I Introduction
Fast and high precision visual tracking is crucial to the success of several robotics applications like visual servoing and autonomous navigation. In recent years, online learning and detection based trackers have become popular [1, 2] due to their robustness to changes in the object’s appearance which makes them better suited to long term tracking. Such trackers, however, are often unsuitable for robotics applications for two reasons. Firstly, they are too slow [3] to allow real time execution of tasks where multiple trackers have to be run simultaneously or tracking is only a small part of a larger system. Secondly, they are not precise enough [3] to provide the exact object pose with sub pixel alignment required for these tasks. As a result, registration based trackers (Sec. IIB
) are more suitable for these applications as being several times faster and capable of estimating higher degreeoffreedom (DOF) transformations like affine and homography.
Though several major advances have been made in this domain since the original Lucas Kanade (LK) tracker was introduced almost thirty five years ago [4], efficient open source implementations of recent trackers are surprisingly difficult to find. In fact, the only such tracker offered by the popular OpenCV library [5], uses a pyramidal implementation of the original algorithm [6]. Similarly, the ROS library [7] currently does not have any package that implements a modern registration based tracker. The XVision system [8] did introduce a full tracking framework including a video pipeline. However, it implements several variants of the same algorithm [9] that only gives reasonable tracking performance with low DOF motion. In addition, it is not well documented and is quite difficult to install on modern systems due to many obsolete dependencies. Even the fairly recent MRPT library [10] includes only a version of the original LK tracker apart from a low DOF particle filter based tracker which is too imprecise and slow to be considered relevant for our target applications.
In the absence of good open source implementations of modern trackers, most robotics research groups either use these out dated trackers or implement their own custom trackers. These, in turn, are often not made publicly available or are tailored to suit very specific needs and so require significant reprogramming to be useful for an unrelated project. To address this need for a tracking library targeted specifically at robotics applications, we introduce Modular Tracking Framework (MTF)^{1}^{1}1available at http://webdocs.cs.ualberta.ca/~vis/mtf/  a generic system for registration based tracking that provides highly efficient implementations for a large subset of trackers introduced in literature to date and is designed to be easily extensible with additional methods.
MTF conceptualizes a registration based tracker as being composed of three semi independent sub modules  Search Method (SM), Appearance Model (AM) and State Space Model (SSM)  where the former is treated as a way to use the functionality in the other two  through a well defined interface  to solve the tracking problem. Such an approach can help to address another urgent need in this field  to unify the myriad of contributions made in the last three decades so they can be better understood. When a new registration based tracker is introduced in literature, it often contributes to only one or two of these sub modules while using existing methods for the rest. In such cases, this breakdown can provide a model within which the contributions of the new tracker can be clearly demarcated and thus studied better [3, 11]. By following this decomposition closely through extensive use of generic programming, MTF provides a convenient interface to plug in a new method for any sub module and test it against existing methods for the other two. This will not only help to compare the new method against existing ones in a more comprehensive way but also make it immediately available to any project that uses MTF. To facilitate the latter, MTF provides a simple ROS interface^{2}^{2}2available at https://gitlab.com/vis/mtf_bridge for seamless integration with robotics systems.
An existing system that is quite similar to MTF in functionality is the template tracker module of the Visual Servoing Platform (ViSP) library [12] that includes 4 SMs, 3 AMs and 6 SSMs though not all combinations work. MTF offers several advantages over ViSP. Firstly, SMs and AMs in ViSP are not implemented as independent modules, rather each combination of methods has its own class. This makes it difficult to add a new method for either of these sub modules and combine it with existing methods for the others. Secondly, MTF has several more AMs than ViSP as well as three stochastic SMs  NN [13], Particle Filter [14] and RANSAC [15]. Lastly, MTF is significantly faster than ViSP  both ICLK and FCLK (Sec. IIC) with SSD/Homography are nearly 30 times^{3}^{3}32400 vs 87 fps and 615 vs 22 fps respectively on an Intel Core i5 system faster. This is mainly because MTF uses the Eigen library  known to be one of the fastest [16]  for all mathematical computations.
To summarize, following are our main contributions:

Present a fast tracking library for robotics applications that is also easy to extend due to its modular design.

Provide a unifying formulation for registration based tracking to establish the theoretical basis for its design.

This can be seen as an extension of the framework reported in [17] with newer methods.

Rest of this paper is organized as follows: Section II introduces the mathematical basis for the design of MTF while section III describes the class structure of MTF along with specifications for important functions. Section IV presents several SMs as examples of using the functionality described in section III to implement the theory of section II. Finally, section V presents a couple of use cases for MTF before concluding in section VI with ideas for future extensions.
Ii Theoretical Background
Iia Notation
Let refer to an image captured at time
treated as a smooth function of real values using sub pixel interpolation
[18] for non integral locations. The patch corresponding to the tracked object’s location in is denoted by where with being the Cartesian coordinates of pixel .Further, denotes a warping function of parameters that represents the set of allowable image motions of the tracked object by specifying the deformations that can be applied to to align with . Examples of include homography, affine, similitude, isometry and translation [19].
Finally is a function of parameters that measures the similarity between two patches  the reference or template patch and a candidate patch . Examples of with include sum of squared differences (SSD) [9]
, sum of conditional variance (SCV)
[20], normalized cross correlation (NCC) [21], mutual information (MI) [18] and cross cumulative residual entropy (CCRE) [22]. So far, the only examples with , to the best of our knowledge, are those with an illumination model (ILM) [23, 24] where is expressed as with accounting for differences in lighting conditions under which and were captured.IiB Registration based tracking
Using this notation, registration based tracking can be formulated (Eq 1) as a search problem where the goal is to find the optimal parameters that maximize the similarity, measured by , between the target patch and the warped image patch , that is,
(1) 
As has been observed before [19, 22], this formulation gives rise to an intuitive way to decompose the tracking task into three modules  the similarity metric , the warping function and the optimization approach. These can be designed to be semi independent in the sense that any given optimizer can be applied unchanged to several combinations of methods for the other two modules which in turn interact only through a well defined and consistent interface. In this work, we refer to these respectively as AM, SSM and SM.
IiC Gradient Descent and the Chain Rule
Though several types of SMs have been reported in literature, gradient descent based methods [4] are most widely used due to their speed and simplicity. As mentioned in [17], the LK tracker can be formulated in four different ways depending on which image is searched for the warped patch  or  and how is updated in each iteration  additive or compositional. The four resultant formulations are thus called Forward Additive (FALK) [4], Inverse Additive (IALK) [9], Forward Compositional (FCLK) [25] and Inverse Compositional (ICLK) [26]. There is also a more recent approach called Efficient Second order Minimization (ESM) [27] that tries to make the best of both ICLK and FCLK by using information from both and .
What all these methods have in common is that they solve Eq 1 by estimating an incremental update to the optimal parameters at time using some variant of the Newton method as:
(2) 
where and respectively are estimates for the Jacobian and the Hessian of w.r.t.
. For any formulation that seeks to decompose this class of trackers (among others) in the aforementioned manner, the chain rule for first and second order derivatives is indispensable and the resultant decompositions for
and are given by Eqs. 3 and 4 respectively, assuming (or ) for simplicity.(3) 
(4) 
with and . It follows that the AM computes terms involving and (, , and ) while the SSM computes those with (, ). Further, these generic expressions do not give the whole scope of the decompositions since the exact forms of and as well as the way these are split vary for different variants of LK. The reader is referred to [17] for more details though formulations relevant to the functions in MTF (Tables I and II), including several extensions to [17], are also presented in the appendix.
IiD Stochastic Search
A limitation of gradient descent type SMs is that they are prone to getting stuck in local maxima of especially when the object’s appearance changes due to factors like occlusions, motion blur or illumination variations. An alternative approach to avoid this problem is to use stochastic search so as to cover a larger portion of the search space of . There are currently three main categories of such methods in our framework  particle filters (PF) [14], nearest neighbor (NN) [13] and RANSAC [15].
These SMs work by generating a set of random samples for
and evaluating the goodness of each by some measure of similarity with the template. Their performance thus depends mostly on the number and quality of stochastic samples used. While the former is limited only by the available computational resources, the latter is a bit harder to guarantee for a general SSM/AM. For methods that draw samples from a Gaussian distribution, the quality thereof is determined by the covariance matrix used and, to the best of our knowledge, no widely accepted method exists to estimate it in the general case. Most works either use heuristics or perform extensive hand tuning to get acceptable results
[14].Given this, a reasonable way to decompose these methods to fit our framework is to delegate the responsibility of generating the set of samples and estimating its mean entirely to the SSM and AM while letting the latter evaluate the suitability of each sample by providing the likelihood of the corresponding patch. Since the definition of what constitutes a good sample and how the mean of a sample set is to be evaluated depends on the SSM/AM, as do any heuristics for generating these samples (like the variance for each component of ), such a decomposition ensures both theoretical validity and good performance in practice.
Iii System Design
As shown in the class diagram in Fig. 1, MTF closely follows the decomposition described in the previous section and has three abstract base classes corresponding to the three sub modules  SearchMethod, AppearanceModel and StateSpaceModel. ^{4}^{4}4For brevity, these will be referred to as SM, AM and SSM respectively with the font serving to distinguish the classes from the corresponding concepts. Of these, only SM is a generic class that is templated on specializations of the other two classes. A concrete tracker, defined as a particular combination of the three sub modules, thus corresponds to a subclass of SM that has been instantiated with subclasses of AM and SSM.
It may be noted that SM itself derives from a non generic base class called TrackerBase for convenient creation and interfacing of objects corresponding to heterogeneous trackers, including those external to MTF, so that they can be run simultaneously and their results combined to create a tracker more robust than any of its components. Allowing a diverse variety of trackers to integrate seamlessly is one of the core design objectives of MTF and this is emphasized by having such trackers derive from a separate base class called CompositeBase. Since individual registration based trackers are well known to be prone to failures and more than three decades of research has failed to make significant improvements in this regard, the composite approach seems to be one of the more promising ones [15]. MTF has thus been designed to facilitate work in this direction.
A particular SM in our formulation is defined only by its objective  to find the that maximizes the similarity measure defined by the AM. Thus, different implementations of SM can cover a potentially wide range of methods that have little in common. As a result, SM is the least specific of these classes and only provides functions to initialize, update and reset the tracker along with accessors to obtain its current state. In fact, an SM is regarded in this framework simply as one way to use the methods provided by the other two sub modules to accomplish the above objective with the idea being to abstract out as much computation from the SM to the AM/SSM as possible so as to make for a general purpose tracker. Therefore, this section describes only AM and SSM in detail while some of the SMs currently available in MTF are presented in the next section as examples of using the functionality described here to carry out the search in different ways.
Another consequence of this conceptual impreciseness of SM is that a specific SM may use only a small subset of the functionality provided by AM/SSM. For instance, gradient descent type SMs do not use the random sampling functions of SSM and conversely, stochastic SMs do not use the differential functions required by the former. This has two further implications. Firstly, the functionality set out in AM and SSM is not fixed but can change depending on the requirements of an SM, i.e. if a new SM is to be implemented that requires some functionality not present in the current specifications, the respective class can be extended to support it  as long as such an extension makes logical sense within the definition of that class. Secondly, it is not necessary for all combinations of AMs and SSMs to support all SMs. For instance a similarity measure does not need to be differentiable to be a valid AM as long as it is understood that it cannot be used with SMs that require derivatives.
In the broadest sense, the division of functionality between AM and SSM described next can be understood as AM being responsible for everything to do with the image , the sampled patch and the similarity computed using it while SSM handles the actual points at which the patch is sampled along with the warping function that defines it in terms of the state parameters .
Iiia AppearanceModel
This class can be divided into three main parts with each defined as a set of variables dependent on and with a corresponding initialize and update function for each. The division is mainly conceptual and methods in different parts are free to interact with each other in practice. Table I presents a brief specification of some important methods in AM.
IiiA1 Image Operations
This part, abstracted into a separate class called ImageBase, handles all pixel level operations on the image like extracting the patch using sub pixel interpolation and computing its numerical gradient and Hessian .
Though AM bears a composition or ”has a” relationship with ImageBase, in practice the latter is actually implemented as a base class of the former to maintain simplicity of the interface and allow a specializing class to efficiently override functions in both classes. Moreover, having a separate class for pixel related operations means that AMs like SCV and ZNCC that differ from SSD only in using a modified version of or (thus deriving from SSDBase in Fig. 1), can implement the corresponding mapping entirely within the functions defined in ImageBase and be combined easily with other AMs besides SSD.
IiiA2 Similarity Functions
This is the core of AM and handles the computation of the similarity measure and its derivatives and w.r.t. both and . It also provides interfacing functions to use inputs from SSM to compute the derivatives of w.r.t. SSM parameters using the chain rule. As a notational convention, all interfacing functions, including those in SSM, are prefixed with cmpt.
The functionality specific to is abstracted into a separate class called IlluminationModel so it can be combined with any AM to add photometric parameters [23, 24] to it. This class provides functions to compute and its derivatives including , , , and . These are called from within AM to compute the respective derivatives w.r.t. so that the concept of ILM is transparent to the SM. It should be noted that AM is designed to support with arbitrary of which ILM is a special case. It also supports learning to update the object’s appearance, as present, for instance, in PCA [28].
Since several of the functions in this part of AM involve common computations, there exist transitive dependency relationships between them as depicted in Fig. 2 to avoid repeating these computations when multiple quantities are needed by the SM. What this means is that a function lower down in the dependency hierarchy may delegate part of its computations to any function higher up in the hierarchy so that the latter must be called before calling the former if correct results are to be expected.
IiiA3 Distance Feature
This part is designed specifically to enable integration with the FLANN library [32] that is used by the NN based SM. It provides two main functions:

A feature transform that maps the pixel values extracted from a patch
into a feature vector that contains the results of all computations in
that depend only on . 
A highly optimized distance functor that computes a measure of the distance or dissimilarity between and (typically ) given the distance features and as inputs.
The main idea behind the design of these two components is to place as much computational load as possible on so that is as fast as possible with the premise that the former is called mostly during initialization when the sample dataset is to be built while the latter is called online to find the best matches for a candidate patch in the dataset.
IiiB StateSpaceModel
This class has a simpler internal state than AM and can be described by only three main variables at any time  sampled grid points , corresponding corners and state parameters . It may be noted (Fig. 1) that, though SSM is designed to support any arbitrary , most SSMs currently implemented are subsets of the planar projective transform and so derive from ProjectiveBase that abstracts out the functionality common to these.
Functions in SSM can be divided into two categories:
IiiB1 Warping Functions
This is the core of SSM and provides a function to transform a regularly spaced grid of points representing the target patch into a warped patch that captures the tracked object’s motion in image space. It also allows for the compositional inverse of to be computed (invertState) to support inverse SMs. Further, there are functions to compute the derivatives of w.r.t. both and but, unlike AM, SSM does not store these as state variables, rather their computation is implicit in the interfacing functions that compute and using chain rule. This design decision was made for reasons of efficiency since and
are large and often very sparse tensors and computing these separately not only wastes a lot of memory but is also very computationally inefficient.
Finally, there are four ways to update the internal state: incrementally using additive (additiveUpdate) or compositional (compositionalUpdate) formulations, or outright by providing either the state vector (setState) or the corresponding corners (setCorners) that define the current location of the patch. There are no complex dependencies in SSM  the correct performance of interfacing functions and accessors depends only on one of the update functions being called every iteration. Table II lists the functionality of some important methods in this part.
Function  Inputs  Output/Result 

compositionalUpdate  
additiveUpdate  
invertState  
cmptPixJacobian  (Eq. 5)  
cmptWarpedPixJacobian  (Eq. 6, 7)  
cmptApproxPixJacobian  (approx) (Eq. 9, 10)  
cmptPixHessian  (Eq. 15)  
cmptWarpedPixHessian  (Eq. 16, 17)  
cmptApproxPixHessian  (approx) (Eq. B) 
IiiB2 Stochastic Sampler
This part is provided to support stochastic SMs and offers following functionality to this end:

generate small random incremental updates to (generatePerturbation
) by drawing these from a zero mean normal distribution with either user provided or heuristically estimated (
estimateStateSigma) variance. 
generate stochastic state samples using the given state transition model  currently random walk (additiveRandomWalk) and first order auto regression (additiveAutoRegression1) are supported. There are also compositional variants.

estimate the mean of a set of samples of (estimateMeanOfSamples)
Iv Examples of Search Methods
This section presents pseudo codes for several SMs to exemplify the usage of functions described in the previous section. Following are some points/conventions to be noted:

am and ssm respectively refer to instances of AM and SSM (or rather of specializations thereof)

only one iteration of the update function is shown and has been assumed for simplicity

different algorithms refer each other to save space and emphasize the parts they have in common

variables used to store the results of computations are not described explicitly but their meaning should be clear from their names and context
V Use Cases
This section presents two simple use cases for MTF:

Track an object in an image sequence (Algorithm 6)

Estimate the trajectory of a UAV in a large satellite image from images it took while flying over the region covered in the map (Algorithm 7).
It should be noted that MTF comes with an input module with wrappers for image capturing functions in OpenCV, ViSP and XVision as well as a preprocessing module with wrappers for OpenCV image filtering functions. Raw images acquired by the former need to be passed to the latter before they can be passed to the tracker. GaussianSmoothing used in the algorithms is an example of the latter while the former is assumed to have been initialized with the appropriate input source. Also, though only a couple of combinations of SM, AM and SSM are shown here, these can be replaced by virtually any combination of methods. Finally, Algorithm 7 assumes that UAV images have approximately the same size as the corresponding region in the satellite image. Due to space limitations, experimental tracking results are not included here  these are available in [3] and [11].
Vi Conclusions and Future Work
This paper presented a modular and extensible open source framework for registration based tracking that provides highly efficient C++ implementations for several well established trackers that will hopefully address practical tracking needs of the wider robotics community. It also formulated a novel method to decompose registration based trackers into sub modules and extended a previously published unifying formulation to account for more recent developments.
MTF is still a work in progress with several promising avenues of future extensions including novel composite SMs, incorporating better parameterization as well as deep learning in AMs, addition of non rigid SSMs and using motion learning to generate better stochastic samples for SSMs.
Appendix: and for different variants of LK
a Jacobian
Denoting with for conciseness ( and is constant in this context) and letting denote an estimate of to which an incremental update is sought, the formulations for used by FALK and FCLK are:
(5) 
(6) 
where in Eq. 6 refers to the gradient of warped using , i.e. is first warped back to the coordinate frame of using to obtain whose gradient is then computed at . It can be further expanded [17] as:
(7) 
Since is usually the most computationally intensive part of and , the so called inverse methods approximate this with the gradient of for efficiency as this only needs to be computed once. The specific expressions for these methods are:
(8) 
(9) 
where the middle two terms in Eq. 9 are derived from Eqs. 5 and 7 by assuming [9] that perfectly aligns with , i.e. so that
(10) 
In its original paper [27], ESM was formulated as using the mean of the pixel gradients and to compute but, as this formulation is only applicable to SSD, we consider a generalized version [34, 21] that uses the difference between FCLK and ICLK Jacobians:
(11) 
B Hessian
For clarity and brevity, evaluation points for the various terms have been omitted in the equations that follow as being obvious from analogy with the previous section.
It is generally assumed [17, 27] that the second term of Eq. 4 is too costly to compute and too small near convergence to matter and so is omitted to give the so called Gauss Newton Hessian
(12) 
Though works very well for SSD (and in fact even better than [17, 18]), it is well known [18, 21] to not work well with other AMs like MI, CCRE and NCC for which an approximation to the Hessian after convergence has to be used by assuming perfect alignment or . We refer to the resultant approximation as the Self Hessian and, as this substitution can be made by setting either or , we get two forms which are respectively deemed to be the Hessians for ICLK and FCLK:
(13) 
(14) 
It is interesting to note that has the exact same form as for SSD (since ) so it seems that interpreting Eq. 12 as the first order approximation of Eq. 4, as in [17, 18], is incorrect and it should instead be seen as a special case of .
differs from only in the way and are computed for the two as given in Eqs. 15 and 16 respectively.
(15) 
(16) 
where can be expanded by differentiating Eq. 7 as:
(17) 
is identical to except that and are used to approximate and . The expression for the former is in Eq. 9 while that for the latter can be derived by differentiating both sides of Eq. 7 after substituting Eq. 10:
which gives:
(18) 
where the second equality again follows from Eq. 7 and 10. Finally, the ESM Hessian corresponding to the Jacobian in Eq. 11 is the sum of FCLK and ICLK Hessians:
(19) 
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