A Variable-Resolution Probabilistic Three-Dimensional Model for Change Detection

Abstract

Given a set of high-resolution images of a scene, it is often desirable to predict the scene's appearance from viewpoints not present in the original data for purposes of change detection. When significant 3-D relief is present, a model of the scene geometry is necessary for accurate prediction to determine surface visibility relationships. In the absence of an a priori high-resolution model (such as those provided by LIDAR), scene geometry can be estimated from the imagery itself. These estimates, however, cannot, in general, be exact due to uncertainties and ambiguities present in image data. For this reason, probabilistic scene models and reconstruction algorithms are ideal due to their inherent ability to predict scene appearance while taking into account such uncertainties and ambiguities. Unfortunately, existing data structures used for probabilistic reconstruction do not scale well to large and complex scenes, primarily due to their dependence on large 3-D voxel arrays. The work presented in this paper generalizes previous probabilistic 3-D models in such a way that multiple orders of magnitude savings in storage are possible, making high-resolution change detection of large-scale scenes from high-resolution aerial and satellite imagery possible. Specifically, the inherent dependence on a discrete array of uniformly sized voxels is removed through the derivation of a probabilistic model which represents uncertain geometry as a density field, allowing implementations to efficiently sample the volume in a nonuniform fashion.

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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING1
A Variable-Resolution Probabilistic
Three-Dimensional Model for Change Detection
Daniel Crispell, Member, IEEE, Joseph Mundy, and Gabriel Taubin, Fellow, IEEE
Abstract—Given a set of high-resolution images of a scene, it is
often desirable to predict the scene’s appearance from viewpoints
not present in the original data for purposes of change detection.
When significant 3-D relief is present, a model of the scene ge-
ometry is necessary for accurate prediction to determine surface
visibilityrelationships.Intheabsenceofanapriorihigh-resolution
model (such as those provided by LIDAR), scene geometry can
be estimated from the imagery itself. These estimates, however,
cannot, in general, be exact due to uncertainties and ambiguities
present in image data. For this reason, probabilistic scene models
and reconstruction algorithms are ideal due to their inherent
ability to predict scene appearance while taking into account
such uncertainties and ambiguities. Unfortunately, existing data
structures used for probabilistic reconstruction do not scale well
to large and complex scenes, primarily due to their dependence
on large 3-D voxel arrays. The work presented in this paper
generalizes previous probabilistic 3-D models in such a way that
multiple orders of magnitude savings in storage are possible, mak-
ing high-resolution change detection of large-scale scenes from
high-resolution aerial and satellite imagery possible. Specifically,
the inherent dependence on a discrete array of uniformly sized
voxels is removed through the derivation of a probabilistic model
which represents uncertain geometry as a density field, allowing
implementations to efficiently sample the volume in a nonuniform
fashion.
Index Terms—Computer vision, data structures, remote
sensing.
I. INTRODUCTION
H
civil application domains due to the advancing technology of
remote sensing and increasing availability of aerial platforms
(including unmanned aerial vehicles). As the immense volume
of produced imagery data grows over time, automated process-
ing algorithms become ever more important.
IGH-RESOLUTION aerial imagery is quickly becoming
a ubiquitous source of information in both defense and
A. Change Detection
One application wherein the collected imagery is frequently
used as input for is change detection, where a new im-
age is collected and must be compared with the “expected”
Manuscript received December 1, 2010; revised March 18, 2011; accepted
April 22, 2011.
D. Crispell is with the National Geospatial-Intelligence Agency, Springfield,
VA 22150 USA (e-mail: dancrispell@gmail.com).
J. Mundy and G. Taubin are with Brown University, Providence, RI 02912,
USA (e-mail: mundy@lems.brown.edu; taubin@brown.edu).
Digital Object Identifier 10.1109/TGRS.2011.2158439
appearance of the scene given the previously observed images.
Some change detection algorithms operate at large scales,
typically indicating changes in land-cover type (e.g., forests,
urban, and farmland). Due to the increasing availability of
high-resolution imagery, however, interest in higher resolution
and intraclass change detection is growing. The precise def-
inition of “change” is application dependent in general, and
in many cases, it is easier to define what does not constitute
valid change [1], [2]. Typically, changes in appearance due
to illumination conditions, atmospheric effects, viewpoint, and
sensor noise are not desired to be reported. Various classes of
methods for accomplishing this have been attempted, a survey
of which was given by Radke et al. [1] in 2005 (which includes
the joint histogram-based method [3] used for comparison by
Pollard et al. [2]). One common assumption relied on by most
of these methods, as well as more recent approaches [4], [5],
is an accurate registration of pixel locations in the collected
image to corresponding pixel locations in the base imagery.
When the scene being imaged is relatively flat or the 3-D
structure is known a priori, “rubber sheeting” techniques can
be used to accomplish this. When the scene contains significant
3-D relief viewed from disparate viewpoints, however, tech-
niques based on this assumption fail due to their inability to
predict occlusions and other viewpoint-dependent effects [2].
High-resolution imagery exacerbates this problem due to the
increased visibility of small-scale 3-D structure (trees, small
buildings, etc.) which is ubiquitous over much of the planet.
In this case, a 3-D model of the scene is necessary for accurate
change detection from arbitrary viewpoints. There have been
some previous works using 3-D models for change detection.
Huertas and Nevatia [6] matched image edges with projections
of 3-D building models with some promise but relied on the
manual step of model creation. Eden and Cooper’s method [7]
based on the automatic reconstruction of 3-D line segments
avoids the manual model creation step, although it is unable
to detect change due to occlusion since only a “wireframe”
(as opposed to surface) model is constructed. Pollard et al.
[2] proposed a probabilistic reconstruction method for change
detection and is the basis for the work presented here.
B. Probabilistic Models
Computing exact 3-D structure based on 2-D images is, in
general, an ill-posed problem. Bhotika et al. [8] characterized
the sources of difficulty as belonging to one of two classes:
scene ambiguity and scene uncertainty. Scene ambiguity ex-
ists due to the existence of multiple possible photo-consistent
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2 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING
Fig. 1.
resolution representation where needed (i.e., near surfaces) with far less data
than required in (a) fixed grid models. Representing surface probabilities using
the proposed method allows for variable-resolution models to be used for
probabilistic 3-D modeling approaches to change detection.
Variable-resolution models (b) such as octrees allow for high-
scenes and is a problem even in the absence of any sensor noise
or violations of assumptions built into the imaging and sensor
model. In the absence of prior information regarding the scene
structure, there is no reason to prefer one possible reconstruc-
tion over another. The term “scene uncertainty,” on the other
hand, is used to describe all other potential sources of error in-
cluding sensor noise, violations of certain simplifying assump-
tions (e.g., Lambertian appearance), and calibration errors. The
presence of scene uncertainty typically makes reconstruction
of a perfectly photo-consistent scene impossible. Probabilistic
models allow the scene ambiguity and uncertainty to be ex-
plicitly represented, which, in turn, allows the assignment of
confidence values to visibility calculations, expected images,
and other data extracted from the model. A probabilistic model
can also be used to determine which areas of the scene require
further data collection due to low confidence.
C. Contributions
The work presented in this paper generalizes previous prob-
abilistic reconstruction models in such a way that multiple
orders of magnitude savings in storage are possible, making
precise representation and change detection of large-scale out-
door scenes possible. Specifically, the inherent dependence on
a discrete array of uniformly sized voxels is removed through
the derivation of a new probabilistic representation based on a
density field model. The representation allows for implementa-
tions which nonuniformly sample the volume, providing high-
resolution detail where needed (e.g., near surfaces) and coarser
resolutions in areas containing little information (e.g., in empty
space) (Fig. 1). Additionally, it represents the first probabilistic
volumetric model to provide a principled way to take viewing
ray/voxel intersection lengths into account, enabling higher ac-
curacy modeling and rendering. The proposed model combined
with the reconstruction and associated algorithms comprise
a practical system capable of automatically detecting change
and generating photo-realistic renderings of large and complex
scenes from arbitrary viewpoints based on image data alone.
D. Outline
The remainder of this paper is laid out as follows. In
Section II, a brief survey of related work in the fields of 3-D
reconstruction is given. Section III describes the theoretical
foundations of the proposed model. Sections IV and V de-
scribe the implementation using an octree data structure and
the reconstruction algorithms, respectively. The application
of the reconstructed models for change detection are dis-
cussed in Section VI, followed by the paper’s conclusion in
Section VII.
II. RELATED WORK
There is a large body of previous work in the computer vi-
sion community involving the automatic reconstruction of 3-D
models from imagery, a brief overview of which is given here.
The bulk of the representations used are not probabilistic in
nature and are discussed in Section II-A. Existing probabilistic
methods are discussed in Section II-B.
A. Deterministic Methods
Three-dimensional reconstruction from images is one of the
fundamental problems in the fields of computer vision and
photogrammetry, the basic principles of which are discussed in
many texts including [9]. Reconstruction methods vary both in
the algorithms used and the type of output produced.
Traditional stereo reconstruction methods take as input two
(calibrated) images and produce a depth (or height) map as
output. A comprehensive review of the stereo reconstruction
literature as of 2002 is given by Scharstein and Szeliski [10].
While highly accurate results are possible with recent methods
[11], [12], the reconstruction results are limited to functions of
the form f(x,y) and cannot completely represent general 3-D
scenes on their own.
Many multiview methods are capable of computing 3-D
point locations as well as camera calibration information
simultaneously using the constraints imposed by feature
matches across multiple images (so called “structure from
motion”). One example of a such a method is presented by
Pollefeyes et al. [13], who use tracked Harris corner [14]
features to establish correspondences across frames of a video
sequence. Brown and Lowe [15] and Snavely et al. [16] use
scale invariant feature transform features [17] for the same
purpose with considerable success. Snavley et al. have shown
their system capable of successfully calibrating data sets con-
sisting of hundreds of images taken from the Internet. The
output of feature-based matching methods (at least in an initial
phase) is a discrete and sparse set of 3-D elements which are
not directly useful for the purpose of appearance prediction
since some regions (e.g., those with homogeneous appearance)
will be void of features and, thus, also void of reconstructed
points. It is possible to estimate a full surface mesh based
on the reconstructed features [18], [19], but doing so requires
imposing regularizing constraints to fill in “holes” correspond-
ing to featureless regions. Methods based on dense matching
techniques avoid the hole-filling problem but are dependent on

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CRISPELL et al.: VARIABLE-RESOLUTION PROBABILISTIC MODEL FOR CHANGE DETECTION3
smoothness and ordering assumptions to perform the matching.
The methods presented in this paper are not dependent on any
assumptions regarding the 3-D scene geometry yet produce a
dense model suitable for high-resolution change detection.
B. Probabilistic Methods
As discussed in Section I-B, probabilistic 3-D models have
the desirable quality of allowing a measure of uncertainty
and ambiguity in the reconstructed model to be explicitly
represented. Probabilistic methods are also capable of produc-
ing a complete representation of the modeled surfaces while
making no assumptions about scene topology or regularizing
constraints.
There exists in the literature several distinct methods for
reconstructing a probabilistic volumetric scene model based on
image data, all based on discrete voxel grid models. Although
the methods vary in their approach, the goal is the same: to
produce a volumetric representation of the 3-D scene, where
each voxel is assigned a probability based on the likelihood
of it being contained in the scene. The algorithms grew out of
earlier “voxel coloring” algorithms [20]–[23] in which voxels
are removed from the scene based on photometric consistency
and visual hull constraints. Voxel coloring methods are prone
to errors due to scene uncertainty; specifically, violations of
the color consistency constraint often manifest themselves as
incorrectly carved “holes” in the model [24]. To combat these
errors, probabilistic methods do not “carve” voxels but rather
assign each a probability of existing as part of the model.
Broadhurst et al. [25] assign a probability to each voxel based
on the likelihood that the image samples originated from a
distribution with small variance rather than make a binary
decision. Similarly, Bhotika et al. [8] carve each voxel with a
probability based on the variance of the samples in each of a
large number of runs. The final voxel probability is computed
as the probability that the voxel exists in a given run.
In addition to uncertainty due to noise and other unmodeled
phenomenon, any reconstruction algorithm must also deal with
scene ambiguity, the condition which exists when multiple
photo-consistent reconstructions are possible given a set of
collected images. If certain a priori information about the scene
is available, the information may be used to choose the photo-
consistent reconstruction which best agrees with the prior. The
reconstruction algorithm presented in this paper is as general
as possible and, thus, does not utilize any such prior. Another
approach is to define a particular member of the set of photo-
consistent reconstructions as “special” and aim to recover that
member. This is the approach taken by Kutulakos and Seitz [21]
and Bhotika et al. [8]. Kutulakos and Seitz define the photo
hull as the tightest possible bound on the true scene geometry,
i.e., the maximal photo-consistent reconstruction. They show
that, under ideal conditions, the photo hull can be recovered
exactly, while Bhotika et al. present a stochastic algorithm
for probabilistic recovery of the photo hull in the presence of
noise. The photo hull provides a maximal bound on the true
scene geometry but does not contain any information about the
distribution of possible scene surfaces within the hull. A third
approach is to explicitly and probabilistically represent the full
Fig. 2.
subdivided to achieve higher resolution. Assuming that no further information
about the voxel is obtained, the occlusion probabilities P(QA1), P(QB1), and
P(QC1) should not depend on the level of subdivision and should be equal to
P(QA2), P(QB2), and P(QC2), respectively.
(Left) Three viewing rays pass through a voxel, which (right) is then
range of possible scene reconstructions. Broadhurst et al. [25]
aim to reconstruct such a representation, as well as Pollard et al.
[2] for the purpose of change detection. Pollard et al. use
an online Bayesian method to update voxel probabilities with
each observation. Because a model which fully represents both
scene ambiguities and uncertainties and is capable of change
detection is desired, the model and algorithms presented in this
paper are based on this approach.
One quality that current volumetric probabilistic reconstruc-
tion methods all share is that the voxel representation is inher-
ently incapable of representing the true continuous nature of
surface location uncertainty. Using standard models, occlusion
can only occur at voxel boundaries, since each voxel is modeled
as being either occupied or empty. A side effect of this fact is
that there is no principled way to take into account the length
of viewing ray/voxel intersections when computing occlusion
probabilities, which limits the accuracy of the computations.
These limitations are typically handled in practice by the use of
high-resolution voxel grids, which minimize the discretization
effects of the model. Unfortunately, high-resolution voxel grids
are prohibitively expensive, requiring O(n3) storage to repre-
sent scenes with linear resolution n.
1) Variable-Resolution Probabilistic Methods: The benefits
of an adaptive variable-resolution representation are clear: In
theory, a very highly effective resolution can be achieved
without the O(n3) storage requirements imposed by a regular
voxel grid. One hurdle to overcome is the development of a
probabilistic representation which is invariant to the local level
of discretization. A simple example is informative.
Consider a single voxel pierced by three distinct viewing
rays, as shown in Fig. 2 (left). After passing through the single
voxel, the viewing rays have been occluded with probabilities
P(QA1), P(QB1), and P(QC1), respectively. Given that no
further information about the volume is obtained, the occlusion
probabilities should not change if the voxel is subdivided to
provide finer resolution, as shown in Fig. 2 (right). In other
words, the occlusion probabilities should not be inherently tied
to the level of discretization.
Using traditionalprobabilistic
P(QB1) = P(QC1) = P(X ∈ S), where P(X ∈ S) is the
probability that the voxel belongs to the set S of occupied
voxels. Upon subdivision of the voxel, eight new voxels are
created, each of which must be assigned a surface probability
P(Xchild∈ S). Whatever the probability chosen, it is assumed
to be constant among the eight “child” voxels since there is
no reason for favoring one over any of the others. Given that
methods,
P(QA1) =

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4 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING
rays A, B, and C pass through four child voxels, two child
voxels, and one child voxel, respectively, the new occlusion
probabilities are computed as the probability that any of the
voxels passed through belong to the set S of surface voxels.
This is easily solved using De Morgan’s Laws by instead
computing the complement of the probability that all voxels
passed through are empty
P(QA2) =1 − (1 − P(Xchild∈ S))4
P(QB2) =1 − (1 − P(Xchild∈ S))2
P(QC2) =P(Xchild∈ S).
(1)
(2)
(3)
Obviously, the three occlusion probabilities cannot be equal
to the original values, i.e., P(X ∈ S), except in the trivial cases
P(X ∈ S) = P(Xchild∈ S) = 0 or P(X ∈ S) = P(Xchild∈
S) = 1. This simple example demonstrates the general impos-
sibility of resampling a traditional voxel-based probabilistic
model while maintaining the semantic meaning of the original.
This presents a major hurdle to generalizing standard proba-
bilistic 3-D models to variable-resolution representations.
The methods proposed in this paper solve the problems
associated with resolution dependence by modeling surface
probability as a density field rather than a set of discrete voxel
probabilities. The density field is still represented discretely in
practice, but the individual voxels can be arbitrarily subdivided
without affecting occlusion probabilities since the density is
a property of the points within the voxel and not the voxel
itself. Occlusion probabilities are computed by integrating the
density field along viewing rays, providing a principled way to
take voxel/viewing ray intersection lengths into account. The
derivation of this density field is presented in Section III.
III. OCCLUSION DENSITY
In order to offset the prohibitively large storage costs and
discretization problems of the regular voxel grid on which tradi-
tional probabilistic methods are based, a novel representation of
surface probability is proposed in the form of a scalar function
termed the occlusion density. The occlusion density at a point
in space can be thought of as a measure of the likelihood
that the point occludes points behind it along the line of sight
of a viewer, given that the point itself is unoccluded. More
precisely, the occlusion density value at the point is a measure
of occlusion probability per unit length of a viewing ray which
is passing through the point.
If the occlusion density is defined over a volume, proba-
bilistic visibility reasoning can be performed for any pair of
points within the volume. In the case where surface geometry
exists and is known completely [e.g., scenes defined by a
surface mesh or digital elevation model (DEM)], the occlusion
density is defined as infinite at the surface locations and zero
elsewhere.
Given a ray in space defined by its origin point q and a unit
direction vector r, the probability of each point x along the
ray being visible from q may be computed. It is assumed here
that q is the position of a viewing camera and r represents a
viewing ray of the camera, but the assumption is not necessary
in general. Points along the line of sight may be parameterized
by s, the distance from q
x(s) = q + sr,s ≥ 0.
(4)
Given the point q and viewing ray r, a proposition Vsmay be
defined as follows:
Vs≡ “The point along r at distance s is visible from q.” (5)
The probability P(Vs) is a (monotonically decreasing) function
of s and can be written as such using the notation vis(s)
vis(s) ≡ P(Vs).
(6)
Given a segment of r with arbitrary length ? beginning at the
point with distance s from q, the segment occlusion probabil-
ity P(Q?
s + ? is not visible, given that the point at distance s is visible
s) is defined as the probability that the point at distance
P?Q?
s
?=P(¯Vs+?|Vs)
=1 − P(Vs+?|Vs).
(7)
Using Bayes’ theorem
P?Q?
s
?= 1 −P(Vs|Vs+?)P(Vs+?)
P(Vs)
.
(8)
Substituting vis(s) for the visibility probability at distance s
and recognizing that P(Vs|Vs+ds) = 1
P?Q?
P?Q?
If an infinitesimal segment length ? = ds is used, (9) can be
written as
s
?=1 −vis(s + ?)
?=vis(s) − vis(s + ?)
vis(s)
s
vis(s)
.
(9)
P?Qds
P?Qds
ds
s
?=−∂vis(s)
?
vis(s)
(10)
s
= −vis?(s)
vis(s).
(11)
The left-hand side of (11) is a measure of occlusion probability
per unit length and defines the occlusion density at point x(s).
The estimation of the occlusion density value is discussed in
Section V
α(x(s)) ≡ −vis?(s)
vis(s).
(12)
2) Visibility Probability Calculation: The visibility proba-
bility of the point at distance s along a viewing ray can be
derived in terms of the occlusion density function along the

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CRISPELL et al.: VARIABLE-RESOLUTION PROBABILISTIC MODEL FOR CHANGE DETECTION5
Fig. 3.
interest. (b) (Top) (Arbitrary) Example plot of occlusion density along the
viewing ray. (Bottom) Resulting visibility probability along the viewing ray.
(a) Viewing ray originating at the camera pierces the volume of
ray by integrating both sides of (12) with respect to a dummy
variable of integration t
s
?
0
?
0
?
0
α(t)dt =
s
?
0
−∂vis(t)
vis(t)
−
s
α(t)dt = [ln(vis(t)) + c]s
0
−
s
α(t)dt = ln(vis(s)) − ln(vis(0)).
(13)
Finally, by recognizing that vis(0) = 1 and placing both sides
of (13) in an exponential
vis(s) = e−?s
Equation (14) gives a simple expression for the visibility
probability in terms of the occlusion density values along the
viewing ray.
An example of corresponding occlusion density and visi-
bility functions is shown in Fig. 3, which depicts a camera
ray piercing a theoretical volume for which the occlusion
density is defined at each point within the volume. The value
of the occlusion density α(s) as a function of distance along
the camera ray is plotted, indicating two significant peaks in
surface probability. The resulting visibility probability function
is plotted directly below it.
3) Occlusion Probability: Substituting (14) back into (9), a
simplified expression of a segment’s occlusion probability is
obtained
?= 1 − e−?s+?
4) Relationship With Discrete Voxel Probabilities: The key
theoretical difference between the discrete voxel probabilities
P(X) of existing methods and the preceding formulation of
occlusion density is the interpretation of the probability values.
Because existing methods effectively model the probability that
a voxel boundary is occluding (whether they are defined as such
or not), the path length of the viewing ray through the voxel
0α(t)dt.
(14)
P?Q?
s
s
α(t)dt.
(15)
Fig. 4.
theocclusion density and appearance model are approximated as being constant
within each cell.
Camera ray parameterized by s cuts through cells in an octree. Both
is irrelevant. By contrast, the occlusion probabilities P(Q?
represent the probability that the viewing ray is occluded at
any point on the interval [s,s + ?]. The path length ? becomes
important when one moves away from high-resolution regular
voxel grids to variable-resolution models because its value
may vary greatly depending on the size of the voxel and the
geometry of the ray-voxel intersection.
s)
IV. IMPLEMENTATION: OCTREE REPRESENTATION
In order to make practical use of the probabilistic model
described in Section III, a finite-sized representation which is
able to associate both an occlusion density and appearance
model with each point in the working volume is needed. Details
are presented in this section of an octree-based implementation
which approximates the underlying occlusion density and ap-
pearance functions as piecewise constant.
Most real-world scenes contain large slowly varying regions
of low occlusion probability in areas of “open space” and
high quickly varying occlusion probability near “surfacelike”
objects. It therefore makes sense to sample α(x) in a nonuni-
form fashion. The proposed implementation approximates both
α(x) and the appearance model as being piecewise constant,
with each region of constant value represented by a cell of an
adaptively refined octree. The implementation details of the
underlying octree data structure itself are beyond the scope
of this paper; the reader is referred to Samet’s comprehensive
treatment [26]. Fig. 4 shows a viewing ray passing through a
volume which has been adaptively subdivided using an octree
data structure and the finite-length ray segments that result from
the intersections with the individual octree cells.
Each cell in the octree stores a single occlusion density
value α and appearance distribution pA(i). The appearance
distribution represents the probability density function of the
pixel intensity value resulting from the imaging of the cell.
The occlusion density value and appearance distribution are
assumed to be constant within the cell. Note that this piecewise-
constant assumption can be made arbitrarily accurate since, in
theory, any cell in which the approximation is not acceptable
can always be subdivided into smaller cells. In practice, how-
ever, the amount of useful resolution in the model is limited by
the resolution of the input data used to construct it.

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6 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING
A. Appearance
In addition to the occlusion density, it is necessary to define
appearance information for points in space in order to per-
form modeling and rendering operations. Here, “appearance”
describes the predicted pixel value of an imaging sensor, given
that the point is visible from the sensor. In general, many
factors contribute to this value: lighting conditions, viewing
angle, particularities of the imaging sensor, and others. There
is a large body of work in the computer graphics literature
that is focused on modeling these factors’ effect on an object’s
appearance, a comprehensive survey of which is given by
Dorsey et al. [27]. For practical reasons, the appearance model
used in this work is relatively simple; it is independent of
viewing direction and is modeled using a mixture of Gaussian
distributions. It is assumed that for a given point x, a single
distribution pA(c,x) describes the probability density of the
imaged value c of x. In the case of grayscale imagery, c
is a scalar value and the distribution is 1-D. In the case of
multi-band imagery (e.g., RGB color), c is an n-dimensional
vector and the distribution is defined over the n appearance
dimensions. When dealing with data such as satellite imagery
in which vastly different lighting conditions may occur, a
separate appearance model is used for each predefined range of
illumination conditions (i.e., sun positions). When performing
change detection, only the relevant appearance model for the
given sun position is considered. This is the approach taken by
Pollard et al. [2].
V. RECONSTRUCTION FROM IMAGERY
Pollard et al. [2] estimate the probability that each voxel X
of the model belongs to the set S of “surface voxels.” This
“surface probability” is denoted as P(X ∈ S) or simply P(X)
for convenience. The voxel surface probabilities are initialized
with a predetermined prior and updated with each new ob-
servation using an online Bayesian update equation (16). The
update equation determines the posterior surface probabilities
of each of a series of voxels along a camera ray, given their
prior probabilities P(X) and an observed image D
P(X|D) = P(X)P(D|X)
P(Dt).
(16)
The marginal (P(Dt)) and conditional (P(D)|X)) probabil-
ities of observing D can be expanded as a sum of probabilities
along the viewing ray. In practice, a single camera ray is
traversed for each pixel in image t and all pixel values are
assumed to be independent.
Rather than a camera ray intersecting a series of voxels, the
equation can be generalized to a series of N intervals along a
ray parameterized by s, the distance from the camera center.
The ith interval is the result of the intersection of the viewing
ray with the ith octree cell along the ray, and it has length ?i.
(See Fig. 4.) The interval lengths resulting from the voxel–ray
intersections are irrelevant in the formulation of Pollard et al.
because the occlusion probabilities are fixed and equal to the
discrete voxel surface probabilities P(Xi). The surface prob-
ability of the ith voxel is replaced by P(Q?i
si), the occlusion
probability (15) of the ith interval. The integral is replaced by a
multiplication due to the assumption that occlusion density has
constant value αiwithin the cell
P?Q?i
si
?= 1 − e−αi?i.
(17)
The probability that the start point of the ith interval is visible
from the camera is denoted by the term vis(si) and can be
simplified using the piecewise-constant model as follows:
⎛
vis(si) = exp
⎝−
i−1
?
j=0
αj?j
⎞
⎠.
si|D) is computed
(18)
The ith posterior occlusion probability P(Q?i
by following closely the formulation of Pollard et al. [2] and
generalizing to a series of intervals with varying lengths
P?Q?
si|D?= P?Q?
si
? P?D|Q?
si
?
P(D)
.
(19)
In order to simplify computations, the term prei, which rep-
resents the probability of observing D taking into account
segments 0 to i − 1 only is defined as follows:
?
prei≡
i−1
?
j=0
P
Q?
sj
?
vis(sj)pAj(cD).
(20)
The term pAj(cD) is the probability density of the viewing ray’s
corresponding pixel intensity value cDgiven by the appearance
model of the ith octree cell along the ray. Equation (19) can
then be generalized as
vis∞≡
N−1
?
i=0
?1 − P?Q?
?prei+ vis(si)pAi(cD)
si
??
(21)
P?Q?
si|D?=P?Q?
si
pre∞+ vis∞pA∞(cD)
(22)
where pre∞represents the total probability of the observation
based on all voxels along the ray. The probability of the ray
passing unoccluded through the model is represented by vis∞
and is computed based on (21). The term pA∞(cD) represents
the probability density of the observed intensity given that the
ray passes unoccluded through the volume and can be thought
of as a “background” appearance model. In practice, portions
of the scene not contained in the volume of interest may be
visible in the image. In this case, the background appearance
model represents the predicted intensity distribution of these
points and is nominally set to a uniform distribution. Note that
the denominator of (22) differs from the update equation of
Pollard et al. [2] due to consolidation of the “pre” and “post”
terms into pre∞and the addition of the “background” term
vis∞pA∞(cD).
Equation (22) provides a method for computing the posterior
occlusion density of a viewing ray segment but must be related
back to the cell’s occlusion density value to be of use. This is

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CRISPELL et al.: VARIABLE-RESOLUTION PROBABILISTIC MODEL FOR CHANGE DETECTION7
Fig. 5.
duration of the sequence, moving objects such as vehicles dominate the changes from image to image.
Four representative frames from the “downtown” video sequence filmed over Providence, RI. Images courtesy of Brown University. Because of the short
accomplishedbyassumingthatαi= α(t)isconstantwithinthe
segment (si<= t < si+ ?i) and by solving (17) for α
αi=−log?1 − P?Q?
si
??
?i
.
(23)
In general, however, multiple viewing rays from an image will
pass through a single octree cell. (This happens any time a
cell projects to more than one pixel in the image.) In order
to reconcile the multiple posterior probabilities, an occlusion
density value is calculated by computing the probability P(¯Q)
of occlusion by any of the K viewing ray segments within a
cell and relating it to the total combined length¯? =?
observations with longer corresponding segment lengths
k?kof
all of the segments. This has the effect of giving more weight to
P(¯ Q) = 1 −
K−1
?
k=0
(1 − P(Qk)).
(24)
The posterior occlusion probability corresponding to the
segment of the kth camera ray passing through the cell is
denoted P(Qk) for convenience
?K−1
K−1
?
−
k=0
K−1
?
The previous occlusion density value of each cell is replaced
withthe ¯ αvaluecomputedusingallK viewingraysofthegiven
image which pass through it.
¯ α =
−log
?
k=0
(1 − P(Qk))
?
k=0
?k
¯ α =
K−1
?
log(1 − P(Qk))
k=0
?k
.
(25)
A. Appearance Calculation
In addition to computing the occlusion density of a given
octree cell, a reconstruction algorithm must also estimate the
cell’s appearance model. Pollard et al. update the Gaussian
mixture model of each voxel with each image using an on-
line approach based on Stauffer and Grimson’s adaptive back-
ground modeling algorithm [28], and a similar approach is
used here. Because the proposed multiresolution model allows
for the possibility of large cells which may project to many
pixels of the image, a weighted average of pixel values is
computed for each cell prior to the update, with the weights
proportional to the segment lengths of the corresponding view-
ing ray/octree cell intersections. The cell’s Gaussian mixture
model is then updated using the single weighted average pixel
value.
B. Adaptive Refinement
Upon initialization, the model consists of a regular 3-D array
of octrees, each containing a single (i.e., root) cell. At this
stage, the model roughly corresponds to a regular voxel grid
at a coarse resolution. As more information is incorporated
into the model, however, the sampling of regions with high oc-
clusion density may be refined. The proposed implementation
subdivides a leaf cell into its eight children when its maximum
occlusion probability P(Qmaxi) reaches a global threshold.
The maximum occlusion probability of cell i is a function of
thelongestpossiblepaththroughthecell(i.e.,between opposite
corners) and the cell’s occlusion density
P (Qmaxi) = 1 − exp(−?maxiαi).
(26)
The occlusion densities and appearance models of the refined
cells are initialized to the value of their parent cell. This process
is executed after each update to the model.

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8 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING
Fig. 6.
algorithms. Original image courtesy of Brown University. (Bottom) ROC
curves for change detection using the proposed 3-D model and a fixed grid
voxel model.
(Top) Ground truth segmentation used to evaluate the change detection
Fig. 7.
that the moving vehicles have low probability, as well as some false detects
around building edges, presumably due to small errors in the camera model.
Original image courtesy of Brown University.
Changes detected by the proposed algorithm are marked in black. Note
VI. CHANGE DETECTION
Given a new image of a previously modeled scene, it is
often useful to detect regions in the image which represent
deviations from the expected intensity values. Pollard et al. [2]
demonstrated their system, which was the first to use proba-
bilistic 3-D information for the purposes of change detection,
Fig. 8.
to evaluate change detection. Because there is a low probability of a mov-
ing vehicle being at a particular position on the roads, they appear empty.
(Top) Image generated using the proposed 3-D model. (Bottom) Image gen-
erated using a fixed grid voxel model. The proposed variable-resolution model
allows for finer details to be represented, as is visible in the expanded crops.
Expected images generated from the viewpoint of the image used
to be superior to previous 2-D approaches. In particular, the
capability to model a 3-D structure allowed apparent changes
due to viewpoint-based occlusions to be correctly ignored. A
popular 2-D approach [3] consistently marked these regions
as change, leading to high false positive rates. The proposed
system utilizes the estimated probabilistic 3-D model in a
similar fashion, but with the added advantage of higher res-
olution capability over broader areas that the multiresolution
representation provides.
Given a model constructed using the methods proposed in
Section V and a viewpoint defined by a camera model, a
probability distribution pA(c) for each pixel in the image can

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CRISPELL et al.: VARIABLE-RESOLUTION PROBABILISTIC MODEL FOR CHANGE DETECTION9
Fig. 9.
changes are a result of moving vehicles on the roads. (Right) Changes detected using the proposed algorithm. Original image copyright Digital Globe.
(Left) Training image of the Haifa Street region from 2002. (Middle) Ground truth changes marked in one of the test images from 2007. Most of the
Fig. 10.
test images from 2007. There is a variety of changes resulting in the placement of large storage containers and other construction apparatus. (Right) Changes
detected using the proposed algorithm. Original image copyright Digital Globe.
(Left) Training image of the region surrounding the construction of the U.S. embassy from 2002. (Middle) Ground truth changes marked in one of the
be generated by integrating the appearance information of each
voxel along the pixel’s corresponding viewing ray R
pA(c) =
?
i∈R
vis(si)(1 − e−αi?i)pAi(c).
(27)
Pixel values with a low probability density are “unexpected.”
A binary “change mask” can therefore be created by thresh-
olding the image of pixel probability densities. The receiver
operating characteristic (ROC) curves in Figs. 6, 11, and 12
show the rate of ground-truth change pixels correctly marked
as change versus the rate of pixels incorrectly identified as
change. The plotted curves indicate how these values vary as
the threshold τ is varied.
Two distinct collection types are investigated: full motion
video collected from an aerial platform and satellite imagery.
The full motion video was collected using a high-definition
(1280 × 720 pixels) video camera from a helicopter flying over
Providence, RI, U.S. A few representative frames are shown in
Fig. 5. The model was updated using 175 frames of a sequence
(in random order) in which the helicopter made a full 360◦
loop around a few blocks of the downtown area, and the change
detection algorithm was then run on an image (not used in the
training set) that contains some moving vehicles (Fig. 7).
The ROC curve in Fig. 6 demonstrates the advantage that
higher resolution 3-D modeling provides to the change detec-
tion algorithm. In order to better visualize the higher resolution
capability of the proposed model, a synthetic image can be
generated by computing the expected intensity value E[pA(c)]
of each pixel based on the computed probability density distri-
butions of each pixel. Fig. 8 shows expected images generated
using the proposed model and the fixed grid model. Small
features such as individual building windows and rooftop air-
conditioning units are visible using the proposed model but
blend into the background using the fixed grid model. The
resolution of the fixed grid model can, in theory, always be
increased to match the capabilities of the variable-resolution
model, but doing so quickly becomes impractical due to the
O(n3) storage requirements. The fixed grid model of the
“downtown” sequence is nearly 50% larger than the variable-
resolution model while providing half the effective resolution.
A fixed grid model equaling the effective resolution of variable-
resolution model would require approximately 18 GB (1000%
larger than the variable-resolution model).
The satellite imagery used for experimentation was collected
by Digital Globe’s Quickbird satellite over Baghdad, Iraq, from
2002 to 2007 and are the same data sets used by Pollard et al.
[2] in their change detection experiments. Two areas of interest
are focused on the following: a region with some high-rise
buildings along Haifa Street and the region surrounding the
construction site of the new U.S. embassy building. A manually
determined translation is applied to bring the supplied camera
model to within approximately one pixel of reprojection error.
The images have a nominal resolution of approximately 0.7-m
GSD. Although the 3-D structure is less pronounced than in
the video sequence, it is still sufficient to pose a challenge to
2-D change detection algorithms, as shown by Pollard et al.
The “haifa” and “embassy” models were updated with each of

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10 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING
Fig. 11.
octrees in the variable-resolution model time to reach their optimal subdivision levels.
(a) Change detection ROC curve for the “haifa” data set after (a) one pass and (b) five passes of the 28 training images. The additional passes allow the
Fig. 12.
the variable-resolution model has again reached the effective resolution of the fixed grid model.
(a) Change detection ROC curve for the “embassy” data set after (a) one pass and (b) five passes of the 25 training images. After the additional passes,
28 and 25 images, respectively, taken between 2002 and 2006.
Although this number of images appears to be sufficient to
train the fixed-grid model, a larger number of images allows
the cells of the variable-resolution model time to reach optimal
subdivision levels. In order to accommodate this requirement,
five passes of the images were used to update the models
to simulate a larger number of collected images. The change
detection algorithm was then run on previously unobserved
images taken during 2007. Figs. 9 and 10 show a representative
training image and the ground truth changes for one of the
2007 images for the two experiments. Figs. 11 and 12 show the
resulting change detection ROC curves resulting after training
the model using one pass of the images and after five passes.
Because the variable-resolution models are initialized at a much
coarser level than the fixed grid model, their performance suf-
fers inthe period before the proper subdivision atsurface voxels
occurs. After five passes, however, the effective resolution has
TABLE I
COVERAGE AND MODEL SIZES FOR THE AERIAL VIDEO AND SATELLITE
IMAGE TEST SETS. THE RESOLUTION LISTED FOR THE
VARIABLE-RESOLUTION MODELS INDICATES THAT
OF THE FINEST OCTREE SUBDIVISION LEVEL
reached that of the fixed grid models while needing a fraction
of the memory and disk space. Table I lists the area modeled,
effective resolution, and storage requirements of each of the
modelsdiscussed.Fig.13providesavisualizationofthestorage

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CRISPELL et al.: VARIABLE-RESOLUTION PROBABILISTIC MODEL FOR CHANGE DETECTION11
Fig. 13.
effective resolution. The “downtown” model requires less memory per voxel in
both cases because only a single appearance distribution is needed to model the
constant illumination condition across video frames.
Storage costs of the models normalized by the number of voxels of
costs of the models, normalized by their respective effective
resolutions.
VII. CONCLUSION
The proposed novel probabilistic 3-D model allows for rep-
resentations which sample the 3-D volume in a nonuniform
fashion by providing a principled method for taking into ac-
count voxel/viewing ray intersection lengths. The model is
a generalization of previous probabilistic approaches which
makes feasible the exploitation of currently available high-
resolution remote sensing imagery. Experiments have shown
that this capability provides a distinct advantage for the ap-
plication of change detection over previously existing models.
Future work will focus on appearance models that can more
efficiently model large changes in illumination conditions, as
well as methods to extract surface information (e.g., DEMs)
from the probabilistic models.
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Daniel Crispell (M’11) received the B.S. degree in
computer engineering from Northeastern University,
Boston, MA, in 2003 and the M.S. and Ph.D. degrees
in engineering from Brown University, Providence,
RI, in 2005 and 2010, respectively.
While at Brown University, his research focused
on camera-based devices for 3-D geometry cap-
ture, aerial video registration, and 3-D modeling
and rendering from aerial and satellite imagery.
He is currently a Visiting Scientist at the National
Geospatial-Intelligence Agency, Springfield, VA.

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12IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING
Joseph Mundy received the B.S. and Ph.D. de-
grees in electrical engineering from Rensselaer Poly-
technic Institute, Troy, NY, in 1963 and 1969,
respectively.
He joined General Electric Global Research in
1963. In his early career at GE, he carried out re-
search in solid state physics and integrated circuit
devices. In the early 1970s, he formed a research
group on computer vision with emphasis on indus-
trial inspection. His group developed a number of in-
spection systems for GE’s manufacturing divisions,
including a system for the inspection of lamp filaments that exploited syntactic
methods in pattern recognition. During the 1980s, his group moved toward
more basic research in object recognition and geometric reasoning. In 1988,
he was named a Coolidge Fellow, which awarded him a sabbatical at Oxford
University, Oxford, U.K. At Oxford, he collaborated on the development of
theory and application of geometric invariants. In 2002, he retired from GE
Global Research and joined the School of Engineering, Brown University,
Providence, RI. At Brown University, his research is in the area of video
analysis and probabilistic computing.
Gabriel Taubin (M’86–F’01) received the Licenci-
ado en Ciencias Matemáticas degree from the Uni-
versity of Buenos Aires, Buenos Aires, Argentina,
and the Ph.D. degree in electrical engineering from
Brown University, Providence, RI.
In 1990, he joined IBM, where during a 13-year
career in the Research Division, he held various
positions, including Research Staff Member and Re-
search Manager. In 2003, he joined the School of
Engineering, Brown University, as an Associate Pro-
fessor of Engineering and Computer Science. While
on sabbatical fromIBMduringthe2000–2001academic year,hewas appointed
Visiting Professor of Electrical Engineering at the California Institute of Tech-
nology, Pasadena. While on sabbatical from Brown during the spring semester
of 2010, he was appointed Visiting Associate Professor of Media Arts and
Sciences at Massachusetts Institute of Technology, Cambridge. He serves as
a member of the editorial board of the Geometric Models journal. He has made
significant theoretical and practical contributions to the field now called Digital
Geometry Processing: to 3-D shape capturing and surface reconstruction and to
geometric modeling, geometry compression, progressive transmission, signal
processing, and display of discrete surfaces. The 3-D geometry compres-
sion technology that he developed with his group was incorporated into the
MPEG-4 standard and became an integral part of IBM products.
Prof. Taubin is the current Editor-in-Chief of the IEEE COMPUTER GRAPH-
ICS AND APPLICATIONS MAGAZINE and has served as Associate Editor of
the IEEE TRANSACTIONS OF VISUALIZATION AND COMPUTER GRAPHICS.
He was named IEEE Fellow for his contributions to the development of
3-D geometry compression technology and multimedia standards, won the
Eurographics 2002 Günter Enderle Best Paper Award, and was named IBM
Master Inventor.