Semantic-Shift for Unsupervised Object Detection David Liu, Tsuhan Chen Department of Electrical and Computer Engineering Carnegie Mellon University 5000 Forbes Ave, Pittsburgh, PA, 15213 {dliu,tsuhan}@cmu.edu

Abstract The bag of visual words representation has attracted a lot of attention in the computer vision community. In particular, Probabilistic Latent Semantic Analysis (PLSA) has been applied to object recognition as an unsupervised technique built on top of the bag of visual words representation. PLSA, however, does not explicitly consider the spatial information of the visual words. In this paper, we propose an iterative technique, where a modified form of PLSA provides location and scale estimates of the foreground object through the estimated latent semantic. In return, the updated location and scale estimates will improve the estimate of the latent semantic. We call this iterative algorithm Semantic-Shift. We show results with significant improvements over PLSA.

(a)

(b)

(c)

(d)

Figure 1. (a) Original image (b) Result of PLSA (c,d) Result of Semantic-Shift at the 6th and 26th iteration during learning.

1. Introduction

shown to outperform classical clustering methods such as k-means. PLSA has earlier been used in the text and linguistic domains. In PLSA, a document is considered as a mixture of “topics”, and each topic consists of a mixture of words. The power of PLSA originates from the fact that topics can be learned in an unsupervised manner given a set of document-word pairs. In the image understanding domain, documents are analogous to images, and words are analogous to visual words. Applying PLSA to pairs of imagevisual words is hence capable of extracting latent topics from a bunch of images. One important drawback of PLSA, however, is that the set of document-word pairs ignores the geometric layout of words in an image. In other words, if we arbitrarily shuffle the local features in the image around, we get the same latent topics! As a result, the performance of PLSA still leaves room for improvement. Here we propose Semantic-Shift to explicitly take spatial structure into account. Semantic-Shift consists of a modified version of PLSA, which has an extra spatial distribu-

Unsupervised image understanding systems have many advantages compared to supervised systems due to the difficulty of image annotation. First, an image may consist of many objects in a complex layout. So far there is no common approach to annotating images at the object level. Second, there are many visual illusions showing that different people may have different understandings of an image. Third, object level annotation in a supervised system requires manually labelling the location and categorization of each object in images, which is very time consuming. It is very expensive to collect large mount of accurate annotated images for constructing a supervised image understanding system. On the contrary, training an unsupervised system does not need annotated images. Considering the abundance of images available on the Internet, unsupervised learning methods provide a promising direction. One unsupervised learning method called Probabilistic Latent Semantic Analysis (PLSA) [8] has recently been applied to the object recognition domain [11][5][10] and 1

and M-step of PLSA can be found in [8]. Here we restate the results: E-step: P (zk |di , wj ) ∝ P (wj |zk )P (zk |di ) (a)

M-step:

(b)

Figure 2. Graphical models of (a) PLSA and (b) Semantic-Shift

P (wj |zk ) ∝

X

(2a)

n(di , wj )P (zk |di , wj )

(2b)

n(di , wj )P (zk |di , wj )

(2c)

i

tion component, and after every iteration of ExpectationMaximization [4], the probability of each word belonging to a specific topic (i.e., the latent semantic of each word) is being updated. As a result, the location and scale estimates of the foreground object are shifted. Fig. 1(c) and (d) show the of Semantic-Shift after the 6th and 26th iteration. More explanations of this figure can be found under Fig. 3. We will describe Semantic-Shift in detail in Section 3, and detail the image representation in Section 4. Before that, we will briefly review PLSA in Section 2. In Section 5, we will show that Semantic-Shift performs significantly better than PLSA.

2. Probabilistic Latent Semantic Analysis PLSA is a model which describes the association of documents and words through a latent topic variable. PLSA is illustrated in Fig. 2 as a graphical model. Suppose the training data consists of D images (or documents in the text and linguistic domains), {d1 , ..., dD }. Each image is considered as a mixture of topics: P (zk |di ) is the probability of topic zk occurring in image di . Assume there are a predefined number of Z latent topics, {z1 , ..., zZ }. Using inference methods, it is then possible to infer latent variables of the model. Each topic is further considered as a mixture of words: P (wj |zk ) is the probability of word wj occurring in topic zk . We denote W as the total number of (visual) words, {w1 , ..., wW }. The joint p.d.f. of document, topics and words is formulated as P (di , wj , zk ) = P (wj |zk )P (zk |di )P (di ).

(1)

The prior probability P (di ) is modelledPas a multinomial distribution of words: P (di ) ∝ j n(di , wj ), where n(d, w) is the image-word co-occurrence table, and n(di , wj ) denotes the number of occurrences of wj in di .

2.1. Learning and Inference To estimate the distribution of latent variables P (wj |zk ) and P (zk |di ) that maximize the likelihood of the joint probability, PLSA employs the standard ExpectationMaximization (EM) algorithm [4]. The EM algorithm consists of two steps: the E-step computes the posterior probabilities for the latent variables; the M-step maximizes the expected complete data likelihood. Derivations of the E-

P (zk |di ) ∝ P (di ) ∝

X

X

j

n(di , wj )

(2d)

j

Note that these equations need normalization to make them probability distributions. In summary, given n(di , wj ), maximum likelihood fitting by the EM algorithm yields P (wj |zk ), P (zk |di ), and P (di ). During inference (test stage), given a test image dquery , the factors P (zk |dquery ) are computed using the “fold-in” technique described in [8]: the EM algorithm is run in the same way as in learning, but now keeping the factors P (wj |zk ) obtained in the learning stage fixed.

3. Semantic-Shift The original PLSA model does not explicitly use spatial information. Since there are no constraints among the locations of the interest points or regions, the semantics of the visual words, P (z|d, w), are often inaccurate. In this section we describe a novel model which respects spatial structure. Some works [7] handle the object location by discretizing the location space. This has the unwanted result that, if discretized too fine, the running time becomes too large (with complexity usually linear in the possible locations); if discretized too coarse, the result can be poor due to quantization effects. In this work we allow interest points to live under their original finest resolution, even in a continuous space, since no quantization is required, and hence none of the problems mentioned above will occur in here.

3.1. Model description The graphical model of Semantic-Shift is shown in Fig. 2. For readers familiar with PLSA it is easier if we follow the tradition and define a co-occurrence table to summarize the observed data. Instead of an image-word co-occurrence table, we define an image-word-position cooccurrence table n(d, w, x), with n(di , wj , xdpi ) denoting the number of occurrences of word wj at position xdpi ∈ {xd1i , ..., xd|dii | } in image di , where |di | denotes the number

of words in image di . In other words, n(di , wj , xdpi ) = 1 if in image di there is a word wj at position xdpi , and n(di , wj , xdpi ) = 0 otherwise. The introduction of n(di , wj , xdpi ) is for the convenience of derivation and expression; in the system, there is no need for implementing this co-occurrence table. We assume that in each image there is no more than one single foreground object. Experiments on the UIUC car dataset (see Fig.8) show that even if there are multiple cars (foreground objects) in one image, Semantic-Shift still can produce good results as long as the following assumptions are satisfied. We make the assumption that the foreground object has no holes and has a convex shape. Both assumptions hold for most objects. The reason we need these assumptions is because we want to model the image area occupied by the object as a Gaussian, which is convex in a 2D image. Since there is no specific reason our model should be confined to a Gaussian except for simplicity, it should be possible to loosen the Gaussian shape assumption so that the model can handle more complicated shapes of objects. We introduce the conditional probability P (x|d, z). The dependence of position x on image d allows the foreground object to have different locations and scales in every image d. In other words, the foreground object is allowed to be at vastly different positions in both learning and testing. The scale can also vary freely over different images. Allowing different locations and scales in different images is desirable, as it provides the basis for scale-invariance and translation-invariance. The dependence of x on z allows the foreground object and the background clutter to have different locations and scales. We model the location distribution of the background clutter as the probability of the complement of that location being foreground. This describes the realistic situation that, at a particular location, the higher the probability of being foreground, the lower the probability of being background.

3.2. Location and Scale Estimation for Foreground Object The conditional probability P (x|d, z) is computed for each topic in each image. Denote the topic zk that corresponds to the foreground object as zF G and call it the foreground topic. Since we want the system to be unsupervised, we need to create an unsupervised rule for deciding which of the two topics, z1 and z2 , is the foreground topic zF G . We achieve this by assuming that the foreground topic has on average a smaller spatial support than the background topic. We call this step foreground topic identification, as described below. In the literature, finding a robust estimate of location and scale under the univariate model assumption is not new. In our experiments, we simply take the weighted mean and weighted standard deviation as estimates of the location

and scale of the foreground object. Specifically, we define the spatial support of a topic zk in an image di as the weighted standard deviation of the positions of all interest points {xd1i , ..., xd|dii | } , where each interest point xdpi is given a weight vp = P (zk |di , wj ), where wj is the visual word corresponding to point xdpi . The weighted standard deviation is defined as vP u |di | u p=1 vp (xdpi − µ ˆik )2 σ ˆik = t (3) P |di | |di |−1 v p p=1 |di | where µ ˆik is the weighted sample mean of the positions of all interest points. After foreground topic identification, we denote the location and scale estimate of the foreground object in image di as µ ˆi,F G and σ ˆi,F G , and the corresponding topic as zF G . We assume the interest points belonging to the foreground topic have a spatial distribution in the form of a Gaussian, P (xdpi |zF G , di ) ≡ N (xdpi |ˆ µi,F G ,ˆ σi,F G ). The spatial distribution of the background is then set to the complement of the former distribution, meaning that the more likely the foreground, the less likely the background, and vice versa.

3.3. Model fitting The goal is to maximize the log-likelihood, XXX n(di , wj , xdpi ) log P (di , wj , xdpi ) L= i

(4)

p

j

where the joint probability P (di , wj , xdpi ) factorizes as P (di , wj , xdpi ) = P (di )P (zk |di )P (wj |zk )P (xdpi |zk , di ) (5) We use a modified version of the EM algorithm where the location and scale of the foreground object are estimated in each iteration. We use E’-step and M’-step to denote the two iteration steps. In each iteration of Semantic-Shift, the posterior probability P (zk |di , wj , xdpi ) is updated as in Eq. (6). This quantity tells us how likely the visual word wj at position xdpi in image di is part of the foreground (or background) object. Using this posterior probability, we can compute the location and scale estimates of the foreground object, as explained in the previous section. This explains Eq. (10). Here is the Semantic-Shift algorithm: E’-step: P (zk |di , wj , xdpi ) ∝ P (zk |di )P (wj |zk )P (xdpi |zk , di ) (6) M’-step: P (wj |zk ) ∝

XX i

p

nijp P (zk |di , wj , xdpi )

(7)

P (zk |di ) ∝

XX j

P (di ) ∝

XX j

4. Image representation nijp P (zk |di , wj , xdpi )

(8)

p

nijp

(9)

p

P (xdpi |zk , di ) updated according to Section 3.2

(10)

where nijp ≡ n(di , wj , xdpi ). Note that these equations need normalization to make them probability distributions. Both learning and inference (test stage) use the above iterative procedure to obtain the conditional probabilities, except that during inference (test stage) the factor P (wj |zk ) is kept fixed and not being updated anymore. After each iteration, the location and scale estimates of the foreground topic are shifted to a new value, and the Gaussian distribution P (xdpi |zk , di ) is updated accordingly. Notice that the E’-step depends on the term P (xdpi |zk , di ), i.e., the “shift” of the location and scale estimates plays a central role in the overall iterative scheme. It is worth mentioning that, even though the location and scale estimates are found on a per image basis, they are actually tightly coupled with all the system parameters across all images, since the same conditional probability table P (w|z) is used by all images. The performance of the iterative scheme depends on the quality of initialization. Since we did not care to find a good initialization scheme for PLSA and Semantic-Shift, the experimental results we show later will come from 50 experiments with random initializations. Here is a flowchart of the overall unsupervised object detection system, which includes Semantic-Shift in Step 4 and 5: Step 1 Compute P (z|d, w), P (w|z), and P (z|d) from training data using PLSA. The first one is used in Step 2 and 3, and the other two are used in Step 4 to initialize Semantic-Shift. Step 2 Foreground topic identification. Step 3 Compute location and scale estimate for each image to initialize P (x|d, z) in Semantic-Shift. Step 4 Compute P (w|z) from training data using Semantic-Shift. Step 5 Run Semantic-Shift on test data. Table 1. Flowchart of system

We use the bag-of-visual words [3] representation, where visual words are the basic units that form the observations of an image. First we need to find a number of regions to generate the visual words from. These regions are determined by running the Canny edge detector and then uniformly sampling points from all edges. In this work we call these points interest points, even though the term usually refers to stable points rather than just samples from edges. Scale Invariant Feature Transform (SIFT) image features [9] around the interest points are computed. This procedure (using the code from [6]) provides a set of local feature vectors. Note that SIFT image features are general and can be applied to a wide range of different objects and tasks [9]. To obtain a finite set of visual words, we perform k-means clustering on all local feature vectors from all training images. The resulting cluster centers form the dictionary of visual words, {w1 , ..., wW }. For each training or test image, its visual words are obtained by choosing the closest wi for each of its local feature vectors. Note that the visual words are neither obtained from labelled data, nor are they specifically designed for the face/non-face or car/non-car tasks in the experiments, implying generality for other objects and the unsupervised nature of this system.

5. Experiments We use two datasets in the experiments. In the Caltech face dataset [12] experiment, we use 450 face images and 451 non-face images and set the number of latent topics to two. Images are resized to a width of 200 pixels and keeping the original aspect ratio, and color information is discarded. Half of the images in both categories are used for unsupervised learning the table P (w|z), and the other half is used for testing. The faces have very limited variation in scale. The UIUC car dataset [1] consists of the following: 550 cropped sideview car images, all of which are used for unsupervised learning; 500 non-car images, from which 382 are randomly chosen for unsupervised learning and the rest for testing; 170 non-cropped sideview car images, all of which are used for testing. In total, there are 550 positive and 382 negative images for learning, and 170 positive and 118 negative images for testing. The numbers are chosen so that the ratio between positive and negative images is the same for learning and for testing. All images are in grayscale. Since the training images are already cropped and contain very little background, there is no need for location and scale estimation during the learning stage. (This is not the case in the Caltech face dataset.) This is done by setting P (xdpi |zF G , di ) to a uniform distribution. In the test stage (inference) the images do contain a lot of background, so the location and scale estimation in Semantic-Shift is enabled.

Figure 3. Learning stage of Semantic-Shift. Estimated topic of each visual word shown in red ellipse (foreground topic) or green plus sign (background topic) after the 1st , 6th , 12th , and 26th iteration. Green ellipses are not drawn for clearer visualization. The sum of red ellipses and green + is fixed for each image.

Figure 4. Test (inference) stage of Semantic-Shift.

The car dataset has larger variation in scale compared to the face dataset. The computation time for Semantic-Shift is around 2 seconds for unsupervised learning and around 1.5 seconds for testing per image on a Pentium-4 machine. The iteration steps in Semantic-Shift are written in MATLAB and not optimized for speed yet. In particular, if the conditional probability tables in the EM step were computed by array multiplications instead of using for loops, significant gains in speed can be expected. The above timing excludes the preprocessing steps of interest points detection and bag of words representation.

5.1. Detecting objects in images To decide the presence/absence of the foreground object in the scene, we compute P (di |zF G ) ∝ P (zF G |di )P (di ) for each image di ; the higher it is, the more likely that image contains a foreground object. Fig. 6 shows that Semantic-Shift performs statistically significantly better than PLSA in the detection task on both the face and car datasets. On the car dataset, Semantic-Shift has a medium area under ROC curve (AUC) over 0.98. The value for PLSA is around 0.90. The medium AUC and other statistics for the boxplot were collected from 50 ex-

periments, where in each experiment the parameters were randomly initialized.

5.2. Labelling the visual words By labelling each visual word with its most likely topic, we can obtain a rough localization of the foreground object (see Fig. 7, 8). The most likely topic of each visual word in PLSA is computed from the posterior in Eq.(2a): z ∗ = arg max P (z|di , wj ) z

(11)

In Semantic-Shift, the most likely topic of each visual word can be similarly obtained from the posterior P (z|di , wj , xdpi ) in Eq. (6). The red and green ellipses in Fig. 7 and 8 represent the inferred most likely topics of each visual word; red indicates that the system labels the particular region as foreground. Comparing PLSA to Semantic-Shift, it can be seen that foreground objects are more precisely located by SemanticShift. Fig. 3, 4, and 5 show how Semantic-Shift evolves over time. In each iteration, the location and scale of the foreground object are re-estimated, and so are the probabilities

Figure 5. Test (inference) stage of Semantic-Shift on the UIUC car dataset. Estimated topic of each visual word shown in red ellipse (foreground topic) or green plus sign (background topic) after the 1st , 3rd , 7th , and 9th iteration. Green ellipses are not drawn for clearer visualization. The sum of red ellipses and green + is fixed for each image.

(a) Boxplot for the Caltech face dataset

(b) Boxplot for the UIUC car dataset

Figure 6. In each boxplot, the left column is PLSA, the right column is Semantic-Shift. Vertical axis is area under ROC curve. SemanticShift has significantly better performance.

P (z|d) and P (w|z). Note that even though the location and scale estimates are computed per image, but the probability P (w|z) is shared by all images, hence the quality of the location and scale estimates in each image actually affects the detection and labelling results in all the other images. We want to emphasize that Semantic-shift is not a post-processing step working on individual images; it is a procedure that tightly couples the location and scale estimates of all images with the global parameter P (w|z). We experimented with various other ways to obtain the location and scale estimates. We found that if we force the topic with larger spatial support to have uniform distribution, the performance is not as good as using the comple-

ment of the Gaussian distribution of the foreground topic. We also tried to force the topic with larger spatial support to have a Gaussian density like the foreground topic has, and with its own location and scale estimates. The motivation for doing so is that background clutter is often also spatially clustered. However, the performance was not as good.

6. Conclusion and Future work The Semantic-Shift model provides a translation and scale invariant basis for unsupervised object detection, i.e., the unlabelled foreground object can have different location and scale in each image in both learning and testing.

formable objects, the spatial configuration becomes harder to model and may require extensive training data. SemanticShift simply considers an object as a blob of visual words without modelling their intra-configuration. Extending the experiments and the framework to more complicated objects is of future interest.

7. Acknowledgement Supported by the Taiwan Merit Scholarship TMS-094-1-A-049 and by the ARDA VACE program.

References

Figure 7. Results on test data. Left column: PLSA. Right column: Semantic-Shift.

Figure 8. Results on test data. Left column: PLSA. Right column: Semantic-Shift.

We have demonstrated its superior performance in detection and localization compared with a recent unsupervised method, PLSA. We expect using robust estimation methods for location and scale estimation would boost the performance of Semantic-Shift further. The weighted mean and weighted standard deviation used now are not robust estimates. In this paper, we do not model the intra-object spatial configuration of interest points such as in the Constellation model [13] or more recent models [7][2]. Although for objects that have consistent spatial relationship between parts modelling the intra-object spatial configuration would certainly boost the performance, but in highly articulated or de-

[1] S. Agarwal, A. Awan, and D. Roth. Learning to detect objects in images via a sparse, part-based representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26:1475–1490, 2004. UIUC car dataset. 4 [2] D. Crandall and D. Huttenlocher. Weakly supervised learning of part-based spatial models for visual object recognition. In European Conference on Computer Vision (ECCV), 2006. 7 [3] C. Dance, J. Willamowski, L. Fan, C. Bray, and G. Csurka. Visual categorization with bags of keypoints. In ECCV International Workshop on Statistical Learning in Computer Vision, 2004. 4 [4] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39:1–38, 1977. 2 [5] L. Fei-Fei and P. Perona. A bayesian hierarchical model for learning natural scene categories. In IEEE Conf. Computer Vision and Pattern Recognition (CVPR), 2005. 1 [6] R. Fergus. ICCV short course 2005, http: //people.csail.mit.edu/fergus/iccv2005/ bagwords.html. 4 [7] A. B. Hillel, T. Hertz, and D. Weinshall. Efficient learning of relational object class models. In IEEE Intl. Conf. on Computer Vision (ICCV), 2005. 2, 7 [8] T. Hofmann. Unsupervised learning by probabilistic latent semantic analysis. Machine Learning, 42:177–196, 2001. 1, 2 [9] D. G. Lowe. Distinctive image features from scale-invariant keypoints. International Journal of Computer Vision, 60:91– 110, 2004. 4 [10] P. Quelhas, F. Monay, J.-M. Odobez, D. Gatica-Perez, T. Tuytelaars, and L. Van Gool. Modeling scenes with local descriptors and latent aspects. In IEEE Intl. Conf. on Computer Vision (ICCV), 2005. 1 [11] J. Sivic, B. Russell, A. Efros, A. Zisserman, and W. Freeman. Discovering objects and their location in images. In IEEE Intl. Conf. on Computer Vision (ICCV), 2005. 1 [12] M. Weber. Caltech face dataset, http://www.vision. caltech.edu/archive.html. 4 [13] M. Weber, M. Welling, and P. Perona. Unsupervised learning of models for recognition. In European Conference on Computer Vision (ECCV), 2000. 7

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notated images for constructing a supervised image under- standing system. .... the same way as in learning, but now keeping the factors. P(wj|zk) ... sponds to the foreground object as zFG and call it the fore- ..... In European Conference on.

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