1
Event-based Camera Pose Tracking using a
Generative Event Model
Guillermo Gallego, Christian Forster, Elias Mueggler, Davide Scaramuzza
Abstract—Event-based vision sensors mimic the operation of
biological retina and they represent a major paradigm shift
from traditional cameras. Instead of providing frames of in-
tensity measurements synchronously, at artiﬁcially chosen rates,
event-based cameras provide information on brightness changes
asynchronously, when they occur. Such non-redundant pieces of
information are called “events”. These sensors overcome some of
the limitations of traditional cameras (response time, bandwidth
and dynamic range) but require new methods to deal with the
data they output. We tackle the problem of event-based camera
localization in a known environment, without additional sensing,
using a probabilistic generative event model in a Bayesian
ﬁltering framework. Our main contribution is the design of the
likelihood function used in the ﬁlter to process the observed
events. Based on the physical characteristics of the sensor and
on empirical evidence of the Gaussian-like distribution of spiked
events with respect to the brightness change, we propose to use
the contrast residual as a measure of how well the estimated
pose of the event-based camera and the environment explain the
observed events. The ﬁlter allows for localization in the general
case of six degrees-of-freedom motions.
Index Terms—Event-based, Dynamic Vision Sensor, generative
model, spiking model, robot localization, pose tracking, Kalman
ﬁlter.
I. INTRODUCTION
Recently, event-based cameras such as the Dynamic Vision
Sensor (DVS) [1] have attracted a lot of attention from both
the robotics and vision communities [2], [3], [4], [5], [6],
[7], [8], [9], [10]. These bio-inspired sensors overcome some
of the limitations of traditional image sensors: they respond
very quickly (within microseconds) to brightness changes,
have very high dynamic range (120 dB compared to 60 dB
of standard cameras), and require low bandwidth [1]. Hence,
they are very promising sensors for high-speed visual appli-
cations in challenging scenes with large brightness contrast.
However, the output of these cameras (a stream of events) is
fundamentally different from that of traditional ones, and so a
paradigm shift is required to design algorithms that exploit the
potential of these vision sensors. Examples of such emerging
event-based algorithms are: event-based optical ﬂow [4], visual
odometry [5], localization [2], [6], Simultaneous Localization
and Mapping (SLAM) [3], [9], mosaicing [7], [8], object
recognition [10], etc.
We address the localization problem of a moving event-
based camera in a known environment. One of the ﬁrst works
The authors are with the Robotics and Perception Group, Department of In-
formatics, University of Zurich, Zurich 8050, Switzerland. http://rpg.iﬁ.uzh.ch
e-mail: guillermo.gallego@iﬁ.uzh.ch.
This research was supported by the National Centre of Competence in
Research (NCCR) Robotics, the ERC-SNSF Starting Grant, and Google
Faculty Research Award.
in this respect is [2], where a particle-ﬁlter system that is
limited to planar motions and 2-D maps was introduced. In the
experiments, they used an upward-looking DVS mounted on a
ground robot moving at low speed. The provided map used for
navigation consisted of line segments on the ceiling. In [5], a
probabilistic ﬁltering approach was designed to localize a DVS
moving on a plane with respect to the temporally closest pair
of frames provided by an additional RGB-D camera attached
to the DVS. An algorithm to track the 6-DOF pose of the DVS
with no additional sensing during high-speed maneuvers was
given in [6]. They used a map consisting of the edges of a
black square of known size and minimized the event-to-line
reprojection distance to estimate the DVS pose.
We propose an implicit Extended Kalman Filter (EKF)
approach [11] to localize the DVS with respect to a given
dense map of the 3-D scene (consisting of geometric and
photometric information) without additional sensing (as in [2],
[6], [8]), just using the information contained in the event
stream. The map is not constrained to consist only of lines,
thus it is more general than those in [2], [6], and it is
also richer in brightness changes than the barcoded scenes
in [5]. We allow for localization in the general case of 6-DOF
motion of the DVS and design the ﬁlter accordingly. Our main
contribution pertains to the design of the likelihood function
used in the correction step of the EKF to process the observed
events (Section III-B), by measuring how well the system
state (DVS pose and velocity) and the map explain an event
from the DVS using a contrast residual. To do so, we ﬁrst
derive a simple yet compelling model for event generation
(Section II-A). The technique is demonstrated on synthetic
and real data in Section IV.
II. DYNAMIC VISION SENSOR (DVS):
GENERATIVE EVENT MODEL
In contrast to standard cameras, which acquire full frames
at ﬁxed rates, event-based vision sensors such as the DVS
(Fig. 1a) have independent pixels that spike events at local
relative brightness changes in continuous time. A visualization
of the output of the DVS is shown in Fig. 1b. Events are
time-stamped with microsecond resolution and transmitted
asynchronously at the time they occur. Each event is a tuple
ek = ⟨xk, yk, tk, pk⟩, where xk, yk are pixel coordinates of the
event, tk is its time-stamp, and pk = ±1 is its polarity (sign
of the brightness change). The sensor’s spatial resolution is
limited1 (128 × 128 pixels), but its 120 dB dynamic range
1A new generation of event-based sensors with VGA resolution (640×480)
is being developed by the group [1].
arXiv:1510.01972v1  [cs.CV]  7 Oct 2015
2
(a)
(b)
(c)
Fig. 1. (a) The Dynamic Vision Sensor (DVS). (b) Space-time visualization of the output of a DVS viewing a rotating dot. Colored dots mark individual
events. Event polarity is not displayed. Noise is visible by isolated points that are not part of the spiral. (c) The contrast of the DVS events empirically
follows a unimodal distribution (e.g. Gaussian-like) centered at a selected threshold C = |∆log(I)| (six threshold settings are shown). Images (b) and (c)
are courtesy of [1].
notably exceeds the 60 dB of high-quality traditional image
sensors.
Next, we provide a generative event model for the DVS
using a principled derivation of the equations that characterize
an ideal sensor. The event model combines several hypothesis
(constant brightness, temporal persistence, etc.) with particular
characteristics of the DVS. The model is at the heart of data
assimilation in our ﬁltering approach for DVS localization.
A. Scene modeling
Assume that objects in the 3-D world are represented
by a surface S with geometric and radiometric properties.
Typically, objects are described by a mesh or depth map
and a corresponding intensity (i.e., “texture”) function (in a
Lambertian context).
The DVS has the same optics as traditional perspective
cameras, therefore, standard models (e.g., pinhole) apply. In
camera coordinates, the projection operation is described by
u = π(X), mapping a 3-D point X = (X, Y, Z)⊤into the
image point u = (u, v)⊤.
Assume a simpliﬁed radiance model where each point on the
surface S has an intensity, IS : S →R, and this is the value
observed by the DVS to trigger events, that is, the intensity
at the image plane corresponds to the intensity deﬁned on the
surface: I(u) .= IS(X) for 3-D points X visible from the
DVS. Hence, the image plane parametrizes both the image I
and the surface S (geometric and photometric properties).
B. 3-D motion and apparent (2-D) motion
The motion of a moving camera is described by a smooth
trajectory in the space of Euclidean transformations, SE(3).
Let the relative motion between the viewing camera and the
scene be described, in the camera coordinate frame, by
dX
dt ≡˙X(t) = −bω(t)X(t) −v(t)
(1)
where ω and v are body angular and linear velocities, respec-
tively, and bω is the cross-product matrix: bab = a × b ∀a, b.
The corresponding apparent motion of the 3-D point X
is described by the velocity of the image point u, which
comprises the image motion ﬁeld. Speciﬁcally, the equation
that relates surface velocity (in the camera frame) to feature
velocity (in normalized coordinates) is (see, e.g. [12], [13, Eq.
5.87]), dropping the t notation:
du
dt ≡˙u = B ξ,
(2)
where twist coordinates ξ = (v⊤, ω⊤)⊤encode the relative
motion and
B =
−Z−1
0
uZ−1
uv
−(1 + u2)
v
0
−Z−1 vZ−1 1 + v2
−uv
−u

(3)
is called interaction matrix, image Jacobian matrix for a
point feature, or feature sensitivity matrix [12], [14, p. 460-
462]. Typically, the surface is assumed to admit a depth map
representation with respect to the camera, and so the depth of
the 3-D point is parametrized in the image plane, Z ≡Z(u, v).
Consequently, B(u, v) is just a function of the surface and the
image point. The motion ﬁeld has two separate components
for translation and rotation.
C. Deterministic generative event model
The standard hypothesis in measuring image motion is that
the intensity structure of local time-varying image regions
are approximately constant under motion for at least a short
duration (temporal persistence). Formally, if ˜I(u, t) is the
space-time image intensity function measured by the DVS,
the total derivative d˜I/dt vanishes for those trajectories u(t)
of constant intensity values, ˜I(u(t), t) = const, that is,
d˜I
dt =
D
∇u ˜I, ˙u
E
+ ∂˜I
∂t = 0,
(4)
where ⟨·, ·⟩is the dot product, ∇u ˜I =
  ∂˜I
∂u, ∂˜I
∂v
⊤are the
ﬁrst partial derivatives with respect to spatial coordinates and
˙u = ( ˙u, ˙v)⊤is the motion ﬁeld.
The DVS senses brightness logarithmically2: ˜I = log(I),
and it generates an event at a location u if the amount of
2Using the chain rule it is easy to verify that, if I ̸= 0, both conditions
dI/dt = 0 and d˜I/dt = 0 are equivalent.
3
intensity (grey level) change ∆log(I) during an interval ∆t
(the time since the previous event at the same location), i.e.,
the contrast
∆log(I) ≈∂log(I)
∂t
∆t
(4)= −⟨∇u log(I), ˙u∆t⟩
(5)
is larger than a threshold C [1], [5] (typically 10-15% relative
brightness change):
|∆log(I)| ≈|⟨∇u log(I), ˙u∆t⟩| ≥C.
(6)
Incorporating polarity, if the contrast ∆log(I) ≥C, a positive
event (pk = +1) is generated; if ∆log(I) ≤−C, a negative
event (pk = −1) is generated; otherwise, no event is ﬁred.
D. Probabilistic generative event model
Equation (6) is a hard decision model for the generation of
events. A more realistic one takes into account sensor noise
and manufacturing mismatches, yielding a soft decision repre-
sented by a smooth probability function. A characterization of
the corresponding probability density averaged over all DVS
pixels is shown in Fig. 6 of [1] (see Fig. 1c), suggesting a
unimodal Gaussian-like distribution, for which they measure
its standard deviation as a function of the threshold C. This
probabilistic generative event model can be included in a
Bayesian ﬁltering approach to process the events, as shown
in the next section, where we adopt the simple yet powerful
ﬁlter given by the Extended Kalman Filter (EKF), which
assumes Gaussian probability distributions to keep a compact
and manageable representation of the posterior probability of
the DVS pose and velocity.
III. BAYESIAN FILTERING APPROACH
A. State-space design
In the popular Bayesian inference framework given by the
EKF [11] we can formulate the DVS localization problem
with respect to a map M as that of estimating the state of
a system deﬁned by its state-space representation (state and
measurement equations).
The state equation is a non-linear function f of the state and
the process noise
xn = f(xn−1, wn),
with
x = (t⊤, r⊤, v⊤, ω⊤)⊤.
(7)
As usual, subscripts {n −1, n} denote temporal references.
The process noise wn is not additive and it is assumed to be
zero-mean multivariate Gaussian distributed with covariance
Qw
n . The state vector describes the DVS pose (position and
orientation) and its velocity: t is the position of the optical
center of the DVS, in world coordinates; r is the rotation vector
parametrizing the orientation of the DVS by means of the
exponential coordinates (as in the ﬁlter proposed by [15]) of
the rotation matrix from the world to the camera frame, R =
exp(br); and the linear and angular velocities (1) (v, ω) are
given in world and camera (body) coordinates, respectively.
We chose the motion model f given by the constant velocity
model, which is typical of SLAM approaches [16]. This ac-
counts for general smooth motions of the DVS. By integration
of the continuous motion over a time interval3 ∆t = tn−tn−1
and discretization, (7) becomes









tn
= tn−1 + (vn−1 + V)∆t,
rn
= (log (exp ((ωn−1 + Ω)∧∆t) exp (rn−1)))∨,
vn
= vn−1 + V,
ωn
= ωn−1 + Ω,
(8)
where the noise is wn
=
(V⊤, Ω⊤)⊤. The log and
exp operators refer to the rotation group, SO(3). ∆R =
exp ((ωn−1 + Ω)∧∆t) is the incremental rotation of angle
θ = ∥ωn−1 + Ω∥∆t around the axis deﬁned by vector
(ωn−1 + Ω), u∧is the cross-product matrix associated to
a 3-vector u, and S∨is the 3-vector associated to a 3 × 3
skew-symmetric matrix S.
B. Implicit measurement equation
In the standard EKF, the likelihood is speciﬁed by an equa-
tion zn = h(xn) + ηn, where observations zn are explicitly
written in terms of the state and the observation noise ηn.
This is the formulation used in classical visual localization
and SLAM, where zn consists of the image coordinates of
sensed map landmarks, and h predicts the observations by
using the camera model to project the landmarks. This design
choice implies Gaussian image coordinate noise, and it may
also be applied to DVS localization [2]. However, it does not
take into account the generative event model (such as (6)). In a
different (non-localization) context, an alternative approach is
given in [8] to estimate the intensity gradient at each pixel: zn
consists of event rates and a generative model is used to write
such explicit dependency. This design choice implies that the
temporal (event-rate) noise is Gaussian, which is an arbitrary
choice.
We depart from the previous explicit models (spatial or
temporal measurements) and propose an implicit measurement
equation
q(zn, xn) = 0
(9)
to quantify how well the event generation model (6) is sat-
isﬁed. This leads to an implicit EKF [17], [18]. Our design
choice assumes that the deviations of the contrast from the
nominal one that ﬁres events is Gaussian, which Fig. 1c
suggests to be. A similar unimodal density function is given
in [8] only for the correction step of rotation tracking.
Assuming constant illumination and independence of the
observations, each event en = ⟨un, vn, tn, pn⟩is caused by
a brightness change at pixel pn = (un, vn)⊤, depending on
both the DVS state xn ≡x(tn) and the map M. Thus,
a more rigorous description than (9) is q = q(zn, xn; M)
because an event is an observation of some map point.
Letting g be a shorthand notation for the spatial gradient
∇u log(I) in (6), we deﬁne the implicit function q as the
difference between the absolute contrast (5) and the nominal
threshold, q = |∆log(I)| −C. Substituting |y| = y sgn(y)
3Here ∆t is the time between prediction steps in the EKF, which may or
may not coincide with the time between events at the same location in (5)
depending on whether events are processed in packets or individually.
4
(a)
(b)
(c)
(d)
Fig. 2. Neighborhood of an event en = ⟨xn, yn, pn, tn⟩triggered by a moving edge. The DVS is moving horizontally to the right (positive X direction).
Top row: positive event (dark-to-bright transition). Bottom row: negative event (bright-to-dark transition). (a) Rendering of the map on the DVS image plane,
˜I(tn −∆tn); the event pn = (xn, yn)⊤is at the center of the patch. The motion ﬁeld ˙u (magenta vectors) points toward the negative X direction. The
image gradient g = ∇u ˜I(tn −∆tn) (perpendicular to the edge) is displayed with cyan vectors. (b) Predicted neighborhood ˜I(tn) ≈˜I(tn −∆tn) + ∆˜I.
(c) Constrast ∆˜I ≈−⟨g, ˙u⟩∆tn. (d) The implicit measurement function q in (10) has the same shape as the absolute contrast, |∆˜I| ≈−pn ⟨g, ˙u⟩∆tn,
which deﬁnes the likelihood that the event was triggered.
for y = ∆log(I) and replacing sgn(y) by the measured
polarity pn, we use (6) to deﬁne
q(zn, xn; M) = −pn ⟨g, ˙u⟩|(pn,xn,Xn) ∆tn −C,
(10)
where ∆tn = tn −tprev is the time span since the previous
event at the same location pn, and the inner product between
the gradient g and the motion ﬁeld ˙u depends on the event
location pn, its corresponding 3-D point Xn and the state
xn. Speciﬁcally, g depends on the DVS pose only (but not
on its velocity) via the perspective projection between the
map Xn and point pn, whereas the motion ﬁeld (2) depends
on both the DVS pose (depth Z of Xn with respect to the
sensor) and velocities (twist coordinates). The gradient g may
be computed by taking the spatial derivatives of the predicted
image intensities I in a neighborhood of the current event
location pn, obtained through rendering the dense map M
according to the DVS pose in the current state. Examples
of the contrast function for positive and negative events are
shown in Fig. 2. Patches of 15 × 15 pixels around the event
location are displayed, but the local analysis of the generative
event model is only reliable close to the center. Fig. 3 reports
the cases of moving edges parallel or almost perpendicular
to the apparent motion, yielding largest and smallest absolute
contrast, respectively.
C. Recursive solution: Implicit EKF equations
Once the system state and measurements equations have
been designed, the update equations of the parameters of the
posterior in the EKF are also determined. The recursive esti-
mation carried out in the EKF is described by the equations in
Algorithm 1. We follow the notation in [11] for the posteriors
and their moments. The DVS pose tracking ﬁlter also assumes
that an accurate estimate of the initial conﬁguration, with
relatively small uncertainties, is given (µ0, Σ0). Let us further
explain the steps of Algorithm 1.
a) Prediction.: In this step, the projection of the posterior
beln−1 ∼N(µn−1, Σn−1) through the kinematic model (8)
gives the predicted posterior beln ∼N(¯µn, ¯Σn) before incor-
porating the measurement. The state mean and error covari-
ance are predicted according to lines (1)-(2) in Algorithm 1.
Uncertainty is propagated through the system by means of the
Jacobians of (8), Fn = ∂f/∂xn−1, Ln = ∂f/∂wn, evaluated
at the current best estimate, (µn−1, wn).
b) Correction.: This is the data assimilation step, where
the predicted posterior beln
∼
N(¯µn, ¯Σn) is combined
with the measurement zn to yield the updated posterior
beln ∼N(µn, Σn). The state mean and error covariance are
corrected according to lines (3)-(7) in Algorithm 1. Events
from the DVS are fed to the generative sensor equation (10)
to produce a residual that drives the update of the ﬁlter
variables. With regard to Figs. 2d and 3d, the correction
step changes the state such that the likelihood at the event
position increases (white region). The innovation process and
its covariance (lines (3)-(4) in Algorithm 1) are obtained by
linearization of the implicit measurement function (10) around
the current best estimate, (zn, ¯µn) (see [17], [18]). Uncertainty
is corrected in the system (up to ﬁrst order) by means of the
Jacobians of (10) (evaluated at (zn, ¯µn; M)), Hn = ∂q/∂xn,
Dn = ∂q/∂zn, with covariance of the measurement noise [17]
Rn := DnQη
nD⊤
n . Since q is a real value, both the noise and
the innovation covariances (Rn and Sn) are scalars.
5
(a)
(b)
(c)
(d)
Fig. 3. Neighborhood of an event triggered by a moving edge. Same notation as in Fig. 2. Top row: at the event location, the image gradient g is parallel to
the motion ﬁeld ˙u. Bottom row: g almost perpendicular to ˙u. Both rows correspond to a negative event.
Algorithm 1 Extended Kalman Filter (EKF) equations for one iteration, (µn−1, Σn−1) →(µn, Σn), with implicit measurement
function q.
1. Mean state (pred.)
¯µn = f(µn−1, wn)
2. Error covar. (pred.)
¯Σn = FnΣn−1F ⊤
n + LnQw
n L⊤
n , with Fn, Ln Jacobians of f.
3. Innovation
νn = −q(zn, ¯µn)
4. Innovation covar.
Sn = Hn ¯ΣnH⊤
n + Rn, with Hn and Rn given by the Jacobians of q.
5. Kalman gain
Kn = ¯ΣnH⊤
n S−1
n
6. Mean state
µn = ¯µn + Knνn
7. Error covar.
Σn = (I −KnHn)¯Σn(I −KnHn)⊤+ KnRnK⊤
n
1) Data association: An additional advantage of our ap-
proach is that there is no data association like in the classical
localization problem (associating predicted measurements to
actual ones), thus removing a challenging sub-problem and
common source of brittleness in localization and mapping with
the EKF. This is a consequence of using a dense map (as
opposed to a set of isolated landmarks) to represent the scene
and to design a measurement equation (10) that exploits such
a representation. There is no data association problem because
a correspondence between the event location and a map point
pn ↔Xn will always exist, and it can be computed via ray-
tracing. The errors caused by a mismatch between the true
surface point ¯Xn that triggered the event and the predicted
one Xn are implicitly taken into account in the EKF via the
innovation (10) and its covariance. For example, the value of
the gradient g in the neighborhood of the event will change
(with some degree of smoothness) and if the predicted value
does not yield the triggering of an event, the EKF adjusts the
state parameters so that a different surface point Xn will be
more likely to trigger the observed event. There is no need to
artiﬁcially search for a 3-D point, close to the predicted one,
that better explains the event.
IV. EXPERIMENTS
A. Synthetic data
The proposed method was tested with synthetic and real
data. The synthetic data was generated using computer graph-
ics software (Blender4) to render images of a given map along
a speciﬁed trajectory. Adjacent images were subtracted, thresh-
olded and randomly sampled to simulate the events generated
by a DVS. We chose a pinhole camera model with intrinsics
identical to the ones of a lens from the real experiments: 2.6
mm lens for a 1/3” sensor. A linear trajectory with constant
acceleration was simulated. Results are reported in Fig. 4.
Groups of 500 events every 8 ms were generated between
adjacent images. The algorithm processed 230k events. This
experiment validated the measurement function (10) since the
kinematic model (8) alone cannot predict the DVS motion.
The results show that the ﬁlter successfully estimated the DVS
pose and velocity, with small relative errors (Fig. 4c).
B. Real data
For the experiment with real data, we mounted the DVS
on a model train that runs on a straight track with constant
4https://www.blender.org/
6
(a)
(b)
(c)
Fig. 4. Constant acceleration experiment. (a) Estimated position. (b) Estimated velocity. (c) Relative errors in position and velocity between simulated trajectory
and estimated one.
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 5. Experiment with approximately constant velocity motion. (a) Visualization of a few events from the DVS (positive events in cyan, negative events in
magenta) used for ﬁlter initialization, overlaid on the rendered map. (b) Time since the last event at each pixel (∆tn in (10)) (c) Normalized histogram of the
absolute contrast in (10) (solid line) and Gaussian ﬁt (dashed line) (cf. Fig. 1c). The mode of the Gaussian corresponds to the threshold C. (d) Innovations
sequence νn. Estimated position (e) and velocity (f) of the event-based camera.
velocity. The DVS faced sideways and observed a planar
scene at a constant distance. The scene contains a pattern
of complex black and white stripes and a set of circles at
known locations; the latter were used for extrinsic calibration.
The DVS was intrinsically calibrated using standard camera
calibration techniques on the imaged points detected from
the projection of an array of blinking LEDS placed in a
checkerboard conﬁguration. Horizontal edges are parallel to
the apparent motion, and, consequently do not trigger events.
The intensities of the map were smoothed to provide non-zero
gradients in the regions near sharp edges that generate events,
hence to smooth the response of the contrast function (10) and
the corresponding likelihood in such regions. Fig. 5 reports
some of the results of this experiment. Fig. 5b shows, for a few
hundreds of events (Fig. 5a), the measured absolute contrast
|∆˜I| ≈−pn ⟨g, ˙u⟩∆tn used in the implicit measurement
function (10). Having the map intensities given in arbitrary
units (log of gray levels) and lacking physical measurements
of the incoming light that the DVS used to trigger the events,
the threshold values in Fig. 1c (≈0.2) are not applicable to the
map, and so a few events are used to estimate the threshold C
corresponding to the given map. The ﬁlter processed about
100k events and successfully estimated the DVS pose and
velocities of the DVS throughout the event stream. Figs. 2
and 3 were also obtained from this experiment.
7
V. CONCLUSION
We have successfully developed an implicit EKF for event-
based camera (DVS) localization based on the contrast resid-
ual (10), which provides a natural measure to deﬁne the
likelihood of an event. For this, we derived a generative
event model that incorporates the physical characteristics of
the DVS. Our algorithm readily matches the asynchronous
nature of the events and allows ﬁlter updates on an event-
by-event basis. An additional advantage of our approach is
that the contrast residual naturally takes into account a dense
map representation of the environment, removing the data-
association sub-problem. In future work, we plan to extend the
developed method to event-based SLAM without additional
sensing.
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