edge and blob models. These models are augmented with a theoretical parametric model of a background feature type, namely the Brownian image model.
Brownian Images: A Generic Background Model Kim S. Pedersen1 and Martin Lillholm1 The IT University of Copenhagen Glentevej 67, DK-2400 Copenhagen NV, Denmark
Abstract. The local structure of a suitably differentiable function can be described through it’s truncated Taylor series. Gaussian scale space theory is a sensible framework for calculating image derivatives or the coefficients of the Taylor series up to any order using the corresponding derivatives of the Gaussian kernel. Zero-crossings of invariant combinations of such regularised derivatives have successfully been used as feature detectors. Recent work shows that feature detection can be stated as a simple problem of classification into the feature types of interest (edges, blobs, etc.). The information used in the classification is the mapping of Taylor series coefficients, i.e. image derivatives, into jet space. The conjecture is that features form classes or clusters in jet space. We introduce the classification framework and present feature detection results based on empirical edge and blob models. These models are augmented with a theoretical parametric model of a background feature type, namely the Brownian image model. The background feature type is not to be considered as the class of all non-modelled feature types, but as a new feature type in itself. The quality of the Brownian image background model is further investigated. The background model leads to classification of images into areas which are plausible featureless background areas. From an intuitive point of view the image structure is more complex in areas classified as features as opposed to the simpler background region. We verify this using a reconstruction scheme with spatially uniform constraints and a simple Gaussian assumption of prior pixel distributions. We are, however, also able to show that reconstructions using a Brownian prior results in a significant gain in reconstruction quality for the Brownian like background areas. Keywords: Brownian images, feature detection, image reconstruction, natural image statistics, background model.
1
Introduction
Geometric image features play an essential role in computer vision as the foundation of many algorithms for solving a wide range of vision problems. The early work by Marr [1] emphasises the importance of image features in computer vision. In [2], we argue for a soft multi-scale image feature detection in which each image point is assigned a probability of being of one of several feature types. Such a probabilistic feature detection would be beneficial in probabilistic higher level processing of images. Probabilistic feature detection has previously been studied by among others Konishi et al. [3], Ren et al. [4], and Laptev et al. [5]. Classical image feature detection, such as edge, blob, and ridge detection, is based on invariant and often non-linear functions of linear filter responses. In [2], we argue that classical feature detection can be substituted with a simple soft classification with linear filter responses as input. The soft classification we propose is based on a Bayesian rationale. By using a classifier we get simultaneous classification into several geometrical image feature types within one framework. We will use the coefficients of the truncated Taylor series, known as the k-jet, calculated using scale space image derivatives, as the description of local image structure. The k-jet is a vector of image derivatives up to order k and is, in the language of pattern recognition (PR), a feature vector or point in our feature space. To avoid confusion we will use the term feature to denote local image structure and jet to denote the feature vector in the PR sense. We give a brief account of the method and results discussed in [2] in order to help the reader appreciate the results presented in this paper. As a proof of concept, we will show results on detection of edges and blobs. We introduce the Brownian image model as an analytical model of generic image background. Hence image background is defined as regions with Brownian behaviour, i.e. spatially correlated Gaussian noise. This background model is considered as a feature which competes with other features such as edges and blobs. In order to investigate the implications of using a Brownian background model, we comment on the pixel difference statistics in background and nonbackground image regions. Furthermore, we study the effect of reconstructing images using two different prior models, the Brownian and the independently identically distributed (i.i.d.) Gaussian model. We use the image reconstruction method proposed in [6]. The conclusion of this study is that we can obtain better reconstruction quality for background regions compared to non-background regions such as edge and blob regions using the Brownian prior. Hence, this suggest that background regions are featureless. The work by Konishi et al. [3] is related to our work, but differ in that they only focus on edges and use log-likelihood ratios of on/off edge probabilities. Furthermore, they use two filter banks different from the scale space jet. In our opinion, the scale space jet is a more natural language when studying local image structure. In Sec. 2 and 3 we set up our classifier. The background and feature modelling are discussed in Sec. 4. We present experimental results on feature detection in
Sec. 5. The background model is investigated in Sec. 6 and finally, we summarise and discuss in Sec. 7.
2
Jet Representation
Through the notion of generalised functions [7], we can regularise the inherently ill-posed problem of calculating derivatives of discrete images I. Using the Gaussian kernel as test function, gives rise to Gaussian scale space theory [8,9,10] where the scale space image L at scale σ is defined as the convolution with the Gaussian kernel Gσ of standard deviation σ: L(x, y; σ) = (Gσ ∗ I)(x, y) , σ ∈ S ⊆ IR+
(1)
with L(x, y; σ = 0) ≡ I(x, y). Furthermore, we get scale normalised (see Sec. 4) derivatives [11] of L by convolution with the corresponding scale normalised derivative of the Gaussian kernel: ∂ n+m Lxn ym (x, y; σ) = σ Gσ ∗ I (x, y) . (2) ∂ xn ∂ y m This framework enables us to calculate the coefficients of the truncated Taylor series up to any finite order k for any given point x = (x, y) of L. As a characterisation of the local image structure in x, we use the k-jet [12], where the k-jet is a functional mapping of scale-space images L(x; σ) into IRN , j k : C ∞ (R2 × IR+ ) 7→ IRN , with N = (k + 2)(k + 1)/2. As local structure should be invariant wrt. luminance, we omit the zeroth order term and represent local structure as a point in an N − 1 dimensional feature space — jet-space — resulting in an N − 1 dimensional feature vector of scale normalised derivatives. We will use the shorthand notation jσ (x) = j k [L](x; σ) (excluding the zeroth order term).
3
Soft Feature Classification
One of the main theses of the paper is that geometrical image features such as edges, ridges, blobs, and background for that matter, are sufficiently separated, in terms of their jet-space representation, such that traditional feature detection can be performed using standard classification techniques. Based on the Bayesian rationale, as in [2], we argue for a classification based feature detection. Doing so allows us to do simultaneous detection of several feature types. Let us assume that we want to detect a set of features f ∈ F. We will represent the local image structure by the scale normalised jet jσ (x) as described in Sec. 2. The conditional probability density that the local structure specified by jσ (x) is of feature type f at scale σ will be denoted by p(f, σ|jσ (x)) and will be called the posterior probability density. Using Bayes theorem, we can write the posterior as p(jσ (x)|f, σ)p(f, σ) p(f, σ|jσ (x)) = . (3) p(jσ (x))
Here, p(jσ (x)|f, σ) is the likelihood of the image structure arising from the feature f at scale σ and p(f, σ) is the prior distribution of the individual features f at scale σ. Assuming that we can specify the terms on the right hand side of Eq. (3), we locally have a probability distribution p(f, σ|jσ (x)) on feature types f and scales σ. We can now detect (i.e. hard classify) features and their inner scales by maximising the posterior, ˆσ (f, ˆ ) = arg
max
(f,σ)∈F ×S
p(f, σ|jσ (x)) .
(4)
In the discrete setting, the classification of features and selection of scale boils down to classification of the elements of the discrete set (f, σ) ∈ F × S0 , where S0 is a discrete (countable) subset of scales S.
4
Modelling Image Features
Having defined the framework, we would now have to either specify the likelihood terms for the individual features as well as the feature prior or directly specify the posteriors. In both cases, we are faced with the choice of either modelling the densities and using parametric estimation techniques, or using purely nonparametric density estimation. To demonstrate the method, we will do experiments based on background, edge, and blob feature types, and use non-parametric density estimates. We introduce an analytical background model by specifying a background likelihood term and extend with point distributions that implicitly model known edge and blob behaviour. We choose to estimate the feature posteriors based on a k-nearest-neighbour (KNN) estimate (see e.g. [13]). We can estimate the joint probability density p(f, σ, jσ (x)) by using the KNN rule. Assume we have a labelled set of jets, each labelled according to a feature f and a scale σ. The density at jσ (x) can be estimated by finding the k-nearest-neighbours and counting the number kf,σ of neighbours labelled as feature f at scale σ. The posterior density p(f, σ|jσ (x)) can be written in terms of the joint density p(f, σ, jσ (x)) as p(f, σ|jσ (x)) = P
kf,σ p(f, σ, jσ (x)) P . = k p(f, σ, j (x)) 0 σ f ∈F σ∈S
(5)
We use the Euclidean metric in jet space in the KNN estimate. This is justified by the fact that we use scale normalised derivatives, hence all derivatives are dimensionless and comparable. 4.1
The Brownian Background Model
Early studies [14,15] of natural image statistics focused on the correlation between pixels and showed that natural images have a power law behaviour of the
Fig. 1. Sample from the Brownian image model.
power spectrum indicating a self similar, and in some situations scale-invariant behaviour of the second order statistics. Classes of natural images with wide variation of motif can be shown to have a scale invariant second order statistics in jet space, which can be modelled by the Brownian image model (see [16]). This result does not indicate that natural images resemble Brownian images (see Fig. 1), but only that the covariance structure of derivatives of natural images is equal to that of derivatives of Brownian images. For instance, it has been shown that various linear filter responses of natural images have empirical distributions which can be modelled by generalised Laplace distributions [17,18,19] or Bessel K-forms [20], i.e. non-Gaussian distributions contrary to the Brownian model. The Brownian image model has previously been studied by among others Mumford et al. [21], who also points out some of the limitations of this model and suggests other stochastic image models in an attempt to remedy these limitations. Despite its limitations, we introduce the Brownian image model as a model of generic image background and investigate its ability to model natural image background. Furthermore, this approach allows us to model the background as an explicit class instead of everything non-feature like; see Sec. 4.2 for more details. The Brownian image model is a scale invariant stochastic field with zero mean Gaussian independent identically distributed spatial increments. This leads to Gaussian distributed partial derivatives of Brownian images. Hence, mapping the Brownian image model into jet space results in a scale invariant Gaussian distribution with an anisotropic covariance structure centred at the origin (see [16]). This results in flat featureless regions being more likely as background than other regions. The likelihood for the background feature class f = B can therefore be written as a Gaussian density 1 1 (6) p(jσ (x)|B, σ) = · exp − jσT (x)Σ−1 jσ (x) , Z 2
where |Σ| denotes the determinant of the covariance matrix Σ and Z is a normalisation constant. An analytic expression for the covariance matrix Σ exists (see [16] for a proof). Let B(x, y; σ) denote the scale space of a Brownian image and Bxn1 ym1 (x, y; σ) denote the scale normalised derivatives, then the covariance matrix of B(x, y; σ) mapped into jet space is given by Σij = E [hBxn1 ym1 , Bxn2 ym2 i] = (−1)
n+m 2 +n2 +m2
σ02 n!m! 2n+m+1 π(n + m)(n/2)!(m/2)!
(7)
whenever both n = n1 + n2 and m = m1 + m2 are even integers, otherwise E [hBxn1 ym1 , Bxn2 ym2 i] = 0 . Where h·, ·i denotes the inner product between two functions and the constant σ02 is the global variance of the Brownian image intensities. 4.2
Modelling Edges and Blobs
As described in the beginning of this section, we have chosen to model edges and blobs implicitly using samples from known edge and blob models/detectors. Specifically, we detect edges and blobs in a large image database and use the k-jet at the detected points as training set for the KNN-classifier. We use Lindeberg’s scale space edge and blob detectors [22,23], where edges are defined as maxima spatially and across scale of the scale normalised derivative in the gradient direction, and blobs as maxima spatially and across scale of the scale normalised Laplacean. As both these detectors are multi-scale by design, the resulting feature samples span several scales in accordance with the classification framework in Sec. 3.
5
Experimental Results on Feature Detection
The KNN-classifier is trained using a subset (excluding images suffering from saturation artifacts, motion blur, major focus problems) of the van Hateren Database [24] of natural images, resulting in 1500 images (see Fig. 2 for examples). For each of these, edges and blobs are detected in the central 512x512 region discarding features that suffer from boundary effects. For each scale σ, we estimate the prior p(f, σ) as the relationship between the number of detected edges and blobs and the total number of pixels (5122 ×1500). The actual number of edge sample points, blob sample points, and background points sampled from the analytical (Eq. (6)) likelihood function reflects this scale-dependent prior; on average the combined edge, blob, and background sample contain 106 points. Finally, we use the scale-normalised 3-jet jσ (x) = j 3 [L](x; σ), as suitable compromise between descriptive power and computational effort, as our feature vector.
Fig. 2. Typical images from the van Hateren natural stimuli collection [24].
In this section, we give a few examples of soft and hard classification of images into edge-, blob-, and background-like regions using the described setup. The top row in Fig. 3 contains an artificial image with both isotropic and anisotropic Gaussian blobs and a blurred step edge. The second image is an example of an everyday scene with many different kinds of structure. Second, third, and fourth row contain the background, edge, and blob posteriors respectively. The last row contains the resulting hard classifications, where blue, green, and red indicate background-, blob-, and edge-like regions respectively. For the artificial image, we see that both the edge and blob posteriors behave as one would expect around the edge and the centre of the blobs. Likewise for the background model, that capture the flat regions as expected — corresponding observations hold for the hard classification where both the edge and the blob centres are classified as one would expect. The very broad edge band around the centre of the blobs is, however, somewhat surprising. The presence of edgelike structure is expected as each Gaussian blob is naturally delineated by an edge but as all the blobs are quite flat, these “ramps” toward their centre does naturally contain more edge-like (first order) structure than typical blob and background points and for lack of a better feature class, they are labelled as edge like points. Finally, the blob posterior has a ringing effect at the outer periphery of the blobs and a corresponding “echo” at the boundary of the edge region; these are regions with pronounced second order structure and as the blob class is the only class with predominant second order structure these regions are, again lacking a better class, labelled as blob-like. For the second image, similar observations hold but are not as pronounced; primarily because of the higher granularity of the image. The background region seems reasonable; especially the distinction between the two persons, where the woman has coarse scale structure in her top and ridge-like highlights in the calves, the suit worn by the man is relatively featureless and is labelled as background. An important observation is the connectedness of the edge like regions in general and specifically around T-junctions. The connectedness of edges around T-junctions is not present in the edge point set (based on the Lindeberg
Fig. 3. Examples of two images and their feature posteriors calculated using five scales. From the top row: Original images, posteriors for background, edges, and blobs, and resulting hard classifications.
edge detector) used in the KNN posterior estimate, but is a product of the soft classification. In Fig. 4, we give an example of using the edge posterior as the basis for calculating an actual edge map. Top left is the original image, top right the edge posterior map masked with the hard classification. Superimposed in red, Lindeberg edges calculated for the original image. Bottom left, Canny edges for the original image without hysteresis, and finally an edge map calculated from the edge posterior using the watershed transform. This example is only calculated for one scale to facilitate comparison and does not demonstrate the full potential of our method (or any of the two others for that matter) but again the main point is that, as for the posterior, the edge map is, in general, well connected, specifically around T-junctions. Finally, the small dot-like artifacts in the bottom right edge map are due to suboptimal watershed calculations and not the quality of the posterior. In a similar, but simpler, fashion the blob posterior can be used to identify actual blob centres as spatial maxima corresponding to the output of a traditional blob detector.
6
Image Reconstruction and the Background Model
In the last section we showed the strength of using the Brownian background model combined with edges and blobs for feature detection. In this section, we will study further the validity of Brownian images as a generic background model. The Brownian background model gives regions that subjectively corresponds to background regions. We start by examining the pixel difference statistics in both background and non-background regions. If the background model really corresponds to featureless regions, we would expect that it is fairly straight forward to obtain a good reconstruction quality in background regions. The difficulty of image reconstruction is in capturing high frequency regions such as feature regions. We therefore, study the effect of using the Brownian and i.i.d. Gaussian reconstruction priors on background and feature regions, and compare the results between the two types of regions. 6.1
Difference Statistics of Background Regions
First, we examine the statistics of pixel differences in image regions labelled as background and regions labelled as features, i.e. blobs and edges. We have made single scale (σ = 1) classifications on 121 (64 × 64 pixels) image patches from the van Hateren database (see Fig. 5 for example of a patch and corresponding classification). The 121 test images are not part of the training set used to train the KNN classifier. We then join regions labelled as edges and blobs into one class and compute the normalised histogram of pixel differences separately for background regions and feature regions (see Fig. 5). The pixel difference distribution for natural background regions are not Gaussian distributed. As expected, this is also the case for feature regions. At first
Fig. 4. An example of extracting an edge map (bottom right) from the edge posterior (top right) compared to Canny (bottom left) and Lindeberg edges (red curves superimposed on edge posterior (top right)). Calculated at a single scale.
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Fig. 5. (Top left) Example of image patch. (Top middle) Single scale (σ = 1) classification. (Top right) Edge and blob class merged into new class. (Bottom left) Histogram of pixel differences in (left) background regions and in (right) feature (edges and blobs) regions. Included in the histograms is the Gaussian distribution with the corresponding variance.
glance, the non-Gaussianity of natural background regions is in conflict with the Gaussianity dictated by the Brownian model. However, this result is in accordance with the remark made in Sec. 4.1 that the Brownian model only models the second order statistics of natural images in jet space, hence natural images does not have to have Gaussian distributed pixel differences in background regions. However, the background regions are classified in accordance with the Brownian model and appear featureless. We seek to verify this through quantification of image reconstruction quality in background and non-background regions using both an i.i.d. Gaussian and a Brownian prior.
6.2
Image Reconstruction
In [25] and [6] a framework for image reconstruction based on local geometrical constraints and simple models of natural image statistics was introduced. As in our setting, local geometry is defined using the k-jet. The reconstruction scheme
is formulated as a constrained functional minimisation: Z K X Ψ (x, J, Jx , Jy ) + λi c i , E= IR
(8)
i=1
where J is the reconstructed image, Ψ is a complexity measure derived from a model of natural image statistics, and the ci ’s are the local geometrical constraints with corresponding Lagrange multipliers λi . In practice, the ci ’s are calculated as the inner product between a spatially and scale localised Gaussian kernel and the original image I: Z H(x)I(y) dx, i = 1, ..., K (9) ci = hH|Ii = IR
∂ ∂x ni ∂y m i
where H(x, y) = Gσi (x − xi , y − yi ). A very naive model of natural images is that pixel intensities are i.i.d. Gaussian. According to [6], this leads to the variance as the complexity measure Ψ and thus a simple linear reconstruction via the pseudo inverse: J = F (F T F )−1 c ,
(10)
where the columns of F are the constraining Gaussian kernels and c the corresponding vector of ci ’s. As described earlier, the Brownian model is a more realistic model of certain aspects of natural image behaviour. An assumption of i.i.d. Gaussian pixel give the Brownian model and leads to a complexity measure Ψ = Rdifference 2 |∇I| dx and a projection based iterative reconstruction scheme — details IR can be found in [6]. We calculate reconstructions of the 121 image patches based on the Brownian image model. As constraints, we use the 2-jet sampled at scale σ = 1.0 in a 4 pixel regular grid (see Fig. 6). This ensures that no areas (such as features) are favoured. Let the root mean square error (RMSE) of the background part of the reconstructed image be defined as s X (L(xi , yi , σ = 1) − J(xi , yi , σ = 1))2 /|CB | (11) B = (xi ,yi )∈CB
where CB is the positions of pixels classified as background and |CB | is the number of background pixels. Similar the RMSE for regions labelled as either blobs or edges is s X (L(xi , yi , σ = 1) − J(xi , yi , σ = 1))2 /|Ceb | (12) eb = (xi ,yi )∈Ceb
where Ceb is the pixels labelled as either blobs or edges and |Ceb | is the amount of pixels. The reconstructed image J(x, y, σ = 1) is compared to the original image at scale σ = 1, L(x, y, σ = 1).
To verify the effect of using the Brownian image model for reconstruction in regions labelled as background compared to other regions (blobs and edge regions) we define the mean ratio of the RMSE as: P j∈M eb /B RRMSE = (13) |M | where M is the set of test image patches and |M | = 121. If the RRMSE ≈ 1, the reconstruction quality in the different regions is similar. Reconstructing images using the Brownian image model as prior leads to a mean ratio of the RMSE of RRMSE ≈ 2.00 with a 95% confidence interval of [1.82, 2.18], which means that the Brownian model leads to a better reconstruction in background regions compared to feature regions (blobs and edges). In order to verify whether this results is specific for the Brownian reconstruction prior we repeat the experiment, but now based on reconstructions with an i.i.d. Gaussian prior. This leads to a mean ratio of the RMSE of RRMSE ≈ 1.57 with a 95% confidence interval of [1.36, 1.78]. A two-sample T-test supports that the mean value ratio of the Brownian model is significantly different from the i.i.d. Gaussian model with the probability p = 0.1% for accepting the null hypothesis (the mean values are equal). Hence, reconstruction of background regions gives a better result compared to feature regions. Furthermore, the Brownian reconstruction prior performs better on average in background regions than the i.i.d. Gaussian prior. Here, we can think of the reconstruction quality obtained with the naive Gaussian model as a measure of the relative difficulty between reconstruction of featureless regions and more complex edge and blob like regions and conclude that the Brownian model yields an actual performance gain for our background regions. We have shown that the Brownian image model, together with edge and blob posteriors, leads to a classification into image regions which we subjectively judge as sensible background regions. The Brownian image model does not capture the structural variation found in feature regions, but seem to be a good model of featureless regions. We can conclude, that from a reconstruction point of view, it make sense to have a large proportion of the total number of constraints at image feature points to fixate the structure in these regions, and then use the Brownian image model to reconstruct featureless (or low frequency) regions of images.
7
Discussion and Summary
We have presented a simple multi-scale scheme for classifying image points into one of several feature types or a generic background class using a KNN-classifier, implicit feature models, and training data from a database of natural images. Examples have been given using edges and blobs but the scheme is in principle extensible to any other feature type such as ridges and corners. Actual feature maps have been calculated from the posterior maps.
Fig. 6. (Left) Example of the 121 test patches used for the reconstruction experiment. (Middle) The grid used to constraint the reconstruction. (Right) Reconstruction of the image seen in this figure.
We argue that using Brownian images as a model of generic image background leads to featureless regions being classified as such. We base this claim on the fact that we obtain a higher image reconstruction quality in background regions compared to non-background regions i.e. feature regions. Furthermore, the Brownian model outperforms the more naive Gaussian model and thus we have a combination of a classification scheme and corresponding generative model. Obviously, a more formal evaluation of the feature detection scheme similar to that presented in e.g. [3] must be carried out to verify the quality of the method. Another interesting development would be to use human “ground truth” training sets such as the segmentation database presented in [26] in place of the models. Conversely, there is much potential space and time saved (and insight gained) by developing parametric feature models in jet space to take the place of large feature samples. Furthermore, the current scale selection scheme is rather naive and could be extended to allow for more than one classification over scale to enable detection of e.g. two concentric blobs of different size; along the line of thought of Lindeberg [23]. Finally, an investigation into natural image models suited for feature dense areas would complement the Brownian background model from a reconstructive point of view.
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