# Cross-Domain Image Matching with Deep Feature Maps

###### Abstract

We investigate the problem of automatically determining what type of shoe left an impression found at a crime scene. This recognition problem is made difficult by the variability in types of crime scene evidence (ranging from traces of dust or oil on hard surfaces to impressions made in soil) and the lack of comprehensive databases of shoe outsole tread patterns. We find that mid-level features extracted by pre-trained convolutional neural nets are surprisingly effective descriptors for this specialized domains. However, the choice of similarity measure for matching exemplars to a query image is essential to good performance. For matching multi-channel deep features, we propose the use of multi-channel normalized cross-correlation and analyze its effectiveness. Our proposed metric significantly improves performance in matching crime scene shoeprints to laboratory test impressions. We also show its effectiveness in other cross-domain image retrieval problems: matching facade images to segmentation labels and aerial photos to map images. Finally, we introduce a discriminatively trained variant and fine-tune our system through our proposed metric, obtaining state-of-the-art performance.

∎

## 1 Introduction

We investigate the problem of automatically determining what type (brand/model/size) of shoe left an impression found at a crime scene. In the forensic footwear examination literature bodziak1999footwear (), this fine-grained category-level recognition problem is known as determining the class characteristics of a tread impression. This is distinct from the instance-level recognition problem of matching acquired characteristics such as cuts or scratches which can provide stronger evidence that a specific shoe left a specific mark.

Analysis of shoe tread impressions is made difficult by the variability in types of crime scene evidence (ranging from traces of dust or oil on hard surfaces to impressions made in soil) and the lack of comprehensive datasets of shoe outsole tread patterns (see Fig. 1). Solving this problem requires developing models that can handle cross-domain matching of tread features between photos of clean test impressions (or images of shoe outsoles) and photos of crime scene evidence. We face the additional challenge that we would like to use extracted image features for matching a given crime scene impression to a large, open-ended database of exemplar tread patterns.

Cross-domain image matching arises in a variety of other application domains beyond our specific scenario of forensic shoeprint matching. For example, matching aerial photos to GIS map data for location discovery SenletICPR2014 (); CosteaBMVC2016 (); DivechaSIGSPATIAL16 (), image retrieval from hand drawn sketches and paintings ChenetalSketch2Photo2009 (); ShrivastavaCrossDomain2011 (), and matching images to 3D models RusselAlignment2011 (). As with shoeprint matching, many of these applications often lack large datasets of ground-truth examples of cross-domain matches. This lack of training data makes it difficult to learn cross-domain matching metrics directly from raw pixel data. Instead traditional approaches have focused on designing feature extractors for each domain which yield domain invariant descriptions (e.g., locations of edges) which can then be directly compared.

Deep convolutional neural net (CNN) features hierarchies have proven incredibly effective at a wide range of recognition tasks. Generic feature extractors trained for general-purpose image categorization often perform surprising well for novel categorization tasks without performing any fine-tuning beyond training a linear classifier sharif2014cnn (). This is often explained by appealing to the notion that these learned representations extract image features with invariances that are, in some sense, generic. We might hope that these same invariances would prove useful in our setting (e.g., encoding the shape of a tread element in a way that is insensitive to shading, contrast reversals, etc.). However, our problem differs in that we need to formulate a cross-domain similarity metric rather than simply training a k-way classifier.

Building on our previous work KongSRF_BMVC_2017 (), we tackle this problem using similarity measures that are derived from normalized cross-correlation (NCC), a classic approach for matching gray-scale templates. For CNN feature maps, it is necessary to extend this to handle multiple channels. Our contribution is to propose a multi-channel variant of NCC which performs normalization on a per-channel basis (rather than, e.g., per-feature volume). We find this performs substantially better than related similarity measures such as the widely used cosine distance. We explain this finding in terms of the statistics of CNN feature maps. Finally, we use this multi-channel NCC as a building block for a Siamese network model which can be trained end-to-end to optimize matching performance.

## 2 Related Work

#### Shoeprint recognition

The widespread success of automatic fingerprint identification systems (AFIS) lee2001advances () has inspired many attempts to similarly automate shoeprint recognition. Much initial work in this area focused on developing feature sets that are rotation and translation invariant. Examples include, phase only correlation gueham2008automatic (), edge histogram DFT magnitudes zhang2005automatic (), power spectral densities de2005automated (); dardi2009texture (), and the Fourier-Mellin transform gueham2008automatic (). Some other approaches pre-align the query and database image using the Radon transform patil2009rotation () while still others sidestep global alignment entirely by computing only relative features between keypoints pairs tang2010footwear (); pavlou2006automatic (). Finally, alignment can be implicitly computed by matching rotationally invariant keypoint descriptors between the query and database images pavlou2006automatic (); wei2014alignment (). The recent study of Richetelli et al. Richetelli2017 () carries out a comprehensive evaluation of many of these approaches in a variety of scenarios using a carefully constructed dataset of crime scene-like impressions. In contrast to these previous works, we handle global invariance by explicitly matching templates using dense search over translations and rotations.

#### One-shot learning

While we must match our crime scene evidence against a large database of candidate shoes, our database contains very few examples per-class. As such, we must learn to recognize each shoe category with as little as one training example. This can be framed as a one-shot learning problem li2006one (). Prior work has explored one-shot object recognition with only a single training example, or “exemplar” malisiewicz2011ensemble (). Specifically in the domain of shoeprints, Kortylewski et al. kortylewski2016probabilistic () fit a compositional active basis model to an exemplar which could then be evaluated against other images. Alternatively, standardized or whitened off-the-shelf HOG features have proven very effective for exemplar recognition hariharan2012discriminative (). Our approach is similar in that we examine the performance of one-shot recognition using generic deep features which have proven surprisingly robust for a huge range of recognition tasks sharif2014cnn ().

#### Similarity metric learning

While off-the-shelf deep features work well sharif2014cnn (), they can be often be fine tuned to improve performance on specific tasks. In particular, for a paired comparison tasks, so-called “Siamese” architectures integrate feature extraction and comparison in a single differentiable model that can be optimized end-to-end. Past work has demonstrated that Siamese networks learn good features for person re-identification, face recognition, and stereo matching zbontar2015computing (); parkhi2015deep (); xiao2016learning (); deep pseudo-Siamese architectures can even learn to embed two dissimilar domains into a common co-domain zagoruyko2015learning (). For shoe class recognition, we similarly learn to embed two types of images: (1) crime scene photos and (2) laboratory test impressions.

## 3 Multi-variate Cross Correlation

In order to compare two corresponding image patches, we extend the approach of normalized cross-correlation (often used for matching gray-scale images) to work with multi-channel CNN features. Interestingly, there is not an immediately obvious extension of NCC to multiple channels, as evidenced by multiple approaches proposed in the literature fisher1995multi (); martin1979multivariate (); geiss1991multivariate (); popper1974multivariate (). To motivate our approach, we appeal to a statistical perspective.

#### Normalized correlation

Let be two scalar random variables. A standard measure of correlation between two variables is given by their Pearson’s correlation coefficient martin1979multivariate ():

(1) |

where

is the standardized version of (similarly for ) and

Intuitively, the above corresponds to the correlation between two transformed random variables that are “whitened” to have zero-mean and unit variance. The normalization ensures that correlation coefficient will lie between and .

#### Normalized cross-correlation

Let us model pixels from an image patch as corrupted by some i.i.d. noise process and similarly pixels another patch (of identical size) as . The sample estimate of the Pearson’s coefficient for variables is equivalent to the normalized cross-correlation (NCC) between patches :

(2) |

where refers to the set of pixel positions in a patch and means and standard deviations are replaced by their sample estimates.

From the perspective of detection theory, normalization is motivated by the need to compare correlation coefficients across different pairs of samples with non-stationary statistics (e.g., determining which patches are the same as a given template patch where statistics vary from one to the next). Estimating first and second-order statistics per-patch provides a convenient way to handle sources of “noise” that are approximately i.i.d. conditioned on the choice of patch but not independent of patch location.

#### Multivariate extension

Let us extend the above formulation for random vectors where corresponds to the multiple channels of values at each pixel (e.g., for a RGB image). The scalar correlation is now replaced by a correlation matrix. To produce a final score capturing the overall correlation, we propose to use the trace of this matrix, which is equivalent to the sum of its eigenvalues. As before, we add invariance by computing correlations on transformed variables that are “whitened” to have a zero-mean and identity covariance matrix:

(3) | ||||

where:

The above multivariate generalization of the Pearson’s coefficient is arguably rather natural, and indeed, is similar to previous formulations that also make use of a trace operator on a correlation matrix martin1979multivariate (); popper1974multivariate (). However, one crucial distinction from such past work is that our generalization (3) reduces to (1) for . In particular, martin1979multivariate (); popper1974multivariate () propose multivariate extensions that are restricted to return a nonnegative coefficient. It is straightforward to show that our multivariate coefficient will lie between and .

#### Decorrelated channel statistics

The above formulation can be computationally cumbersome for large , since it requires obtaining sample estimates of matrices of size . Suppose we make the strong assumption that all channels are uncorrelated with each other. This greatly simplifies the above expression, since the covariance matrices are then diagonal matrices:

Plugging this assumption into (3) yields the simplified expression for multivariate correlation

(4) |

where the diagonal multivariate statistic is simply the average of per-channel correlation coefficients. It is easy to see that this sum must lie between and .

#### Multi-channel NCC

The sample estimate of (4) yields a multi-channel extension of NCC which is adapted to the patch:

The above multi-channel extension is similar to the final formulation in fisher1995multi (), but is derived from a statistical assumption on the channel correlation.

#### Cross-domain covariates and whitening

Assuming a diagonal covariance makes strong assumptions about cross-channel correlations. When strong cross-correlations exist, an alternative approach to reducing computational complexity is to assume that cross-channel correlations lie within a dimensional subspace, where . We can learn a projection matrix for reducing the dimensionality of features from both patch and which decorrelates and scales the channels to have unit variance:

In general, the projection matrix could be different for different domains (in our case, crime scene versus test prints). One strategy for learning the projection matrices is applying principle component analysis (PCA) on samples from each domain separately. Alternatively, when paired training examples are available, one could use canonical correlation analysis (CCA) MardiaKentBibby1980 (), which jointly learn the projections that maximize correlation across domains. An added benefit of using orthogonalizing transformations such as PCA/CCA is that transformed data satisfies the diagonal assumptions (globally) allowing us to estimate patch multivariate correlations in this projected space with diagonalized covariance matrices of size .

#### Global versus local whitening

There are two distinct aspects to whitening (or normalizing) variables in our problem setup to be determined: (1) assumptions on the structure of the sample mean and covariance matrix, and (2) the data over which the sample mean and covariance are estimated. In choosing the structure, one could enforce an unrestricted covariance matrix, a low-rank covariance matrix (e.g., PCA), or a diagonal covariance matrix (e.g., estimating scalar means and variances). In choosing the data, one could estimate these parameters over individual patches (local whitening) or over the entire dataset (global whitening). In Section 5, we empirically explore various combinations of these design choices which are computationally feasible (e.g., estimating a full-rank covariance matrix locally for each patch would be too expensive). We find a good tradeoff to be global whitening (to decorrelate features globally), followed by local whitening with a diagonal covariance assumption (e.g., MCNCC).

To understand the value of global and per-patch normalization, we examine the statistics of CNN feature channels across samples of our dataset. Fig. 2 and Fig. 3 illustrate how the per-channel normalizing statistics () vary across patches and across channels. Notably, for some channels, the normalizing statistics change substantially from patch to patch. This makes the results of performing local, per-patch normalization significantly different from global, per-dataset normalization.

One common effect of both global and local whitening is to prevent feature channels that tend to have large means and variances from dominating the correlation score. However, by the same merit this can have the undesirable effect of amplifying the influence of low-variance channels which may not be discriminative for matching. In the next section we generalize both PCA and CCA using a learning framework which can learn channel decorrelation and per-channel importance weighting by optimizing a discriminative performance objective.

## 4 Learning Correlation Similarity Measures

In order to allow for additional flexibility of weighting the relevance of each channel we consider a channel-weighted variant of MCNCC parameterized by vector :

(5) |

This per-channel weighting can undo the effect of scaling by the standard deviation in order to re-weight channels by their informativeness. Furthermore, since the features are themselves produced by a CNN model, we can consider the parameters of that model as additional candidates for optimization. In this view, PCA/CCA can be seen as adding an extra linear network layer prior to the correlation calculation. The parameters of such a layer can be initialized using PCA/CCA and then discriminatively tuned. The resulting “Siamese” architecture is illustrated in Fig. 1.

#### Siamese loss:

To train the model, we minimize a hinge-loss:

(6) | ||||

where we have made explicit the function which computes the deep features of two shoeprints and , with , , and representing the parameters for the per-channel importance weighting and the linear projections for the two domains respectively. is the bias and is a binary source label (i.e., when and come from the same source and otherwise). Finally, is the regularization hyperparameter for and is the same for and .

We implement using a deep architecture, which is trainable using standard backpropagation. Each channel contributes a term to the MCNCC which itself is just a single channel (NCC) term. The operation is symmetric in and , and the gradient can be computed efficiently by reusing the NCC computation from the forward pass:

(7) |

#### Derivation of NCC gradient:

To derive the NCC gradient, we first expand it as a sum over individual pixels indexed by and consider the total derivative with respect to input feature

(8) |

where we have have dropped the channel subscript for clarity. The partial derivative , if and only if and is zero otherwise. The remaining partials derive as follows:

Substituting them into Eq. 8, we arrive at a final expression:

(9) |

where we have made use of the fact that is zero-mean.

## 5 Diagnostic Experiments

To understand the effects of feature channel normalization on retrieval performance, we compare the proposed MCNCC measure to two baseline approaches: simple unnormalized cross-correlation and cross-correlation normalized by a single and estimated over the whole 3D feature volume. We note that the latter is equivalent to the cosine distance which is popular in many retrieval applications. We also consider variants which only perform partial standardization and/or whitening of the input features.

#### Partial Print Matching

We evaluate these methods in a setup that mimics the occurrence of partial occlusions in shoeprint matching, but focus on a single modality of test impressions. We extract 512 query patches (random selected pixel sub-windows) from test impressions that have two or more matching tread patterns in the database. The task is then to retrieve from the database the set of relevant prints. As the query patches are smaller than the test impressions, we search over spatial translations (with a stride of 1), using the maximizing correlation value to score the match to the test impression. We do not need to search over rotations as all test impressions were aligned to a canonical orientation. When querying the database, the original shoeprint the query was extracted from is removed (i.e., the results do not include the self-match).

We carry out these experiments using a dataset that contains 387 test
impression of shoes and 137 crime scene prints collected by the Israel National
Police yekutieli2012expert (). As this dataset is not publicly available,
we used this dataset primarily for the diagnostic analysis and for training and
validating learned models. In all of these diagnostic experiments, we use the
256-channel ‘res2bx’ activations from a pre-trained ResNet-50
model^{1}^{1}1Pretrained model was obtained from
http://www.vlfeat.org/matconvnet/models/imagenet-resnet-50-dag.mat. We
evaluated feature maps at other locations along the network, but found those to
performed the best.

#### Global versus local normalization:

Fig. 4 shows retrieval performance in terms of the tradeoff of precision and recall at different match thresholds. In the legend we denote different schemes in square brackets, where the first term indicates the centering operation and the second term indicates the normalization operation. A indicates the absence of the operation. and indicate that standardization was performed using local (i.e., per-exemplar) statistics of features over the entire (3D) feature map. and indicate local per-channel centering and normalization. and indicate global per-channel centering and normalization (i.e., statistics are estimated over the whole dataset). Therefore, simple unnormalized cross-correlation is indicated as , cosine distance is indicated as , and our proposed MCNCC measure is indicated as .

We can clearly see from Fig. 4 that using per-channel statistics estimated independently for each comparison gives substantial gains over the baseline methods. Performing global centering and scaling per channel is substantially better than the straight correlation or cosine distance. In general, removing the mean response is far more important than scaling by the standard deviation. Interestingly, in the case of cosine distance and global channel normalization, scaling by the variance actually hurts performance (i.e., versus and versus respectively). As normalization re-weights channels, we posit that this may be negatively effecting the scores by down-weighing important signals or boosting noisy signals.

#### Channel decorrelation:

Recall that, for efficiency reasons, our multivariate estimate of correlation assumes that channels are largely decorrelated. We also explored decorrelating the channels globally using a full-dimension PCA (which also subtracts out the global mean ). The second panel of Fig. 4 shows a comparison of these decorrelated feature channels (solid curves) relative to baseline ResNet channels (dashed curves). While the decorrelated features outperform baseline correlation (due to the mean subtraction) we found that full MCNCC on the raw features performed better than on globally decorrelated features. This may be explained in part due to the fact that decorrelated features show an even wider range of variation across different channels which may exacerbate some of the negative effects of scaling by .

## 6 Cross-Domain Matching Experiments

In this section, we evaluate our proposed system in settings that closely resembles various real-world scenarios where query images are matched to a database containing images from a different domain than that of the query. We focus primarily on matching crime scene prints to a collection of test impressions, but also demonstrate the effectiveness of MCNCC on two other cross-domain applications: semantic segmentation label retrieval from building facade images, and map retrieval from aerial photos. As in our diagnostic experiments, we use the same pre-trained ResNet-50 model. We use the 256-channel ‘res2bx’ activations for the shoeprint and building facade data and the 1024-channel ‘res4cx’ activations for map retrieval. And as before, we evaluated other activations, but found those to perform best.

### 6.1 Shoeprint Retrieval

In addition to the internal dataset used Section 5, we also evaluated our approach on a publicly available benchmark, the footwear identification dataset (FID-300) kortylewski2014unsupervised (). FID-300 contains 1175 test impressions and 300 crime scene prints. The task here is similar to that of the previous section, but now matching is done across domains. As the crime scene prints are not aligned to a canonical orientation, we search over both translations (with a stride of 2) and rotations (from -20 to +20 with a stride of 4). We compute the local statistics only over the valid support region of the test impression from a predetermined mask associated with each image. The correlation score is similarly computed over the same valid support region .

As mentioned in Sec. 4, we can learn both the linear projections of the features and the importance of each channel for the retrieval task. We demonstrate that such learning is feasible and can significantly improve performance. We use a 50/50 split of the crime scene prints of the Israeli dataset for training and testing, and determine hyperparameters settings using 10-fold cross-validation. In the left panel of Fig. 5 we compare the performance of three different models with varying degrees of learning. The model with no learning is denoted as , with learned per-channel weights is denoted as , with learned projections is denoted as CCA , and with piece-wise learned linear projections and per-channel weights is denoted as CCA . Our final model, CCA ft, jointly fine-tunes the linear projections and the per-channel weights together. The model with learned per-channel importance weights has parameters (a scalar for each channel and a single bias term), and was learned using a SVM solver with a regularization value of . The linear projections (CCA) were learned using , MATLAB’s canonical correlation analysis function. Our final model, CCA ft, was fine-tuned using gradient descent using an L2 regularization value of on the per-channel importance weights and on the linear projections. This full model has 131K parameters ( projections, channel importance, and bias).

As seen in the left panel of Fig. 5, learning per-channel importance weights, , yields substantial gains improvements, outperforming and CCA when recall is less than 0.34. When learning both importance weights and linear projections, we see gains across all recall values as our Siamese network significantly outperforms all other models. However, we observe only marginal gains when fine-tuning the whole model. We expect this is due in part to the small amount of training data which makes it difficult to optimize parameters without overfitting.

We subsequently tested these same models (without any retraining) on the FID-300 benchmark (shown in the right panel of Fig. 5). In this, and in later results, we use cumulative match charactertistic (CMC) rather than precision-recall (PR) because there is only a single correct database item for each query. PR is ill-suited in this single-item recall setting as precision can only decrease once the correct database item has been retrieved. On the other hand, as CMC shows the number of correct matches (recall) as a function of the number of database items reviewed, the curve is more meaningful. On FID-300, we observe the same trend as on the Israeli dataset — models with more learned parameters perform better. However, even without learning (i.e., ) MCNCC significantly outperforms the previously published state-of-the-art approaches of Kortylewski et al. kortylewski2014unsupervised (); kortylewski2016probabilistic (); kortylewski2017model ()

### 6.2 Segmentation Retrieval for Building Facades

To further demonstrate the robustness of MCNCC for cross domain matching, we consider the task of retrieving segmentation label maps which match for a given building facade query image. We use the CMP Facade Database tylecek2013spatial () which contains 606 images of facades from different cities around the world and their corresponding semantic segmentation labels. These labels can be viewed as a simplified “cartoon image” of the building facade by mapping each label to a distinct gray level.

In our experiments, we generate 1657 matching pair by resizing the original 606 images (base + extended dataset) to either or depending on their aspect ratio and crop out non-overlapping patches. We prune this set by removing 161 patches which contain more than 50% background pixels to get our final dataset. Examples from this dataset can be seen in the right panel of Fig. 7. In order treat the segmentation label map as an image suitable for the pre-trained feature extractor, we scale the segmentation labels to span the whole range of gray values (i.e., from to ).

We compare MCNCC (denoted in the legend as ) to three baseline similarity metrics: Cosine, Euclidean distance, and normalized cross-correlation using across-channel local statistics (denoted as ). We can see in the left panel of Fig. 7 that MCNCC performs significantly better than the baselines. MCNCC returns the true matching label map as the top scoring match in 39.2% of queries. In corresponding top match accuracy for normalized cross-correlation using across-channel local statistics is 25.2%, for Cosine similarity is 18.3%, and for Euclidean distance is 6.0%. When learning parameters with MCNCC (denoted as CCA ), using a 50/50 training-test split, we see significantly better retrieval performance (96.4% for reviewing one database item). The right panel of Fig. 7 shows some retrieval results for this model.

### 6.3 Retrieval of Maps from Aerial Imagery

Finally, we evaluate matching performance on the problem of retrieving map data corresponding to query aerial photos. We use a dataset released by Isola et al. isola2017image () that contains 2194 pairs of images scraped from Google Maps. For simplicity in treating this as a retrieval task, we excluded map tiles which consisted entirely of water. Both aerial photos and map images were converted from RGB to gray-scale prior to feature extraction (see the right panel of Fig. 8 for examples). We compare MCNCC to three baseline similarity metrics: Cosine, Euclidean distance, and normalized cross-correlation using across-channel local statistics (denoted as ).

The results are shown in the left panel of Fig. 8. MCNCC outperforms the baseline Cosine and Euclidean distance measures, but this time normalized cross-correlation using across-channel local statistics and Cosine similarity are nearly identical. For top-1 retrieval performance, MCNCC is correct 98.7% of the time, normalized cross-correlation using across-channel local statistics and Cosine similarity are correct 95.8%, and Euclidean distance is correct 28.6% of the time when retrieving only one item. We show example retrieval results for MCNCC in the right panel of Fig. 8. We did not evaluate any learned models in this experiment since the performance of baseline MCNCC left little room for improvement.

## 7 Conclusion

In this work, we proposed an extension to normalized cross-correlation suitable for CNN feature maps that performs normalization of feature responses on a per-channel and per-image basis. The benefits of performing per-image normalization can be explained in terms of spatially local whitening which adapts to non-stationary statistics of the input. Relative to other standard feature normalization schemes (e.g., cosine similarity), per-channel normalization accommodates variation in statistics of different feature channels.

Utilizing MCNCC in combination with CCA provides a highly effective building block for constructing Siamese network models that can be trained in an end-to-end discriminative learning framework. Our experiments demonstrate that even with very limited amounts of data, this framework achieves robust cross-domain matching using generic feature extractors combined with piece-wise training of simple linear feature-transform layers. This approach yields state-of-the art performance for retrieval of shoe tread patterns matching crime scene evidence. We expect our findings here will be applicable to a wide variety of single-shot and exemplar matching tasks using CNN features.

###### Acknowledgements.

We thank Sarena Wiesner and Yaron Shor for providing access to their dataset. This work was partially funded by the Center for Statistics and Applications in Forensic Evidence (CSAFE) through NIST Cooperative Agreement #70NANB15H176.## References

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