel using (11). original image 0th-order 1st-order 1st-order 2nd-order 2nd-order 3rd-order 3rd-order component/ reconstruction component reconstruction component reconstruction component reconstruction mse = 583.7 mse = 0.006 mse = 0 mse = 1317 mse = 37.4 mse = 0.001 mse = 1845 mse = 334.9 mse = 19.0 Figure 1: Higher-order components of toy images. The image components of different orders are created by the corresponding Wiener functionals. They are added up to obtain the different orders of reconstruction. Note that the constant 0-order component and reconstruction are identical. The reconstruction error (mse) is given as the mean squared error between the true grey values of the image and the reconstruction. Although the linear first-order model seems to reconstruct the lines, this is actually not true since the linear model just smoothes over the image (note its large reconstruction error). A correct prediction is only obtained by adding a second-order component to the model. The third-order component is only significant at crossings, corners and line endings. Models of orders 0 . . . 3 were learned from the image by extracting the maximal training set of 76 106 patches of size 5 57. The corresponding image components of order 0 to 3 were computed according to (14). Note the different components generated by the Wiener functionals can also be negative. In Fig. 1, they are scaled to the gray values [0..255]. The behaviour of the models conforms to our intuition: the linear model cannot capture the line structure of the image thus leading to a large reconstruction error which drops to nearly zero when a second-order model is used. The additional small correction achieved by the third-order model is mainly due to discretization effects. Similar to lines, we expect that we need at least a third-order model to predict crossings or corners correctly. This is confirmed by the second and third test image shown in the corresponding row in Fig. 1. Note that the third-order component is only significant at crossings, corners and line endings. The fourth- and fifth-order terms (not shown) have only negligible contributions. The fact that the reconstruction error does not drop to zero for the third image is caused by the line endings which cannot be predicted to a higher accuracy than one pixel. Application to natural images. Are there further predictable structures in natural images that are not due to lines, crossings or corners? This can be investigated by applying our method to a set of natural images (an example of size 80 110 is depicted in Fig. 2). Our 7In contrast to the usual setting in machine learning, training and test set are identical in our case since we are not interested in generalization to other images, but in analyzing the higher-order components of the image at hand. original image 0th-order 1st-order 1st-order 2nd-order 2nd-order component/ reconstruction component reconstruction component reconstruction mse = 1070 mse = 957.4 3rd-order 3rd-order 4th-order 4th-order 5th-order 5th-order reconstruction component reconstruction component reconstruction component mse = 414.6 mse = 98.5 mse = 18.5 6th-order 6th-order 7th-order 7th-order 8th-order 8th-order reconstruction component reconstruction component reconstruction component mse = 4.98 mse = 1.32 mse = 0.41 Figure 2: Higher-order components and reconstructions of a photograph. Interactions up to the fifth order play an important role. Note that significant components become sparser with increasing model order. results on a set of 10 natural images of size 50 70 show an an approximately exponential decay of the reconstruction error when more and more higher-order terms are added to the reconstruction (Fig. 3). Interestingly, terms up to order 5 still play a significant role, although the image regions with a significant component become sparser with increasing model order (see Fig. 2). Note that the nonlinear terms reduce the reconstruction error to almost 0. This suggests a high degree of higher-order redundancy in natural images that cannot be exploited by the usual linear prediction models. 4 Conclusion The implicit estimation of Wiener functionals via polynomial kernels opens up new pos- sibilities for the estimation of higher-order image statistics. Compared to the classical methods such as higher-order spectra, moments or cumulants, our approach avoids the combinatorial explosion caused by the exponential increase of the number of terms to be estimated and interpreted. When put into a predictive framework, multiplicative pixel inter- actions of different orders are easily visualized and conform to the intuitive notions about image structures such as edges, lines, crossings or corners. There is no one-to-one mapping between the classical higher-order statistics and multi- plicative pixel interactions. Any nonlinear Wiener functional, for instance, creates infinitely many correlations or cumulants of higher order, and often also of lower order. On the other 700 Figure 3: Mean square reconstruction error of 600 models of different order for a set of 10 natural images. 500 400 mse 300 200 100 00 1 2 3 4 5 6 7 model order hand, a Wiener functional of order n produces only harmonic phase interactions up to order n + 1, but sometimes also of lower orders. Thus, when one analyzes a classical statistic of a given order, one often cannot determine by which order of pixel interaction it was created. In contrast, our method is able to isolate image components that are created by a single order of interaction. Although of preliminary nature, our results on natural images suggest an important role of statistics up to the fifth order. Most of the currently used low-level feature detectors such as edge or corner detectors maximally use third-order interactions. The investigation of fourth- or higher-order features is a field that might lead to new insights into the nature and role of higher-order image structures. As often observed in the literature (e.g. [2][7]), our results seem to confirm that a large proportion of the redundancy in natural images is contained in the higher-order pixel in- teractions. Before any further conclusions can be drawn, however, our study needs to be extended in several directions: 1. A representative image database has to be analyzed. The images must be carefully calibrated since nonlinear statistics can be highly calibration- sensitive. In addition, the contribution of image noise has to be investigated. 2. Currently, only images up to 9000 pixels can be analyzed due to the matrix inversion required by Eq. 11. To accomodate for larger images, our method has to be reformulated in an iterative algorithm. 3. So far, we only considered 5 5-patches. To systematically investigate patch size effects, the analysis has to be conducted in a multi-scale framework.

This paper proposes a method for computing fast approximations to sup- port vector decision functions in the field of object detection. In the present approach we are building on an existing algorithm where the set of support vectors is replaced by a smaller, so-called reduced set of syn- thesized input space points. In contrast to the existing method that finds the reduced set via unconstrained optimization, we impose a structural constraint on the synthetic points such that the resulting approximations can be evaluated via separable filters. For applications that require scan- ning large images, this decreases the computational complexity by a sig- nificant amount. Experimental results show that in face detection, rank deficient approximations are 4 to 6 times faster than unconstrained re- duced set systems. 1 Introduction It has been shown that support vector machines (SVMs) provide state-of-the-art accuracies in object detection. In time-critical applications, however, they are of limited use due to their computationally expensive decision functions. In particular, the time complexity of an SVM classification operation is characterized by two parameters. First, it is linear in the number of support vectors (SVs). Second, it scales with the number of operations needed for computing the similarity between an SV and the input, i.e. the complexity of the kernel function. When classifying image patches of size h w using plain gray value features, the decision function requires an h w dimensional dot product for each SV. As the patch size increases, these computations become extremely expensive. As an example, the evaluation of a single 20 20 patch on a 320 240 image at 25 frames per second already requires 660 million operations per second. In the past, research towards speeding up kernel expansions has focused exclusively on the first issue, i.e. on how to reduce the number of expansion points (SVs) [1, 2]. In [2], Burges introduced a method that, for a given SVM, creates a set of so-called reduced set vectors (RSVs) that approximate the decision function. This approach has been successfully ap- plied in the image classification domain -- speedups on the order of 10 to 30 have been re- ported [2, 3, 4] while the full accuracy was retained. Additionally, for strongly unbalanced classification problems such as face detection, the average number of RSV evaluations can be further reduced using cascaded classifiers [5, 6, 7]. Unfortunately, the above example illustrates that even with as few as three RSVs on average (as in [5]), such systems are not competitive for time-critical applications. The present work focuses on the second issue, i.e. the high computational cost of the kernel evaluations. While this could be remedied by switching to a sparser image representation (e.g. a wavelet basis), one could argue that in connection with SVMs, not only are plain gray values straightforward to use, but they have shown to outperform Haar wavelets and gradients in the face detection domain [8]. Alternatively, in [9], the authors suggest to com- pute the costly correlations in the frequency domain. In this paper, we develop a method that combines the simplicity of gray value correlations with the speed advantage of more sophisticated image representations. To this end, we borrow an idea from image processing: by constraining the RSVs to have a special structure, they can be evaluated via separable convolutions. This works for most standard kernels (e.g. linear, polynomial, Gaussian and sigmoid) and decreases the average computational complexity of the RSV evaluations from O(h w) to O(r (h + w)), where r is a small number that allows the user to balance be- tween speed and accuracy. To evaluate our approach, we examine the performance of these approximations on the MIT+CMU face detection database (used in [10, 8, 5, 6]). 2 Burges' method for reduced set approximations The present section briefly describes Burges' reduced set method [2] on which our work is based. For reasons that will become clear below, h w image patches are written as h w matrices (denoted by bold capital letters) whose entries are the respective pixel intensities. In this paper, we refer to this as the image-matrix notation. Assume that an SVM has been successfully trained on the problem at hand. Let {X1, . . . Xm} denote the set of SVs, {1, . . . m} the corresponding coefficients, k(, ) the kernel function and b the bias of the SVM solution. The decision rule for a test pattern X reads m f (X) = sgn yiik(Xi, X) + b . (1) i=1 In SVMs, the decision surface induced by f corresponds to a hyperplane in the reproducing kernel Hilbert space (RKHS) associated with k. The corresponding normal m = yiik(Xi, ) (2) i=1 can be approximated using a smaller, so-called reduced set (RS) {Z1, . . . Zm } of size m < m, i.e. an approximation to of the form m = ik(Zi, ). (3) i=1 This speeds up the decision process by a factor of m/m . To find such , we fix a desired set size m and solve min - 2RKHS (4) for i and Zi. Here, RKHS denotes the Euclidian norm in the RKHS. The resulting RS decision function f is then given by m f (X) = sgn ik(Zi,X)+b i=1 . (5) In practice, i, Zi are found using a gradient based optimization technique. Details can be found in [2]. 3 From separable filters to rank deficient reduced sets We now describe the concept of separable filters in image processing and show how this idea extends to a broader class of linear filters and to a special class of nonlinear filters, namely those used by SVM decision functions. Using the image-matrix notation, it will become clear that the separability property boils down to a matrix rank constraint. 3.1 Linear separable filters Applying a linear filter to an image amounts to a two-dimensional convolution of the image with the impulse response of the filter. In particular, if I is the input image, H the impulse response, i.e. the filter mask, and J the output image, then J = I H. (6) If H has size h w, the convolution requires O(h w) operations for each output pixel. However, in special cases where H can be decomposed into two column vectors a and b, such that H = ab (7) holds, we can rewrite (6) as J = [I a] b , (8) since the convolution is associative and in this case, ab = a b . This splits the original problem (6) into two convolution operations with masks of size h1 and 1w, respectively. As a result, if a linear filter is separable in the sense of equation (7), the computational complexity of the filtering operation can be reduced from O(h w) to O(h + w) per pixel by computing (8) instead of (6). 3.2 Linear rank deficient filters In view of (7) being equivalent to rank(H) 1, we now generalize the above concept to linear filters with low rank impulse responses. Consider the singular value decomposition (SVD) of the h w matrix H, H = USV , (9) and recall that U and V are orthogonal matrices of size h h and w w, respectively, whereas S is diagonal (the diagonal entries are the singular values) and has size h w. Now let r = rank(H). Due to rank(S) = rank(H), we may write H as a sum of r rank one matrices r H = s u v i i i (10) i=1 where si denotes the ith singular value of H and ui, vi are the iths columns of U and V (i.e. the ith singular vectors of the matrix H), respectively. As a result, the correspond- ing linear filter can be evaluated (analogously to (8)) as the weighted sum of r separable convolutions r J = si [I ui] vi (11) i=1 and the computational complexity drops from O(h w) to O(r (h + w)) per output pixel. Not surprisingly, the speed benefit depends on r, which can be seen to measure the structural complexity1 of H. For square matrices (w = h) for instance, (11) does not give any speedup compared to (6) if r > w/2. 1In other words, the flatter the spectrum of HH , the less benefit can be expected from (11). 3.3 Nonlinear rank deficient filters and reduced sets Due to the fact that in 2D, correlation is identical with convolution if the filter mask is rotated by 180 degrees (and vice versa), we can apply the above idea to any image filter f (X) = g(c(H, X)) where g is an arbitrary nonlinear function and c(H, X) denotes the correlation between images patches X and H (both of size h w). In SVMs this amounts to using a kernel of the form k(H, X) = g(c(H, X)). (12) If H has rank r, we may split the kernel evaluation into r separable correlations plus a scalar nonlinearity. As a result, if the RSVs in a kernel expansion such as (5) satisfy this constraint, the average computational complexity decreases from O(m h w) to O(m r (h + w)) per output pixel. This concept works for many off-the-shelf kernels used in SVMs. While linear, polynomial and sigmoid kernels are defined as functions of input space dot products and therefore immediately satisfy equation (12), the idea applies to kernels based on the Euclidian distance as well. For instance, the Gaussian kernel reads k(H, X) = exp((c(X, X) - 2c(H, X) + c(H, H))). (13) Here, the middle term is the correlation which we are going to evaluate via separable filters. The first term is independent of the SVs -- it can be efficiently pre-computed and stored in a separate image. The last term is merely a constant scalar independent of the image data. Finally, note that these kernels are usually defined on vectors. Nevertheless, we can use our image-matrix notation due to the fact that the squared Euclidian distance between two vectors of gray values x and z may be written as x - z 2 = X - Z 2 , (14) F whereas the dot product amounts to 1 x z = X 2 + Z 2 - X - Z 2 , (15) 2 F F F where X and Z are the corresponding image patches and F is the Frobenius norm for matrices. 4 Finding rank deficient reduced sets In our approach we consider a special class of the approximations given by (3), namely those where the RSVs can be evaluated efficiently via separable correlations. In order to obtain such approximations, we use a constrained version of Burges' method. In particular, we restrict the RSV search space to the manifold spanned by all image patches that -- viewed as matrices -- have a fixed, small rank r (which is to be chosen a priori by the user). To this end, the Zi in equation (3) are replaced by their singular value decompositions Z S V i Ui i i . (16) The rank constraint can then be imposed by allowing only the first r diagonal elements of Si to be non-zero. Note that this boils down to using an approximation of the form m = S V r ik(Ui,r i,r i,r , ) (17) i=1 with Si,r being r r (diagonal) and Ui,r, Vi,r being h r, w r (orthogonal2) matrices, respectively. Analogously to (4) we fix m and r and find Si,r, Ui,r, Vi,r and i that mini- mize the approximation error = - 2 . r The minimization problem is solved via RKHS 2In this paper we call a non-square matrix orthogonal if its columns are pairwise orthogonal and have unit length. gradient decent. Note that when computing gradients, the image-matrix notation (together with (14) or (15), and the equality X 2 = tr(XX )) allows a straightforward computa- F tion of the kernel derivatives w.r.t. the components of the decomposed RSV image patches, i.e. the row, column and scale information in Vi,r, Ui,r and Si,r, respectively. However, while the update rules for i and Si,r follow immediately from the respective derivatives, care must be taken in order to keep Ui,r and Vi,r orthogonal during optimization. This can be achieved through re-orthogonalization of these matrices after each gradient step. In our current implementation, however, we perform those updates subject to the so-called Stiefel constraints [11]. Intuitively, this amounts to rotating (rather than translating) the columns of Ui,r and Vi,r, which ensures that the resulting matrices are still orthogonal, i.e. lie on the Stiefel manifold. Let S(h, r) be the manifold of orthogonal h r matrices, the (h, r)-Stiefel manifold. Further, let U denote an orthogonal basis for the orthogo- i,r nal complement of the subspace spanned by the columns of Ui,r. Now, given the 'free' gradient G = /Ui,r we compute the 'constrained' gradient ^ G = G - U G U i,r i,r , (18) which is the projection of G onto the tangent space of S(h, r) at Ui,r. The desired rotation is then given [11] by the (matrix) exponential of the h h skew-symmetric matrix ^ G U ) A = t i,r -( ^ G U i,r ^ (19) G U 0 i,r where t is a user-defined step size parameter. For details, see [11]. A Matlab library is available at [12]. 5 Experiments This section shows the results of two experiments. The first part illustrates the behavior of rank deficient approximations for a face detection SVM in terms of the convergence rate and classification accuracy for different values of r. In the second part, we show how an actual face detection system, similar to that presented in [5], can be sped up using rank deficient RSVs. In both experiments we used the same training and validation set. It consisted of 19 19 gray level image patches containing 16081 manually collected faces (3194 of them kindly provided by Sami Romdhani) and 42972 non-faces automatically collected from a set of 206 background scenes. Each patch was normalized to zero mean and unit variance. The set was split into a training set (13331 faces and 35827 non-faces) and a validation set (2687 faces and 7145 non-faces). We trained a 1-norm soft margin SVM on the training set using a Gaussian kernel with = 10. The regularization constant C was set to 1. The resulting decision function (1) achieved a hit rate of 97.3% at 1.0% false positives on the validation set using m = 6910 SVs. This solution served as the approximation target (see equation (2)) during the experiments described below. 5.1 Rank deficient faces In order to see how m and r affect the accuracy of our approximations, we compute rank deficient reduced sets for m = 1 . . . 32 and r = 1 . . . 3 (the left array in Figure 1 illustrates the actual appearance of rank deficient RSVs for the m = 6 case). Accuracy of the re- sulting decision functions is measured in ROC score (the area under the ROC curve) on the validation set. For the full SVM, this amounts to 0.99. The results for our approximations are depicted in Figure 2. As expected, we need a larger number of rank deficient RSVs than unconstrained RSVs to obtain similar classification accuracies, especially for small r. Nevertheless, the experiment points out two advantages of our method. First, a rank as m' = 6 m' = 1 = + + + ... r=full r=full r=3 = + + r=3 r=2 = + r=2 r=1 r=1 = Figure 1: Rank deficient faces. The left array shows the RSVs (Zi) of the unconstrained (top row) and constrained (r decreases from 3 to 1 down the remaining rows) approxi- mations for m = 6. Interestingly, the r = 3 RSVs are already able to capture face-like structures. This supports the fact that the classification accuracy for r = 3 is similar to that of the unconstrained approximations (cf. Figure 2, left plot). The right array shows the m = 1 RSVs (r = full, 3, 2, 1, top to bottom row) and their decomposition into rank one matrices according to (10). For the unconstrained RSV (first row) it shows an approximate (truncated) expansion based on the three leading singular vectors. While for r = 3 the decomposition is indeed similar to the truncated SVD, note how this similarity decreases for r = 2, 1. This illustrates that the approach is clearly different from simply finding un- constrained RSVs and then imposing the rank constraint via SVD (in fact, the norm (4) is smaller for the r = 1 RSV than for the leading singular vector of the r = full RSV). low as three seems already sufficient for our face detection SVM in the sense that for equal sizes m there is no significant loss in accuracy compared to the unconstrained approxima- tion (at least for m > 2). The associated speed benefit over unconstrained RSVs is shown in the right plot of Figure 2: the rank three approximations achieve accuracies similar to the unconstrained functions, while the number of operations reduces to less than a third. Second, while for unconstrained RSVs there is no solution with a number of operations smaller than h w = 361 (in the right plot, this is the region beyond the left end of the solid line), there exist rank deficient functions which are not only much faster than this, but yield considerably higher accuracies. This property will be exploited in the next experiment. 5.2 A cascade-based face detection system In this experiment we built a cascade-based face detection system similar to [5, 6], i.e. a cascade of RSV approximations of increasing size m . As the benefit of a cascaded classifier heavily depends on the speed of the first classifier which has to be evaluated on the whole image [5, 6], our system uses a rank deficient approximation as the first stage. Based on the previous experiment, we chose the m = 3, r = 1 classifier. Note that this function yields an ROC score of 0.9 using 114 multiply-adds, whereas the simplest possible unconstrained approximation m = 1, r = full needs 361 multiply-adds to achieve a ROC score of only 0.83 (cf. Figure 2). In particular, if the threshold of the first stage is set to yield a hit rate of 95% on the validation set, scanning the MIT+CMU set (130 images, 507 faces) with m = 3, r = 1 discards 91.5% of the false positives, whereas the m = 1, r = full can only reject 70.2%. At the same time, when scanning a 320 240 image3, the three separable convolutions plus nonlinearity require 55ms, whereas the single, full kernel evaluation takes 208ms on a Pentium 4 with 2.8 GHz. Moreover, for the unconstrained 3For multi-scale processing the detectors are evaluated on an image pyramid with 12 different scales using a scale decay of 0.75. This amounts to scanning 140158 patches for a 320 240 image. 1 1 0.9 0.9 0.8 0.8 ROC score ROC score r=1 r=1 0.7 r=2 0.7 r=2 r=3 r=3 r=full r=full 0.6 0.6 0 1 2 3 4 10 10 10 10 10 #RSVs (m') #operations (m'r(h+w)) Figure 2: Effect of the rank parameter r on classification accuracies. The left plots shows the ROC score of the rank deficient RSV approximations (cf. Section 4) for varying set sizes (m = 1 . . . 32, on a logarithmic scale) and ranks (r = 1 . . . 3). Additionally, the solid line shows the accuracy of the RSVs without rank constraint (cf. Section 2), here denoted by r = full. The right plot shows the same four curves, but plotted against the number of operations needed for the evaluation of the corresponding decision function when scanning large images (i.e. m r (h + w) with h = w = 19), also on a logarithmic scale. Figure 3: A sample output from our demonstra- tion system (running at 14 frames per second). In this implementation, we reduced the number of false positives by adjusting the threshold of the final classifier. Although this reduces the number of detections as well, the results are still satisfactory. This is probably due to the fact that the MIT+CMU set contains several images of very low quality that are not likely to occur in our setting, using a good USB camera. cascade to catch up in terms of accuracy, the (at least) m = 2, r = full classifier (also with an ROC score of roughly 0.9) should be applied afterwards, requiring another 0.3 2 208 ms 125ms. The subsequent stages of our system consist of unconstrained RSV approximations of size m = 4, 8, 16, 32, respectively. These sizes were chosen such that the number of false positives roughly halves after each stage, while the number of correct detections remains close to 95% on the validation set (with the decision thresholds adjusted accordingly). To eliminate redundant detections, we combine overlapping detections via averaging of posi- tion and size if they are closer than 0.15 times the estimated patch size. This system yields 93.1% correct detections and 0.034% false positives on the MIT+CMU set. The current system was incorporated into a demo application (Figure 3). For optimal performance, we re-compiled our system using the Intel compiler (ICC). The application now classifies a 320x240 image within 54ms (vs. 238ms with full rank RSVs only) on a 2.8 GHz PC. To further reduce the number of false positives, additional bootstrapped (as in [5]) stages need to be added to the cascade. Note that this will not significantly affect the speed of our sys- tem (currently 14 frames per second) since 0.034% false positives amounts to merely 47 patches to be processed by subsequent classifiers.

Tangential neurons in the fly brain are sensitive to the typical optic flow patterns generated during egomotion. In this study, we examine whether a simplified linear model based on the organization principles in tangential neurons can be used to estimate egomotion from the optic flow. We present a theory for the construction of an estimator consisting of a linear combination of optic flow vectors that incorporates prior knowledge about the distance distribution of the environment and about the noise and egomotion statistics of the sensor. The estimator is tested on a gantry carrying an omnidirectional vision sensor. The experiments show that the proposed approach leads to accurate and robust estimates of rotation rates, whereas translation estimates are of reasonable quality, albeit less reliable.

The tangential neurons in the fly brain are sensitive to the typical optic flow patterns generated during self-motion. In this study, we examine whether a simplified linear model of these neurons can be used to esti- mate self-motion from the optic flow. We present a theory for the con- struction of an estimator consisting of a linear combination of optic flow vectors that incorporates prior knowledge both about the distance distri- bution of the environment, and about the noise and self-motion statistics of the sensor. The estimator is tested on a gantry carrying an omnidirec- tional vision sensor. The experiments show that the proposed approach leads to accurate and robust estimates of rotation rates, whereas transla- tion estimates turn out to be less reliable.