![A Probabilistic Model for Recursive Factorized Image Features
Sergey Karayev 1 Mario Fritz 1,2 Sanja Fidler 1,3 Trevor Darrell 1
1 2 3
UC Berkeley and ICSI MPI Informatics University of Toronto
Berkeley, CA Saarbr¨ cken, Germany
u Toronto, ON
{sergeyk, mfritz, sanja, trevor}@icsi.berkeley.edu
Abstract
L1 T1
representation activations L1 T1 activations
Layered representations for object recognition are im- lea
rn inference lea
rn
ing ing
portant due to their increased invariance, biological plau-
Joint Inference
sibility, and computational benefits. However, most of exist- L1 patch
L1 patch
ing approaches to hierarchical representations are strictly L0 T0 activations
representation L0 T0 activations
feedforward, and thus not well able to resolve local ambigu- lea
rn
inference
l ea
ing
ities. We propose a probabilistic model that learns and in- Joint
rn
ing
representation
fers all layers of the hierarchy jointly. Specifically, we sug- w L0 patch w L0 patch
gest a process of recursive probabilistic factorization, and (a) Feedforward visual hierarchies (b) Our full probabilistic model
present a novel generative model based on Latent Dirichlet
Allocation to this end. The approach is tested on a stan- Figure 1: (a) Traditional visual hierarchies are feedforward
dard recognition dataset, outperforming existing hierarchi- with known disadvantages [19]. In contrast, our probabilis-
cal approaches and demonstrating performance on par with tic model (b) learns and infers all layers jointly.
current single-feature state-of-the-art models. We demon-
strate two important properties of our proposed model: 1)
adding an additional layer to the representation increases nents [32]. The increased robustness of such distributed
performance over the flat model; 2) a full Bayesian ap- representations is appealing, and may exist in biological
proach outperforms a feedforward implementation of the systems [23]. It has also been shown that such factoriza-
model. tion leads to state-of-the-art performance on existing object
recognition datasets [5], and allows good performance on
novel datasets [10].
1. Introduction Despite increased representative power, these local fea-
tures are still input directly to global object representations.
One of the most successful and widely used develop- While this approach has yielded some of the best recog-
ments in computer vision has been the rise of low-level lo- nition performance to date [32] , some recent works have
cal feature descriptors such as SIFT [21]. The basic idea of shown that multi-layer intermediate visual representations
such local feature descriptors is to compactly yet discrim- could improve recognition performance by increased ro-
inatively code the gradient orientations in small patches of bustness to variance [22, 9, 33].
an image. These features have been successfully used for This is also in line with current theories of hierarchi-
scene and object recognition by representing densely ex- cal, or layered, processing in the mammalian visual cortex
tracted descriptors in terms of learned visual words—cluster [26]. Indeed, these theories give strong support to the im-
centers in descriptor space [29]. On top of this quantized portance of feedback, which both improves features that are
representation, more global image representations such as being learned and disambiguates local information during
bags of words or spatial pyramids [17] can be assembled. inference. However, most existing hierarchical models are
Recent publications in the field have started re- strictly feedforward (with a few notable exceptions, such
evaluating the hard clustering approach of visual words as [13]), with each layer of the hierarchy operating over the
in favor of “softer” representations that allow a single de- fixed output of the previous layer [27, 25, 18].
scriptor to be represented as a mixture of multiple compo- The aim of this paper is to develop a fully probabilis-
401](https://image.slidesharecdn.com/aprobabilisticmodelforrecursivefactorizedimagefeatures-110428055457-phpapp02/85/A-probabilistic-model-for-recursive-factorized-image-features-1-320.jpg)
![tic hierarchical model to learn and represent visual features tions of this idea have shown promising classification re-
at different levels of complexity. The layers of the hierar- sults [22]. The units in the original HMAX model were
chy are learned jointly, which, we show, has crucial advan- pre-designed, not learned, while the recent improvements
tage in performance over its feedforward counterpart. Our have included random template selection from a training
model is based on Latent Dirichlet Allocation (LDA) [4], corpus [27].
a probabilistic factorization method originally formulated Learning mid-level visual features in a recursive hi-
in the field of text information retrieval. It has been suc- erarchical framework has motivated several recent works
cessfully applied to the modeling of visual words [28], and, on convolutional networks [25, 2], deep Boltzmann Ma-
more recently, to modeling of descriptors in an object detec- chines [12, 18], hyperfeatures [1], fragment-based hierar-
tion task [10]. Here, we extend the latter representation to chies [30], stochastic grammars [34] and compositional ob-
a recursively defined hierarchical probabilistic model, and ject representations [9, 33]. The underlying ideas behind
show its advantage over the flat approach and the feedfor- these approaches are similar, but they differ in the type of
ward implementation of the model. representation used.
The approach is tested on a standard recognition dataset, Convolutional networks stack one or several feature ex-
showing performance that is higher than previous hierar- traction stages, each of which consists of a filter bank layer,
chical models and is on par with current single-descriptor- non-linear transformation layers, and a pooling layer that
type state of the art. We demonstrate two important prop- combines filter responses over local neighborhoods using
erties of our proposed model: 1) adding an additional layer a pooling operation, thereby achieving invariance to small
to the LDA representation increases performance over the distortions [25]. In [1], the authors propose a representa-
flat model; 2) a full Bayesian approach outperforms a feed- tion with “hyperfeatures” which are recursively formed as
forward implementation of the model. This highlights the quantized local histograms over the labels from the pre-
importance of feedback in hierarchical processing, which is vious layer, which in turn are quantized local histograms
currently missing from most existing hierarchical models. over smaller spatial areas in images. Compositional hierar-
Our probabilistic model is a step toward nonparametric ap- chies [9, 33, 24] and stochastic grammars [34] define ob-
proaches to distributed coding for visual representations. jects in terms of spatially related parts which are then re-
cursively defined in terms of simpler constituents from the
2. Related work layers below.
There is strong biological evidence for the presence of Most of these models process information in a feedfor-
object-specific cells in higher visual processing areas. It is ward manner, which is not robust to local ambiguities in
believed that the complexity of represented shape gradually the visual input. In order to disambiguate local informa-
increases as visual signal travels from the retina and LGN tion, contextual cues imposed by more global image rep-
through V1 and higher cortical areas [26]. While the con- resentation may need to be used. A probabilistic Bayesian
nectivity between different visual areas is far from being un- framework offers a good way of recruiting such top-down
derstood, there is evidence for a hierarchical organization of information [19] into the model.
the visual pathway, in which units at each level respond to In this paper, we propose a recursive Bayesian prob-
the output of the units at the level below, aggregated within abilistic model as a representation for the intermediately
a larger spatial area [6]. complex visual features. Our model is based on Latent
We will refer to computational modeling of a hierar- Dirichlet Allocation [4], a latent factor mixture model ex-
chy of stacked layers of increasing complexity as recur- tensively used in the field of text analysis. Here, we ex-
sive models. The main idea behind these approaches is tend this representation to a recursively defined probabilis-
to achieve high-level representation through recursive com- tic model over image features and show its advantage over
bination of low-level units, gradually encoding larger and the flat approach as well as a feedforward implementation.
larger spatial areas in images. This allows more efficient Due to overlapping terminology, we emphasize that
parametrization of image structures, and potentially better our recursive LDA model is inherently different from
recognition performance. Hierarchical-LDA [3] which forms a hierarchy of topics
A number of recursive models have been proposed. The over the same vocabulary. In contrast, our hierarchy is
HMAX model formulated the increase in shape complex- over recursively formed inputs, with a fixed base vocabu-
ity as the interaction of two types of operations: template lary. Similar structure is seen in the Pachinko Allocation
matching and max pooling [27]. Units performing these op- Model [20], which generalizes the LDA model from a sin-
erations are stacked in layers, where the template-matching gle layer of latent topic variables to an arbitrary DAG of
units on the bottom of each layer receive inputs from small variables. As we explain later, this structure is similar to
patches of the image, and the pooling units on top of each our model, but crucially, it is missing a spatial grid that al-
layer output more complex visual features. Implementa- lows us to model image patches of increasing spatial sup-
402](https://image.slidesharecdn.com/aprobabilisticmodelforrecursivefactorizedimagefeatures-110428055457-phpapp02/85/A-probabilistic-model-for-recursive-factorized-image-features-2-320.jpg)
![α θ
(z1 ,·) X1
χ ∈R θ (d)
X1
z1 z1
x1 χ x1
φ(z1 ,·,·) ∈ RX1 ×T0
T1 X1
T0 L1 β1 φ1 z0
T1
z0
χ(z0 ,·) ∈ RX0 x0
X0
χ x0
φ(z0 ,·,·) ∈ RX0 ×T0
X 0
T0 w
L0 β0 φ0 w
V T0 N (d)
D
(a) (b)
Figure 2: a) Concept for the Recursive LDA model. b) Graphical model describing two-layer RLDA. Wednesday, November 10, 2010
p(w, z0,1 |x0,1 , α, β0,1 ) = p(w, z1 , z0 , φ1 , φ0 , θ, χ1 , χ0 |x1 , x0 , α, β1 , β0 )dχ0 dχ1 dφ0 dφ1 dθ (2)
θ φ1 φ0 χ1 χ0
= p(w, z1 , z0 , φ1 , φ0 , θ|x1 , x0 , α, β1 , β0 )dφ0 dφ1 dθ (3)
θ φ1 φ0
= p(θ|α)p(z1 |θ)dθ p(φ1 |β1 )p(z0 |φ1 , z1 , x1 )dφ1 p(w|φ0 , z0 , x0 )p(φ0 |β)dφ0 (4)
θ φ1 φ0
top layer intermediate layer evidence layer
L−1
p(w, z1,...,L |x1,...,L , α, β1,...,L ) = p(θ|α)p(zL |θ)dθ p(φl |βl )p(zl−1 |φl , zl , xl )dφl p(w|φ0 , z0 , x0 )p(φ0 |β)dφ0 (5)
θ l=1 φl φ0
top layer L layer l evidence layer
Figure 3: RLDA conditional probabilities for a two-layer model, which is then generalized to an L-layer model.
all variables x are observed and the deterministic transfor- different numbers of components at each of the layers and
mation between φ and χ has probability one. Our formula- explore single-layer, feed-forward (FLDA), and fully gen-
tion now closely resembles the Latent Dirichlet Allocation erative (RLDA) models, showing that 1) classification per-
but adds an additional layer in between that also performs formance improves with an additional layer over the single-
spatial grouping via the variables x. This formulation easily layer model, and 2) the fully generative RLDA model im-
generalizes to L layers, as shown in equation (5). proves over the feed-forward-only FLDA analog.
The derivation of the Gibbs Sampling equations is analo- Our work seeks to improve local features for low and
gous to derivations for LDA (for an excellent reference, see mid-level vision independent of any specific object recog-
[11]), with the addition of the observed position variables nition methods, and we do not innovate in that regard. We
x. Due to space constraints, the derivations are presented in note that we test our model using only a single feature, and
the supplementary materials. compare it only to other single-descriptor approaches, fo-
cusing on hierarchical models.
4. Experiments
4.1. Implementation
To evaluate the performance of our probabilistic recur-
sive model, we test on the classification dataset Caltech The feature representation of an image for our approach
101 [7], which has been the most common evaluation of are SIFT descriptors of size 16 × 16 pixels, densely ex-
hierarchical local descriptors. We show performance for tracted from the image with a stride of 6 pixels. Individ-
404](https://image.slidesharecdn.com/aprobabilisticmodelforrecursivefactorizedimagefeatures-110428055457-phpapp02/85/A-probabilistic-model-for-recursive-factorized-image-features-4-320.jpg)
![Table 1: Comparison of RLDA with 128 components at layer L1 and 1024 components at layer L2 to other hierarchical
approaches. Under layers, “bottom” refers to the L1 features of the RLDA model, “top” to L2 features, and “both” denotes
the feature obtained by stacking both layers.
Approach Caltech-101
Model Layer(s) used 15 30
RLDA (1024t/128b) bottom 56.6 ± 0.8% 62.7 ± 0.5%
Our Model RLDA (1024t/128b) top 66.7 ± 0.9% 72.6 ± 1.2%
RLDA (1024t/128b) both 67.4 ± 0.5 73.7 ± 0.8%
Sparse-HMAX [22] top 51.0% 56.0%
CNN [16] bottom – 57.6 ± 0.4%
CNN [16] top – 66.3 ± 1.5%
Hierarchical CNN + Transfer [2] top 58.1% 67.2%
Models CDBN [18] bottom 53.2 ± 1.2% 60.5 ± 1.1%
CDBN [18] both 57.7 ± 1.5% 65.4 ± 0.4%
Hierarchy-of-parts [8] both 60.5% 66.5%
Ommer and Buhmann [24] top – 61.3 ± 0.9%
Table 2: Results for different implementations of our model with 128 components at layer L1 and 128 components at L2 .
For LDA models, “bottom” refers to using SIFT patches as input, while “top” refers to using 4 × 4 SIFT superpatches.
Approach Caltech-101
Model Basis size Layer(s) used 15 30
LDA 128 “bottom” 52.3 ± 0.5% 58.7 ± 1.1%
RLDA 128t/128b bottom 55.2 ± 0.3% 62.6 ± 0.9%
LDA 128 “top” 53.7 ± 0.4% 60.5 ± 1.0%
128-dim
FLDA 128t/128b top 55.4 ± 0.5% 61.3 ± 1.3%
models
RLDA 128t/128b top 59.3 ± 0.3% 66.0 ± 1.2%
FLDA 128t/128b both 57.8 ± 0.8% 64.2 ± 1.0%
RLDA 128t/128b both 61.9 ± 0.3% 68.3 ± 0.7%
ual descriptors were processed by our probabilistic mod- 128 components at the bottom layer, and 128 and 1024
els, and results of inference were used in a classification components at the top layer.
framework described in section 4.2. Because LDA requires
discrete count data and SIFT dimensions are continuous- 3. RLDA: The full model was trained on the same size
valued, normalization of the maximum SIFT value to 100 patches as FLDA described above: SIFT descriptors in
tokens was performed; this level of quantization was shown a 4 × 4 spatial arrangement, with model parameters set
to maintain sufficient information about the descriptor. accordingly. We tested 128 components at the bottom,
We trained and compared the following three types of and 128 and 1024 components at the top layer.
models: 4.2. Evaluation
1. LDA: LDA models with various numbers of compo- The setup of our classification experiments follows the
nents (128, 1024, and 2048) trained on 20K randomly Spatial Pyramid Match, a commonly followed approach in
extracted SIFT patches. We also trained LDA models Caltech-101 evaluations [17]. A spatial pyramid with 3 lay-
on “superpatches” consisting of 4 × 4 SIFT patches, to ers of 4 × 4, 2 × 2, and 1 × 1 grids was constructed on
give the same spatial support as our two-layer models. top of our features. Guided by the best practices outlined
in a recent comparison of different pooling and factoriza-
2. FLDA: The feed-forward model first trains an LDA tion functions [5], we used max pooling for the spatial pyra-
model on SIFT patches, as above. Topic activations mid aggregation. For classification, we used a linear SVM,
are output and assembled as 4 × 4 superpatches. An- following the state-of-the-art results of Yang et al. [32].
other LDA model is learned on this input. We tested Caltech-101 is a dataset comprised of 101 object categories,
405](https://image.slidesharecdn.com/aprobabilisticmodelforrecursivefactorizedimagefeatures-110428055457-phpapp02/85/A-probabilistic-model-for-recursive-factorized-image-features-5-320.jpg)
![70 70
65 65
Average % correct
Average % correct
60
60
55
55
50
RLDA,top,128t/128b 50 RDLA,both,128t/128b
45
FLDA,top,128t/128b RLDA,top,128t/128b
40
LDA,"top",128b 45 FLDA,both,128t/128b
LDA,"bottom",128b FLDA,top,128t/128b
35 40
5 10 15 20 25 30 5 10 15 20 25 30
Number of training examples Number of training examples
(a) Caltech-101: 128 topic models, LDA-SIFT vs FLDA vs RLDA (b) Caltech-101: 128 topic models, L2 vs L1 +L2 models
Figure 4: Comparison of classification rates on Caltech-101 of the one-layer model (LDA-SIFT) with the feed-forward
(FLDA) and full generative (RLDA) two-layer models for different number of training examples, all trained with 128 at L1
and 128 at L2 layers. We also compare to the stacked L1 and L2 layers for both FLDA and RLDA, which performed the best.
with differing numbers of images per category [7]. Follow- FLDA model, which in turn always outperforms the one-
ing standard procedure, we use 30 images per category for layer model.
training and the rest for testing. Mean accuracy and stan- We have also tested the classification performance of
dard deviation are reported for 8 runs over different splits stacking both L1 and L2 features into a feature vector. This
of the data, normalized per class. feature contains more information, as both spatially larger
We first tested the one-layer LDA model over SIFT fea- and smaller features are taken into account. This is a stan-
tures. For 128 components we obtain classification accu- dard practice for hierarchical models [18, 14]. The results,
racy of 58.7%. Performance increases as we increase the presented in Table 2 (under “both”) and in Figure 4b, show
number of components: for 1024 topics the performance is that using information from both layers improves perfor-
68.8%, and for 2048 it is 70.4% (all numbers here and on mance further, by about 3% for FLDA and 2% for RLDA.
refer to the case of 30 training examples). We also compared the performance of just the first
The emphasis of our experiments was to evaluate the layer L1 obtained with the RLDA model to the single-
contribution of additional layers to classification perfor- layer model (which also forms the first layer of the FLDA
mance. We tested the FLDA and RLDA models (described model). Interestingly, the decomposition learned in L1
in 4.1) in the same regime as described above. of the RLDA model outperforms the single layer model
As an initial experiment, we constrained the number of (62.6% vs 58.7%). This demonstrates that learning the lay-
topics to 128 and trained three models: single-layer, FLDA, ers jointly is beneficial to the performance of both bottom
and RLDA. The two-layer models were trained with 128 and top layers, separately examined.
topics on the bottom layer, corresponding to SIFT descrip- Following this evaluation, we also learned two-layer
tors, and 128 on the top. models with 1024 topics in the top layer and 128 in the
First, only the top-layer topics were used for classifica- bottom layer. The results hold, but the difference between
tion. The dimensionality of vectors fed into the classifier the models becomes smaller—we hypothesize that single-
was therefore the same for the two-layer and the single- feature performance on the dataset begins saturating at these
layer models. We present the results in Table 2. levels. RLDA top layer performs at 72.6% vs. 72.5% for
Both of the two-layer models improved on the one-layer FLDA and 68.8% for the single-layer model. The bottom
model, obtaining classification rates of 61.3% for FLDA layer of RLDA achieves 62.7% classification accuracy vs
and 66.0% for RLDA, compared to the single-layer results 62.6% for FLDA. Using both layers, RLDA gets 73.7%
of 58.7%, when trained on SIFT patches, and 60.6%, when while FLDA gets 72.9%.
trained on 4 × 4 SIFT “superpatches” (which are reported in The comparison with related hierarchical models is given
Table 2 under LDA “top”). Detailed results across different in Table 1, which shows that while the baseline single-
numbers of training examples are shown in Figure 4a. The layer LDA-SIFT model performs worse than most of the
figure shows that the RLDA model always outperforms the approaches, the proposed RLDA model outperforms the ex-
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![isting work by more than 5%.
Our best performance of 73.7% compares well to the
current best performing approaches: sparse coding with
73.2% [32] and 75.7% [5] (using increased spatial support
(a) FLDA
and much denser sampling), and LCC [31] with 73.5%.
State-of-the art single-feature performance has been shown
by keeping all training data in a non-parametric classifica-
tion scheme, obtaining 78.5% [15].
We note that in the existing literature on hierarchical
models, most authors report best performance with just one
learned layer, while adding another layer seems to decrease
the recognition performance [27, 25, 8]. In contrast, in our
model, the performance increases by adding another layer
to the representation. The most important result, however,
(b) RLDA
is that the full Bayesian model as proposed in the paper out-
performs the feed-forward approach. This supports the idea
that inference carried out in the Bayesian approach results
in more stable estimations of the feature activations, and
thus better and more robust recognition performance. This
highlights the importance of feedback in hierarchical recog-
nition. top layer bottom layer
4.3. Role of Feedback in the Model Figure 5: Comparison of the (a) feed-forward and (b)
full generative two-layer models in terms of components
Additional evidence for the crucial role of feedback in learned, visualized as average image patches. While FLDA
the model comes from visualization of the average image learns only localized edge orientations, RLDA learns more
patches corresponding to top- and bottom-layer components complex spatial structures.
learned by the two models. Figure 5 shows that the full gen-
erative RLDA model uses lower-layer components in a no-
tably different way than the feed-forward model, and learns
and that the full generative process performs better than the
different, more complex spatial structures at the top layer.
feed-forward approach. This emphasizes the necessity of
In the feed-forward model, the bottom-layer topics are
feedback in hierarchical visual recognition models.
in essence orientation filters. The second layer does not
impose any additional structure on them, and therefore the Probabilistic models are robust, make modular combi-
top-layer topics appear to be the same simple orientations, nations easy, and form the basis of possible non-parametric
localized within a bigger spatial support. In the fully gener- extensions. The theorized goal of hierarchical approaches
ative RLDA model, the top-layer components seem to rep- to vision is reaching object-level representations. With
resent more interesting and potentially discriminative spa- additional layers and variables, our model can be developed
tial structures. from merely a source of mid-level features to a full-scale
We also found that the RLDA bottom-layer activations object recognition method. With a non-parametric exten-
exhibit stronger correlations between topic activations in sion, the number of components would be inferred from the
neighboring patches, which suggests that the model allows training data, which would appear especially important at
bottom layer inference to represent continuous structures the part- and object-level representation. These topics are
across subpatches. subject of future work.
5. Conclusions
We presented a probabilistic model for visual features
of increasing complexity and spatial support. The layers Acknowledgements: Various co-authors of this work
of the hierarchy are trained jointly. We demonstrate per- were supported in part by a Feodor Lynen Fellowship
formance that is among the recent best in single-descriptor granted by the Alexander von Humboldt Foundation; by
approaches, and that outperforms existing hierarchical ap- awards from the US DOD and DARPA, including contract
proaches. Most importantly, we show that adding an- W911NF-10-2-0059; by NSF awards IIS-0905647 and IIS-
other layer to our model significantly improves performance 0819984; by EU FP7-215843 project POETICON; and by
(something that is rarely true for layered models in vision), Toyota and Google.
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A probabilistic model for recursive factorized image features
- 1. A Probabilistic Model for Recursive Factorized Image Features Sergey Karayev 1 Mario Fritz 1,2 Sanja Fidler 1,3 Trevor Darrell 1 1 2 3 UC Berkeley and ICSI MPI Informatics University of Toronto Berkeley, CA Saarbr¨ cken, Germany u Toronto, ON {sergeyk, mfritz, sanja, trevor}@icsi.berkeley.edu Abstract L1 T1 representation activations L1 T1 activations Layered representations for object recognition are im- lea rn inference lea rn ing ing portant due to their increased invariance, biological plau- Joint Inference sibility, and computational benefits. However, most of exist- L1 patch L1 patch ing approaches to hierarchical representations are strictly L0 T0 activations representation L0 T0 activations feedforward, and thus not well able to resolve local ambigu- lea rn inference l ea ing ities. We propose a probabilistic model that learns and in- Joint rn ing representation fers all layers of the hierarchy jointly. Specifically, we sug- w L0 patch w L0 patch gest a process of recursive probabilistic factorization, and (a) Feedforward visual hierarchies (b) Our full probabilistic model present a novel generative model based on Latent Dirichlet Allocation to this end. The approach is tested on a stan- Figure 1: (a) Traditional visual hierarchies are feedforward dard recognition dataset, outperforming existing hierarchi- with known disadvantages [19]. In contrast, our probabilis- cal approaches and demonstrating performance on par with tic model (b) learns and infers all layers jointly. current single-feature state-of-the-art models. We demon- strate two important properties of our proposed model: 1) adding an additional layer to the representation increases nents [32]. The increased robustness of such distributed performance over the flat model; 2) a full Bayesian ap- representations is appealing, and may exist in biological proach outperforms a feedforward implementation of the systems [23]. It has also been shown that such factoriza- model. tion leads to state-of-the-art performance on existing object recognition datasets [5], and allows good performance on novel datasets [10]. 1. Introduction Despite increased representative power, these local fea- tures are still input directly to global object representations. One of the most successful and widely used develop- While this approach has yielded some of the best recog- ments in computer vision has been the rise of low-level lo- nition performance to date [32] , some recent works have cal feature descriptors such as SIFT [21]. The basic idea of shown that multi-layer intermediate visual representations such local feature descriptors is to compactly yet discrim- could improve recognition performance by increased ro- inatively code the gradient orientations in small patches of bustness to variance [22, 9, 33]. an image. These features have been successfully used for This is also in line with current theories of hierarchi- scene and object recognition by representing densely ex- cal, or layered, processing in the mammalian visual cortex tracted descriptors in terms of learned visual words—cluster [26]. Indeed, these theories give strong support to the im- centers in descriptor space [29]. On top of this quantized portance of feedback, which both improves features that are representation, more global image representations such as being learned and disambiguates local information during bags of words or spatial pyramids [17] can be assembled. inference. However, most existing hierarchical models are Recent publications in the field have started re- strictly feedforward (with a few notable exceptions, such evaluating the hard clustering approach of visual words as [13]), with each layer of the hierarchy operating over the in favor of “softer” representations that allow a single de- fixed output of the previous layer [27, 25, 18]. scriptor to be represented as a mixture of multiple compo- The aim of this paper is to develop a fully probabilis- 401
- 2. tic hierarchical model to learn and represent visual features tions of this idea have shown promising classification re- at different levels of complexity. The layers of the hierar- sults [22]. The units in the original HMAX model were chy are learned jointly, which, we show, has crucial advan- pre-designed, not learned, while the recent improvements tage in performance over its feedforward counterpart. Our have included random template selection from a training model is based on Latent Dirichlet Allocation (LDA) [4], corpus [27]. a probabilistic factorization method originally formulated Learning mid-level visual features in a recursive hi- in the field of text information retrieval. It has been suc- erarchical framework has motivated several recent works cessfully applied to the modeling of visual words [28], and, on convolutional networks [25, 2], deep Boltzmann Ma- more recently, to modeling of descriptors in an object detec- chines [12, 18], hyperfeatures [1], fragment-based hierar- tion task [10]. Here, we extend the latter representation to chies [30], stochastic grammars [34] and compositional ob- a recursively defined hierarchical probabilistic model, and ject representations [9, 33]. The underlying ideas behind show its advantage over the flat approach and the feedfor- these approaches are similar, but they differ in the type of ward implementation of the model. representation used. The approach is tested on a standard recognition dataset, Convolutional networks stack one or several feature ex- showing performance that is higher than previous hierar- traction stages, each of which consists of a filter bank layer, chical models and is on par with current single-descriptor- non-linear transformation layers, and a pooling layer that type state of the art. We demonstrate two important prop- combines filter responses over local neighborhoods using erties of our proposed model: 1) adding an additional layer a pooling operation, thereby achieving invariance to small to the LDA representation increases performance over the distortions [25]. In [1], the authors propose a representa- flat model; 2) a full Bayesian approach outperforms a feed- tion with “hyperfeatures” which are recursively formed as forward implementation of the model. This highlights the quantized local histograms over the labels from the pre- importance of feedback in hierarchical processing, which is vious layer, which in turn are quantized local histograms currently missing from most existing hierarchical models. over smaller spatial areas in images. Compositional hierar- Our probabilistic model is a step toward nonparametric ap- chies [9, 33, 24] and stochastic grammars [34] define ob- proaches to distributed coding for visual representations. jects in terms of spatially related parts which are then re- cursively defined in terms of simpler constituents from the 2. Related work layers below. There is strong biological evidence for the presence of Most of these models process information in a feedfor- object-specific cells in higher visual processing areas. It is ward manner, which is not robust to local ambiguities in believed that the complexity of represented shape gradually the visual input. In order to disambiguate local informa- increases as visual signal travels from the retina and LGN tion, contextual cues imposed by more global image rep- through V1 and higher cortical areas [26]. While the con- resentation may need to be used. A probabilistic Bayesian nectivity between different visual areas is far from being un- framework offers a good way of recruiting such top-down derstood, there is evidence for a hierarchical organization of information [19] into the model. the visual pathway, in which units at each level respond to In this paper, we propose a recursive Bayesian prob- the output of the units at the level below, aggregated within abilistic model as a representation for the intermediately a larger spatial area [6]. complex visual features. Our model is based on Latent We will refer to computational modeling of a hierar- Dirichlet Allocation [4], a latent factor mixture model ex- chy of stacked layers of increasing complexity as recur- tensively used in the field of text analysis. Here, we ex- sive models. The main idea behind these approaches is tend this representation to a recursively defined probabilis- to achieve high-level representation through recursive com- tic model over image features and show its advantage over bination of low-level units, gradually encoding larger and the flat approach as well as a feedforward implementation. larger spatial areas in images. This allows more efficient Due to overlapping terminology, we emphasize that parametrization of image structures, and potentially better our recursive LDA model is inherently different from recognition performance. Hierarchical-LDA [3] which forms a hierarchy of topics A number of recursive models have been proposed. The over the same vocabulary. In contrast, our hierarchy is HMAX model formulated the increase in shape complex- over recursively formed inputs, with a fixed base vocabu- ity as the interaction of two types of operations: template lary. Similar structure is seen in the Pachinko Allocation matching and max pooling [27]. Units performing these op- Model [20], which generalizes the LDA model from a sin- erations are stacked in layers, where the template-matching gle layer of latent topic variables to an arbitrary DAG of units on the bottom of each layer receive inputs from small variables. As we explain later, this structure is similar to patches of the image, and the pooling units on top of each our model, but crucially, it is missing a spatial grid that al- layer output more complex visual features. Implementa- lows us to model image patches of increasing spatial sup- 402
- 3. port. This allows us to keep the base vocabulary small, • θ(d) ∼ Dir(α) : sample top level mixing proportions while for PAM it would grow with the size of the support in the image. For each document d, N (d) words w are sampled according to: 3. Recursive LDA Model • z1 ∼ Mult(θ(d) ) : sample L1 mixture distribution (z ,·) In contrast to previous approaches to latent factor mod- • x1 ∼ Mult(χ1 1 ) : sample spatial position on L1 eling, we formulate a layered approach that derives progres- given z1 (z ,x ,·) sively higher-level spatial distributions based on the under- • z0 ∼ Mult(φ1 1 1 ) : sample L0 mixture distribution lying latent activations of the lower layer. given z1 and x1 from L1 (z0,·) For clarity, we will first derive this model for two layers • x0 ∼ Mult(χ0 ) : sample spatial position on L0 (L0 and L1 ) as shown in Figure 2a, but will show how it given z0 (z ,x ,·) generalizes to an arbitrary number of layers. For illustra- • w ∼ Mult(φ0 0 0 ) : sample word given z0 and x0 tion purposes, the reader may visualize a particular instance According to the proposed generative process, the joint dis- of our model that observes SIFT descriptors (such as the L0 tribution of the model parameters given the hyperparame- patch in Figure 2a) as discrete spatial distributions on the ters can be factorized as: L0 -layer. In this particular case, the L0 layer models the distribution of words from a vocabulary of size V = 8 gra- D dient orientation bins over a spatial grid X0 of size 4 × 4. p(w, z0,1 , φ0,1 , x0,1 , χ0,1 , θ|α, β0,1 ) = Pφ0 Pφ1 Pd As words in our vocabulary correspond to orientation bins, d=1 their frequency represents the histogram energy in the cor- (1) responding bin of the SIFT descriptor. where The mixture model of T0 components is parameterized T1 by multinomial parameters φ0 ∈ RT0 ×X0 ×V ; in our partic- P φi = p(φi (ti ,·,·) | βi ) p(χi (ti ,·) (ti ,·,·) | φi ) ular example φ0 ∈ RT0 ×(4×4)×8 . The L1 aggregates the t1 =1 mixing proportions obtained at layer L0 over a spatial grid N (d) (z0 ,x0 ,·) X1 to an L1 patch. In contrast to the L0 layer, L1 mod- Pd = p(θ(d) |α) Pz1 Pz0 p(w(d,n) |φ0 ) els a spatial distribution over L0 components. The mixture n=1 (d,n) (d,n) (z1 ,·) model of T1 components at layer L1 is parameterized by Pz1 = p(z1 | θ(d) ) p(x1 | χ1 ) multinomial parameters φ1 ∈ RT1 ×X1 ×T0 . Pz0 = (d,n) p(z0 (z ,x ,·) | φ1 1 1 ) (d,n) (z ,·) p(x0 | χ0 0 ) The spatial grid is considered to be deterministic at each layer and position variables for each word x are observed. We use the superscript in parentheses to index each vari- However, the distribution of words / topics over the grid is able uniquely in the nested plates. Whenever a “·” is speci- not uniform and may vary across different components. We fied, we refer to the whole range of the variable. As an ex- thus have to introduce a spatial (multinomial) distribution χ (t ,·,·) ample, φi i refers to the multinomial parameters of topic at each layer which is computed from the mixture distribu- ti over the spatial grid and the topics of the lower layer L0 . tion φ. This is needed to define a full generative model. The spatial distribution χ0 ∈ RT0 ×X0 and χ1 ∈ RT1 ×X1 The model for a single layer with a grid of size 1 × 1 is are directly computed from φ0 and φ1 respectively by sum- equivalent to LDA, which is therefore a special case of our ming the multinomial coefficients over the vocabulary. recursive approach. 3.2. Learning and inference 3.1. Generative Process For learning the model parameters we infer for each ob- Given symmetric Dirichlet priors α, β0 , β1 and the num- (d,n) ber of mixture components T0 and T1 for layer L1 and L0 served word occurrence w(d,n) the latent allocations z0 (d,n) respectively, we define the following generative process, and z1 , which indicate which mixture distributions were also illustrated in Figure 2b. sampled at L1 and L0 . Additionally, we observe the posi- (d,n) (d,n) Mixture distributions are sampled globally according to: tion variables x0 and x1 , which trace each word oc- currence through the X0 and X1 grids to θ(d) , as visualized • φ1 ∼ Dir(β1 ) and φ0 ∼ Dir(β0 ): sample L1 and L0 in Figure 2a. multinomial parameters • χ1 ← φ1 and χ0 ← φ0 : compute spatial distributions As we seek to perform Gibbs Sampling over the latent from mixture distributions variables z for inference, we condition on the observed vari- ables x and integrate out the multinomial parameters of the For each document, d ∈ {1, . . . , D} top level mixing pro- model. The equations are presented in Figure 3. In equa- portions θ(d) are sampled according to: tion (3) we are able to eliminate all terms referring to χ, as 403
- 4. α θ (z1 ,·) X1 χ ∈R θ (d) X1 z1 z1 x1 χ x1 φ(z1 ,·,·) ∈ RX1 ×T0 T1 X1 T0 L1 β1 φ1 z0 T1 z0 χ(z0 ,·) ∈ RX0 x0 X0 χ x0 φ(z0 ,·,·) ∈ RX0 ×T0 X 0 T0 w L0 β0 φ0 w V T0 N (d) D (a) (b) Figure 2: a) Concept for the Recursive LDA model. b) Graphical model describing two-layer RLDA. Wednesday, November 10, 2010 p(w, z0,1 |x0,1 , α, β0,1 ) = p(w, z1 , z0 , φ1 , φ0 , θ, χ1 , χ0 |x1 , x0 , α, β1 , β0 )dχ0 dχ1 dφ0 dφ1 dθ (2) θ φ1 φ0 χ1 χ0 = p(w, z1 , z0 , φ1 , φ0 , θ|x1 , x0 , α, β1 , β0 )dφ0 dφ1 dθ (3) θ φ1 φ0 = p(θ|α)p(z1 |θ)dθ p(φ1 |β1 )p(z0 |φ1 , z1 , x1 )dφ1 p(w|φ0 , z0 , x0 )p(φ0 |β)dφ0 (4) θ φ1 φ0 top layer intermediate layer evidence layer L−1 p(w, z1,...,L |x1,...,L , α, β1,...,L ) = p(θ|α)p(zL |θ)dθ p(φl |βl )p(zl−1 |φl , zl , xl )dφl p(w|φ0 , z0 , x0 )p(φ0 |β)dφ0 (5) θ l=1 φl φ0 top layer L layer l evidence layer Figure 3: RLDA conditional probabilities for a two-layer model, which is then generalized to an L-layer model. all variables x are observed and the deterministic transfor- different numbers of components at each of the layers and mation between φ and χ has probability one. Our formula- explore single-layer, feed-forward (FLDA), and fully gen- tion now closely resembles the Latent Dirichlet Allocation erative (RLDA) models, showing that 1) classification per- but adds an additional layer in between that also performs formance improves with an additional layer over the single- spatial grouping via the variables x. This formulation easily layer model, and 2) the fully generative RLDA model im- generalizes to L layers, as shown in equation (5). proves over the feed-forward-only FLDA analog. The derivation of the Gibbs Sampling equations is analo- Our work seeks to improve local features for low and gous to derivations for LDA (for an excellent reference, see mid-level vision independent of any specific object recog- [11]), with the addition of the observed position variables nition methods, and we do not innovate in that regard. We x. Due to space constraints, the derivations are presented in note that we test our model using only a single feature, and the supplementary materials. compare it only to other single-descriptor approaches, fo- cusing on hierarchical models. 4. Experiments 4.1. Implementation To evaluate the performance of our probabilistic recur- sive model, we test on the classification dataset Caltech The feature representation of an image for our approach 101 [7], which has been the most common evaluation of are SIFT descriptors of size 16 × 16 pixels, densely ex- hierarchical local descriptors. We show performance for tracted from the image with a stride of 6 pixels. Individ- 404
- 5. Table 1: Comparison of RLDA with 128 components at layer L1 and 1024 components at layer L2 to other hierarchical approaches. Under layers, “bottom” refers to the L1 features of the RLDA model, “top” to L2 features, and “both” denotes the feature obtained by stacking both layers. Approach Caltech-101 Model Layer(s) used 15 30 RLDA (1024t/128b) bottom 56.6 ± 0.8% 62.7 ± 0.5% Our Model RLDA (1024t/128b) top 66.7 ± 0.9% 72.6 ± 1.2% RLDA (1024t/128b) both 67.4 ± 0.5 73.7 ± 0.8% Sparse-HMAX [22] top 51.0% 56.0% CNN [16] bottom – 57.6 ± 0.4% CNN [16] top – 66.3 ± 1.5% Hierarchical CNN + Transfer [2] top 58.1% 67.2% Models CDBN [18] bottom 53.2 ± 1.2% 60.5 ± 1.1% CDBN [18] both 57.7 ± 1.5% 65.4 ± 0.4% Hierarchy-of-parts [8] both 60.5% 66.5% Ommer and Buhmann [24] top – 61.3 ± 0.9% Table 2: Results for different implementations of our model with 128 components at layer L1 and 128 components at L2 . For LDA models, “bottom” refers to using SIFT patches as input, while “top” refers to using 4 × 4 SIFT superpatches. Approach Caltech-101 Model Basis size Layer(s) used 15 30 LDA 128 “bottom” 52.3 ± 0.5% 58.7 ± 1.1% RLDA 128t/128b bottom 55.2 ± 0.3% 62.6 ± 0.9% LDA 128 “top” 53.7 ± 0.4% 60.5 ± 1.0% 128-dim FLDA 128t/128b top 55.4 ± 0.5% 61.3 ± 1.3% models RLDA 128t/128b top 59.3 ± 0.3% 66.0 ± 1.2% FLDA 128t/128b both 57.8 ± 0.8% 64.2 ± 1.0% RLDA 128t/128b both 61.9 ± 0.3% 68.3 ± 0.7% ual descriptors were processed by our probabilistic mod- 128 components at the bottom layer, and 128 and 1024 els, and results of inference were used in a classification components at the top layer. framework described in section 4.2. Because LDA requires discrete count data and SIFT dimensions are continuous- 3. RLDA: The full model was trained on the same size valued, normalization of the maximum SIFT value to 100 patches as FLDA described above: SIFT descriptors in tokens was performed; this level of quantization was shown a 4 × 4 spatial arrangement, with model parameters set to maintain sufficient information about the descriptor. accordingly. We tested 128 components at the bottom, We trained and compared the following three types of and 128 and 1024 components at the top layer. models: 4.2. Evaluation 1. LDA: LDA models with various numbers of compo- The setup of our classification experiments follows the nents (128, 1024, and 2048) trained on 20K randomly Spatial Pyramid Match, a commonly followed approach in extracted SIFT patches. We also trained LDA models Caltech-101 evaluations [17]. A spatial pyramid with 3 lay- on “superpatches” consisting of 4 × 4 SIFT patches, to ers of 4 × 4, 2 × 2, and 1 × 1 grids was constructed on give the same spatial support as our two-layer models. top of our features. Guided by the best practices outlined in a recent comparison of different pooling and factoriza- 2. FLDA: The feed-forward model first trains an LDA tion functions [5], we used max pooling for the spatial pyra- model on SIFT patches, as above. Topic activations mid aggregation. For classification, we used a linear SVM, are output and assembled as 4 × 4 superpatches. An- following the state-of-the-art results of Yang et al. [32]. other LDA model is learned on this input. We tested Caltech-101 is a dataset comprised of 101 object categories, 405
- 6. 70 70 65 65 Average % correct Average % correct 60 60 55 55 50 RLDA,top,128t/128b 50 RDLA,both,128t/128b 45 FLDA,top,128t/128b RLDA,top,128t/128b 40 LDA,"top",128b 45 FLDA,both,128t/128b LDA,"bottom",128b FLDA,top,128t/128b 35 40 5 10 15 20 25 30 5 10 15 20 25 30 Number of training examples Number of training examples (a) Caltech-101: 128 topic models, LDA-SIFT vs FLDA vs RLDA (b) Caltech-101: 128 topic models, L2 vs L1 +L2 models Figure 4: Comparison of classification rates on Caltech-101 of the one-layer model (LDA-SIFT) with the feed-forward (FLDA) and full generative (RLDA) two-layer models for different number of training examples, all trained with 128 at L1 and 128 at L2 layers. We also compare to the stacked L1 and L2 layers for both FLDA and RLDA, which performed the best. with differing numbers of images per category [7]. Follow- FLDA model, which in turn always outperforms the one- ing standard procedure, we use 30 images per category for layer model. training and the rest for testing. Mean accuracy and stan- We have also tested the classification performance of dard deviation are reported for 8 runs over different splits stacking both L1 and L2 features into a feature vector. This of the data, normalized per class. feature contains more information, as both spatially larger We first tested the one-layer LDA model over SIFT fea- and smaller features are taken into account. This is a stan- tures. For 128 components we obtain classification accu- dard practice for hierarchical models [18, 14]. The results, racy of 58.7%. Performance increases as we increase the presented in Table 2 (under “both”) and in Figure 4b, show number of components: for 1024 topics the performance is that using information from both layers improves perfor- 68.8%, and for 2048 it is 70.4% (all numbers here and on mance further, by about 3% for FLDA and 2% for RLDA. refer to the case of 30 training examples). We also compared the performance of just the first The emphasis of our experiments was to evaluate the layer L1 obtained with the RLDA model to the single- contribution of additional layers to classification perfor- layer model (which also forms the first layer of the FLDA mance. We tested the FLDA and RLDA models (described model). Interestingly, the decomposition learned in L1 in 4.1) in the same regime as described above. of the RLDA model outperforms the single layer model As an initial experiment, we constrained the number of (62.6% vs 58.7%). This demonstrates that learning the lay- topics to 128 and trained three models: single-layer, FLDA, ers jointly is beneficial to the performance of both bottom and RLDA. The two-layer models were trained with 128 and top layers, separately examined. topics on the bottom layer, corresponding to SIFT descrip- Following this evaluation, we also learned two-layer tors, and 128 on the top. models with 1024 topics in the top layer and 128 in the First, only the top-layer topics were used for classifica- bottom layer. The results hold, but the difference between tion. The dimensionality of vectors fed into the classifier the models becomes smaller—we hypothesize that single- was therefore the same for the two-layer and the single- feature performance on the dataset begins saturating at these layer models. We present the results in Table 2. levels. RLDA top layer performs at 72.6% vs. 72.5% for Both of the two-layer models improved on the one-layer FLDA and 68.8% for the single-layer model. The bottom model, obtaining classification rates of 61.3% for FLDA layer of RLDA achieves 62.7% classification accuracy vs and 66.0% for RLDA, compared to the single-layer results 62.6% for FLDA. Using both layers, RLDA gets 73.7% of 58.7%, when trained on SIFT patches, and 60.6%, when while FLDA gets 72.9%. trained on 4 × 4 SIFT “superpatches” (which are reported in The comparison with related hierarchical models is given Table 2 under LDA “top”). Detailed results across different in Table 1, which shows that while the baseline single- numbers of training examples are shown in Figure 4a. The layer LDA-SIFT model performs worse than most of the figure shows that the RLDA model always outperforms the approaches, the proposed RLDA model outperforms the ex- 406
- 7. isting work by more than 5%. Our best performance of 73.7% compares well to the current best performing approaches: sparse coding with 73.2% [32] and 75.7% [5] (using increased spatial support (a) FLDA and much denser sampling), and LCC [31] with 73.5%. State-of-the art single-feature performance has been shown by keeping all training data in a non-parametric classifica- tion scheme, obtaining 78.5% [15]. We note that in the existing literature on hierarchical models, most authors report best performance with just one learned layer, while adding another layer seems to decrease the recognition performance [27, 25, 8]. In contrast, in our model, the performance increases by adding another layer to the representation. The most important result, however, (b) RLDA is that the full Bayesian model as proposed in the paper out- performs the feed-forward approach. This supports the idea that inference carried out in the Bayesian approach results in more stable estimations of the feature activations, and thus better and more robust recognition performance. This highlights the importance of feedback in hierarchical recog- nition. top layer bottom layer 4.3. Role of Feedback in the Model Figure 5: Comparison of the (a) feed-forward and (b) full generative two-layer models in terms of components Additional evidence for the crucial role of feedback in learned, visualized as average image patches. While FLDA the model comes from visualization of the average image learns only localized edge orientations, RLDA learns more patches corresponding to top- and bottom-layer components complex spatial structures. learned by the two models. Figure 5 shows that the full gen- erative RLDA model uses lower-layer components in a no- tably different way than the feed-forward model, and learns and that the full generative process performs better than the different, more complex spatial structures at the top layer. feed-forward approach. This emphasizes the necessity of In the feed-forward model, the bottom-layer topics are feedback in hierarchical visual recognition models. in essence orientation filters. The second layer does not impose any additional structure on them, and therefore the Probabilistic models are robust, make modular combi- top-layer topics appear to be the same simple orientations, nations easy, and form the basis of possible non-parametric localized within a bigger spatial support. In the fully gener- extensions. The theorized goal of hierarchical approaches ative RLDA model, the top-layer components seem to rep- to vision is reaching object-level representations. With resent more interesting and potentially discriminative spa- additional layers and variables, our model can be developed tial structures. from merely a source of mid-level features to a full-scale We also found that the RLDA bottom-layer activations object recognition method. With a non-parametric exten- exhibit stronger correlations between topic activations in sion, the number of components would be inferred from the neighboring patches, which suggests that the model allows training data, which would appear especially important at bottom layer inference to represent continuous structures the part- and object-level representation. These topics are across subpatches. subject of future work. 5. Conclusions We presented a probabilistic model for visual features of increasing complexity and spatial support. The layers Acknowledgements: Various co-authors of this work of the hierarchy are trained jointly. We demonstrate per- were supported in part by a Feodor Lynen Fellowship formance that is among the recent best in single-descriptor granted by the Alexander von Humboldt Foundation; by approaches, and that outperforms existing hierarchical ap- awards from the US DOD and DARPA, including contract proaches. Most importantly, we show that adding an- W911NF-10-2-0059; by NSF awards IIS-0905647 and IIS- other layer to our model significantly improves performance 0819984; by EU FP7-215843 project POETICON; and by (something that is rarely true for layered models in vision), Toyota and Google. 407
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