Optical Normal Flow Estimation on Log-polar Images. A Solution for Real-Time Binocular Vision

Introduction

The development of computational solutions for simulationof visual behavior deals with the problem of integration andco-operation between different computational processes forcontrol of active vision systems. In general these computa-tional processes could emulate different and specificartificial visual behavior and their integration presents dis-tinct facets and problems. The first is the interaction andco-operation between different control systems, and thesecond is the use of common feedback information pro-vided by images captured.

One example is when a human observer has to accomplisha navigational task such as driving a car. In that case the

entire field of view is used to estimate the 3D motion and tocontrol the vehicle. However, it is known that the resolutionof the human eye decreases towards the periphery of theretina. This fact does not prevent the observer from detectingalarming events and from exploiting the image periphery tosense and estimate the ego-motion. If, during the observer’snavigational task, another vehicle (or object) is approaching,an estimate of the relative motion is calculated, including thedirection of a possible impact, in order to enable an appropri-ate reaction. During that phase the observer directs his gazetowards the moving vehicle (or object) and pursues it for ashort time interval. From this scenario it is evident that bothmotion detection and estimation are important for artificialsimulation of a navigational task. This simulation is morecomplete if it is accomplished with space-variant resolution.

1077-2014/97/030213 + 16 $25.00/ri960053 © 1997 Academic Press Limited

Optical Normal Flow Estimation on Log-polar Images. A Solution for Real-TimeBinocular Vision

his paper is concerned with a computational solution for normal optical flow estimation usingspace-variant image sampling. The article describes one solution for the problem based on log- Tpolar images, including the algorithm implementation on digital signal processing hardware. The

log-polar structure achieves high data compressing rates by maintaining a high resolution area—thefovea—and a space-variant resolution—the retinal periphery. This sampling scheme enables the imple-mentation of selective processing, characteristic of attentive vision. The solution was developed to beused with an active vision system to pursue moving objects, assuming that the cameras could alsomove. Some properties of the log-polar structure are also described and the main results of the imple-mentation are presented.

© 1997 Academic Press Limited

Jorge Dias, Helder Araújo, Carlos Paredes and Jorge Batista

Instituto de Sistemas e Robótica-Coimbra Pole,Departamento de Engenharia Electrotécnica,

Universidade de Coimbra, Largo Marquês de Pombal, 3000 Coimbra, Portugal

Real-Time Imaging 3, 213–228 (1997)

In this article we address this problem by proposing analgorithm for the estimation of optical normal flow on log-polar images. This article describes the advantages of thissampling scheme, its properties and a fast solution fornormal flow estimation using this type of space-variant datastructure.

Fast algorithms for optical flow estimation are crucial toimplement solutions for visual tracking or problems relatedwith it. These include surveillance, automated guidancesystems and robot manipulation [1–5].

Papanikolopoulos proposed a tracking process by using acamera mounted on a manipulator for tracking objects witha trajectory parallel to the image plane [1]. The informationsupplied by the vision system is processed by an opticalflow based algorithm that is accurate enough to track theobjects in real-time. The efficient use of windows in theimage improves the performance of the method. Allen alsoproposed a similar control process for tracking movingobjects, but extended to 3D [2].

The concept of visual behavior and active/animate visionand the work around this subject originated from severalpublications that relate concepts from robotics and alsofrom computer vision. Coombs studied the real-time imple-mentation of a gaze holding process using binocular vision[3]. The system used a binocular system mounted on amanipulator and Coombs studied different control schemesfor visual gaze holding using the information from twoimages. These studies included predictive control schemesto avoid delays in the control system [3,6]. The work ofCoombs [7] was one of the first to show real-time perfor-mance with an active vision system. However, this solutionhas some constraints regarding the type of targets that canbe pursued. These constraints are related with the zero-dis-parity filtering used to extract the target.

The problem of integration of different visual processeshas also been studied by Ballard, and in Grosso et al. [4]the authors proposed an approach based on multiple inde-pendent processes and on different vision strategies, tocontrol different degrees of freedom of an eye-head systemsupported by a manipulator. In that system, visual fixationwas an essential feature to control the system, but the per-formance of the system in achieving that state is notreported.

The co-operation between visual processes is also impor-tant and Pahlavan [8,9] considered the co-operation betweenfocusing and vergence for dynamic fixation. More recentlyUhlin [10] proposed a system based on the concept of an

open architecture, where the integration of differentmodules is used to fixate and pursue a moving target. Thework describes the architecture of the system and, as anopen architecture, integrates different techniques proposedby other authors. In that work special and expensive hard-ware is used to perform the task.

There are also other results reporting the use of visioninformation in real-time processes For example, Burtreports a real-time feature detection algorithm based onhierarchical pyramid data structures for surveillance androbotics [11]. These hierarchical data structures have beenused by Pahlavan [8] and Uhlin [10] in experiments forreal-time tracking with an artificial vision system. Also,Murray developed a monocular active system for real-timepursuit [12]. This implementation relies on the extractionand tracking of corners, but its performance on very pat-terned or textured environments is not reported.

More recently other image data structures, based on thelog-polar mapping, have been proposed for active visionvergence control [13,14]. Our work follows the same line,and we believe that log-polar sampling, by its properties, isa good alternative for the image data structure providinggood mechanisms for real-time processing when comparedwith pyramidal structures applied to regular sampledimages.

Searching for a Computational Solution for ActiveVision

The visual fixation is a state of our visual system where theobject of our attention is projected on to the central part ofthe human eye (fovea). Fixation is a consequence of thecontrol of our visual system to achieve that state. The fixa-tion state is part of more elaborate visual behaviors such asgaze-changing behaviors and gaze-holding behaviors[15,16].

The gaze-changing behavior includes behavior for pur-suit in which the eye is moved smoothly to track movingtargets, and saccades, which abruptly shifts the gaze fromone target to another. The two types of behavior are dis-tinct. The pursuit movements, in general, attempt to matchthe target velocity, independently of the target’s position.The saccadic movements ignore target velocity and attemptto match target position without processing the images dur-ing the trajectory of the eyes. Vergence movement is a thirdtype of gaze-changing behavior. If the objects are to beviewed with both eyes simultaneously, their related anglesmust be changed to achieve the fixation state. These angles

214 J. DIAS, H. ARAÚJO, C. PAREDES AND J. BATISTA

depend on the distance and orientation of the object relativeto our visual system. This is accomplished by a distinctslow eye movement behavior named vergence. The move-ment is driven by the retinal disparity vergence or bychanges in the distance where the eyes are focused (accom-modative vergence) [15].

Gaze-holding behavior uses movements to maintain theprojection of the environment in the retina stationary,despite our locomotion or head’s movements. Two types ofbehavior, named reflexes, are used to stabilize the images[15]. The optokinetic and vestibulo-ocular reflexes stabilizethe eyes and the vestibulocollic reflex stabilizes the head.

In the gaze-changing behavior the fixation state is anintegrated part of the process. That feature is essential forvisual sensing and, from the artificial systems point ofview, it is important to define the computational approachto achieve that state. That could be part of the solution formore elaborate processes such as body orientation [4],vergence control [8], pursuit simulation with mobile robots[17] or gaze-holding [3]. Computationally, these types ofbehavior can be defined as processes and these processescan be described graphically with states and transitionsbetween states. One example of such a description is givenin Figure 1. The figure represents the process diagram of asystem that simulates pursuit using an artificial system. Thetarget of interest is moving relative to the world in front ofthe cameras and the vision system must perform move-ments to hold the target in the image center. Two types ofmovements, known as pursuit and vergence, are used.These movements are the consequence of the control per-formed by the process that includes the fixation of thevisual system on the object. Previously, the system exe-cutes a saccadic movement and stabilizes the target in theimage center (fixation stabilization). After achieving fixa-tion the system changes to the state named pursuit. Figure 2represents the geometrical model of the active vision sys-tem. We assume that the pan angle for each camera can becontrolled separately through the angles ul and ur and theirtilt by using the c angle. The angles ul, ur and c can beused as variables to control the active vision system duringa process of visual fixation. These are the basic variablesnecessary to control in order to perform the simulation ofthis visual task. If the active vision system exhibits redun-dancy on some degrees of freedom, the correspondentvariables can be related in order to achieve the goal (see[17] for an example).

Many advantages can be achieved from this fixation pro-cess, where the selected target is always in the image center(the foveal region in mammals). This avoids the segmenta-

tion of all the image to select the target and allows the useof relative coordinate systems, which simplifies the spatialdescription of the target itself (relationship between theobserver reference system and the object reference system).Another advantage is that the computational effort is con-centrated in the image center to perform the correctexecution of the process. This advantage suggested a differ-ent type of structure for the sensor used in image sampling.Examples of sensors based on these retina like structures aredescribed by Manzotti et al. [13], Sandini et al. [18] andSchwartz [19]. Fortunately, this structure can be simulatedby re-mapping the image array using a log-polar transformand look-up tables simulating this sampling.

Space-variant Image Sampling for Active Vision

The implementation of visual behaviors in artificial systemsis strongly related to the performance of the image analysisalgorithms used. To achieve a good performance it isnecessary to develop fast algorithms for image processingand control, but exhibiting stability and robustness, to fit

OPTICAL NORMAL FLOW ESTIMATION ON LOG-POLAR IMAGES 215

Saccade movement

Vergencemovement

Fixation achievedSmooth pursuitand vergencemovements

Rest

Vergencestabilization

Pursuit

Figure 1. State diagram of the pursuit process.

y

{El}

z

xy z

Right eye

Left eye

O

x O

{Er}

{C}

+

+

θl

θg

ψ+θr

Figure 2. The geometric model used to represent the activevision system. The angles ul and ur control the vergence of eachcamera and can be independently controlled. Angle ug is respons-ible for the gaze of the system and can also be controlledindependently. The angle c is responsible for the cameras’ tilt.We assume that the system can, by calibration, be adjusted toalign the different degrees of freedom.

the goals of the vision system activity. This could include theintegration of the set of different algorithms running in thecomputational structure used in the active vision system andtheir adjustment to the geometry of the mechanical system.

The performance of these algorithms has a significantimpact on the system performance and sampling period.The sampling period can affect the stability of the systemand how rapidly disturbances are responded to. So it isalways desirable to develop fast algorithms and efficientdata structures to achieve the goal of a short samplingperiod. This section describes algorithms for real-timeimage processing operating on image data structures basedon the principles of log-polar transformation. The proper-ties and potentialities of these data structures for real-timeimage processing are described in this section.

One interesting feature of the human visual system is thetopological transformation [18,19] of the retinal image intoits cortical projection. In our own human vision system, aswell as in those of other animals, it has been found that theexcitation of the cortex can be approximated by a log-polarmapping of the eye’s retinal image. In other words, the realworld projected in the retinas of our eyes is reconfiguredonto the cortex by a process similar to log-polar mappingbefore it is examined by our brain [19].

In the human visual system the cortical mapping is per-formed through a space-variant sampling strategy, with thesampling period increasing almost linearly with the dis-tance from the fovea. Within the fovea the sampling periodbecomes almost constant. This retino-cortical mapping canbe described through a transformation from the retinalplane (r, u) onto the cortical plane (log(r), u) as shown inFigure 3. This transformation presents some interestingproperties as scale and rotation invariance about the originin the Cartesian plane which are represented by shiftsparallel to real and imaginary axes, respectively. This trans-formation is applied just on the non-foveal part of a retinal

image. If (x, y) are Cartesian coordinates and (r, u) are thepolar coordinates, by denoting as z = x + jy = reju a point inthe complex plane, the complex logarithmic (or log-polar)mapping is

(1)

for every z ≠ 0 where Real(w) = ln(r) and Im(w) = u + 2kp.However, we constrain angle u to the range of (0, 2p). Thislogarithmic mapping is a known conformal mapping pre-serving the angle of intersection of two curves.

Log-polar Mapping and its Properties

Log-polar mapping can be performed from regular imagesensors by using different types of space-variant samplingstructures [20–24]. For example, the mapping described inMassone et al. [20] is characterized by a linear relationshipbetween the sampling period and the eccentricity, definedas the distance from the image center.

The space-variant sampling structure described byMassone et al. [20] can be modified into a more simplifiedsampling structure, as is illustrated in Figure 4.

This spatial variant geometry of the sampling points isalso obtained through a regular tessellation and a samplinggrid formed by concentric circles with Nang samples overeach circle. The number of samples for each circle isalways constant and they differ by the arc

(2)

between samples.

For a given Nang, the radius rr of the circle i is expressedby an equation of the type

2πNang

w ln z= ( ),


y

x

θ

ρ

Figure 3. Log-polar transformation. Any point (xi, yi) in theimage plane (left) can be expressed in terms of (r, u) in the corti-cal plane (right), by (lnb(r), u).

Figure 4. Graphical representation of a simpler sampling struc-ture. The figure represents a structure with Nang = 60 angularsamples.

(3)

with i = 0…Ncirc and rfovea > 0 representing the minimumradius of the sampling circles.

In this case the radius basis b is expressed by

(4)

with i = 0…Ncirc and rfovea the minimum radius of the sam-pling circles.

The concentric circles are sampled with the perioddescribed by expression (2) and each sample covers a patchof the image corresponding to a circle with a radius givenby

(5)

The value rfovea could be chosen equal to the minimumsampling period to cover all the image center without gen-erating oversampling in the retinal plane. This constraint isexpressed by

(6)

The intensity value at each point of the cortical plane isobtained by the mean of the intensity values inside thecircle centered at the sampling point (rri, uri). Thereforethis model is similar to the models described by Spiegel etal. [22], Weiman [21,24] and Wallace et al. [23].

This simplified version of space-variant structure needsless storage than the one described by Massone et al. [20],as we can verify in Table 1.

Log-polar PropertiesThe log-polar mapping has a number of important proper-ties that make it useful as a sampling structure. Themapping of two regular patterns as shown in Figure 5results in similarly regular patterns in the other domain.From Figure 5(a) the concentric circles in the image planebecome vertical lines in the cortical plane. A single circlemaps to a single vertical line, since the constant radius r atall angles u of the circle gives a constant rc coordinate forall uc coordinates. Similarly an image of radial lines whichhave constant angle but variable radius results in a map ofhorizontal lines.

These mapping characteristics are fundamental for someproperties such as rotation and scaling invariance. Rotationand scaling result in shifts along the uc and rc axes, respec-tively. For rotation invariance it should be noted that allpossible angular orientations of a point at a given radiuswill map to the same vertical line. Thus, if an object isrotated around the origin between successive images, thiswill result only in a vertical displacement of the mappedimage. This same result is valid for radial lines. As a radialline rotates about the origin, its entire horizontal line map-ping moves only vertically. Also, from Figure 5 we canconclude that as a point moves out from the origin along aradial line, its mapping stays on the same horizontal linemoving from the left to the right. The mappings of the con-centric circles remain vertical lines and only movehorizontally as the circles change in size. These propertieswere fundamental for the development of algorithms forpattern recognition [20,25]. If we model image formationas perspective projection, scaling invariance implies thatchanges in focal length (or in the distance along the opticalaxis) will result in shifts along the rc axis.

ρπfoveaangN

≥ .

TNradius

ri

ang

= πρ.

bN

Nang

ang

=+−

ππ

ρ ρr foveai

ib=


Table 1. Different sampling schemes also require different stor-age in memory.

Reg. Sampling(512 3 512) Log-polar (Nang = 60)

Massone et al. [20] 262144 samples 6540 samples

262144 samples 1860 samplesbN

N

ang

ang

=+

−

π

π

x

y

x

y

<=>

θ

ρ

<=>

θ

ρ

Figure 5. The log-polar mapping applied to regular patterns. (a)Concentric circles in the image plane are mapped into verticallines in the cortical plane. (b) Radial lines in the image plane aremapped into horizontal lines in the cortical plane.

(a)

(b)

Normal Optical Flow on Log-polar

The space-variant resolution and sampling exhibits interest-ing properties for the optical flow. At this point we studysome of these properties of the optical flow and an algo-rithm for normal flow detection is described.

Consider the monocular imaging situation where theobserver and the scene are in motion relative to each other.In order to obtain the equations relating the scene to theimage measurements in a general form we define twocoordinate systems. The coordinate system {CAM} is fixedto the observer and used as reference and another co-ordin-ate system {W} is fixed at a point in the scene.

If this motion is rigid then it can be described by a trans-lational velocity vt = (vx, vy, vz)

T and angular velocity v =(vx, vy, vz)

T relative to a stationary environment. Eachscene point described by P = (X, Y, Z) measured withrespect to a coordinate system OXYZ fixed to the camera,moves relative to the sensor’s frame with velocity V, where

(7)

The 2D motion on an imaging surface is the projection of this3D motion field of the scene points moving relative to that sur-face. This is equivalent to projecting the 3D motion vectors onan image surface of a given shape, generating an image of themotion field. A planar or a spherical imaging surface can beconsidered, but in this work we will use a planar one.

Consider that the image is formed in a plane I orthogonalto the optical axis (equivalent to the Z2axis of the camerareferential—see Figure 6).

If the center of projection is at the origin O and theimage is formed in a plane orthogonal to the Z2axis at dis-tance f (focal length) from O, the relationship between theimage point p = (x, y, f) described and the scene point Punder perspective projection is

(8)

where z is an unit vector and ‘·’ denotes the inner productof vectors. Note that the image for processing is actually asampled version of the continuous image. Therefore it isnecessary to convert the coordinates obtained from Eqn (8)to the image coordinates in pixels. For that we consider therelationship between the units used in the {CAM} referen-tial and the pixels of the image. That relationship isobtained by a calibration process described by Batista et al.[26]. That process enables the estimation of the scale fac-tors for both the x and y axes, the focal length f and thecoordinates of the image center.

If we now differentiate p with respect to time, and sub-stitute for V, we obtain the following equation for v:

(9)

This equation represents the analytical equation of the opti-cal motion field projected in the image plane. The first termof v in Eqn (9) denotes the translational motion componentwhich depends on the depth Z = PT · z, while the secondterm denotes the rotational component which does notdepend on depth, but only on the rotational velocity.

To relate the optical flow field in log-polar coordinateswith Eqn (9), let us write

(10)

where r·, u·, x·, y· stand for the derivatives with respect to

time. Substituting the partial derivatives by their expres-sions we obtain

(11)

Defining

(12)

and g = u, the relationship between the time derivatives of jand r is given by

ξ ρρ

= lnbfovea

˙˙

cos sinsin cos ˙

˙.

ρθ

θ θθ

ρθ

ρ

= −

x

y

˙˙

˙

˙

ρθ

∂ρ∂

∂ρ∂

∂θ∂

∂θ∂

=

x y

x y

x

y

vP z

vT t t=⋅

⋅ ⋅ − + ⋅ × − ×ff

ˆ(( ˆ) ) ( ( ˆ ))z p v p z pωω ωω .

pP z

PT= f

.ˆ,

V v P= − − ×t ω .


y

O(CAM)

ZFOE

X

x(x0,yO)

νzwx

νt

ν

u

νx ωy

νy

ωx

Y

Figure 6. Image formation and motion representation.

(13)

From (11) and using (13) we obtain the relationshipbetween the motion field in Cartesian coordinates (x·, y·) andlog-polar coordinates (j

·, g· ) as

(14)

Assuming that the sensor is approaching the scene withtranslational motion, the point where the axis of translationintersects the retina is called the Focus of Expansion(FOE), since at this point the projection of the translationalmotion field is zero and all projections of the translationalmotion vectors emanate from this point.

For an observer with translational motion with an oppo-site direction, this point has the projection of the motionfield vectors pointing towards it, in which case we speak ofthe Focus of Contraction (FOC). The direction of theseprojected vectors is determined by the location of thisvanishing point and their lengths depend on the 3D posi-tions of the points in the scene.

If the observer wants to translate in the direction towardwhich he or she looks, this means that translation directionmust intersect the retina plane. If the observer is approachingthe scene, that intersection will be a point with characteristicsof a FOE. To construct an autonomous artificial system thatcan be guided, it is necessary to acquire information about itsown motion. That is a prerequisite to perform navigationaltasks. Since we are using an active vision system, we need todevelop algorithms to determine the projected motion field(optical flow field) in the images. The determination of thismotion can be useful, in a more advanced phase, to controlthe system and keep the FOE point around the image centeror, at least, stable in the image.

The images of Figure 7 illustrate the mapping of the opti-cal flow vectors for different types of translational motionof the sensor. Notice that when the sensor translates in thesame direction as the optical axis, the optical flow gener-ated appears as vectors diverging from the image center.The effect in the cortical plane is a set of vectors with thesame orientation.

The relative motion of the observer with respect to thescene gives rise to motion of the brightness patterns in theimage plane. The instantaneous changes of the brightness

pattern in the image plane are analysed to derive the opticalflow field, a two-dimensional vector field (u, v) reflectingthe image displacements.

The optical flow value of each pixel is computed locally—that is, only information from a small spatio-temporal neigh-borhood is used to estimate it. In general, it is not possible tocompute the true velocity of an image point just by observ-ing a small neighborhood. Suppose that we are watching afeature (a piece of contour or a line) at two instants of timeand through a small aperture smaller than the feature (seeFigure 8).

Watching through this small aperture, it is impossible todetermine where each point of the feature has moved to.The only information directly available from local meas-urements is the component of the velocity which isperpendicular to the feature, the normal flow. The compon-ent of the optical flow parallel to the feature cannot bedetermined. This ambiguity is known as the aperture prob-lem and exists independently of the technique employed forlocal estimation of flow. However, in cases where the aper-ture is located around an endpoint of a feature, the truevelocity can be computed, because the exact location of theendpoint at two instants of time can be computed. Thus, theaperture problem exists in regions that have stronglyoriented intensity gradients, and may not exist at locationsof higher-order intensity variations, such as corners.

Any optical flow procedure involves two computationalsteps. In the first, assuming the local conservation of someform of information, only local velocity is computed. In asecond step, in order to compute the other component ofthe optical flow vectors, additional assumptions have to be

˙

˙

cos

lnsin

sin

lncos

˙

˙.

ξγ

γ

γ

γ

γ

=+ −

12 2x y

b bx

y

˙ln

˙

ln

˙.ξ ρ

ρρ= =+

1 12 2b b x y


Forward motion

y

x

y

x

y

x

θ

θ'ρ

θ

θ'ρ

Lateral motion

θ

θ'ρ

Lateral motion

... ... ...

... ... ...

Figure 7. The optical flow vectors for different types of transla-tional motion. For lateral motion the optical flow vectors generatein the cortical plane streamlines of vectors with the same orienta-tion. For forward motion these lines are equal in all the plane.

made. Generally they are related with some kind ofsmoothness on the optical flow values.

The gradient-based approach introduced by Horn et al.[27] is based on the assumption that for a given scene pointthe intensity I at the corresponding image point remainsconstant over a short time instant. This corresponds to a

brightness constancy assumption, = 0, that gives a

relationship that can be used to estimate the flow para-meters directly from the spatial and temporal grey-levelgradients. If a scene point projects onto image point (j, g)at time t and onto the image point (j + dj, g + dg) at time (t + dt), we obtain the optical flow constraint equation [27],

(15)

which relates the flow (u, v) to the partial derivatives (Ij, Ig,It) of the image I. From this constraint alone, without mak-ing any additional assumptions we can only compute thenormal flow un, equivalent to the projection of optical flowon the gradient direction:

(16)

Since our goal is to develop robust algorithms that can per-form successfully in general environments, we abandoned allcomputational processes that are based on unrealistic assump-tions or assumptions that can fail to verify, and the develop-ment of the algorithms was based on normal optical flow.

Let I(j, g, t) denote the image intensity, and consider theoptical flow field v = (u, v) and the motion field vm 5 (um,vm) at the point (j, g), where the normalized local intensitygradient is n = (Ij, Ig)/ . The normal motion field at

point (j, g) is by definition

(17)

If we approximate the differential by its total deriva-

tive we get a relationship between the equations (16) and

(17)

(18)

which shows that the two fields are close to equal when thelocal image intensity gradient DI is high. This confirms thatthe normal flow is a good measurement of the normalmotion field in locations where the intensity gradientexhibits high magnitude [28].

Figure 7 illustrates that a uniform motion in the log-polarplane is not uniform in the retina plane. This fact can beexplored to generate an algorithm that, using this datastructure, detects and determines large motions in theperiphery of the image and smaller motions around theretina center. In the next section an algorithm for findingnormal flow using this data structure is described.

Practical Implementation and Results

The main hardware components of the system are themobile robot and the active vision system. These two basicunits are interconnected by a computer designated MasterProcessing Unit. This unit controls the movements of theactive vision system, communicates with the robot’s on-board computer and is the host computer of the DSP(Digital Signal Processors) image processing hardware.The connections between the different processing units arerepresented in the diagram shown in Figure 9, and a photo-graph of the system is presented in Figure 10.

The hardware for image processing is based onTMS320C40* DSP CPUs, including one frame grabber foracquisition of the images from the two cameras. The active

u uI

dI

dtm nn− =

1

∆,

dI

dt

ud

dt

d

dt

I

I II

d

dtI

d

dtmT

n= =

⋅ =v nξ γ ξ γ

ξ γ, ( , ).∆∆ ∆

1

I Iξ γ2 2+

u IIn t= − 1

∆.

∂∂ξ

∂∂γ ξ γ

Iu

Iv I I u I v It t+ + = + + = 0

dI

dt


Candidate motions ofa point in the edge

Aperture

Final edgeposition

Initial edgeposition

Figure 8. A line feature or contour observed through a smallaperture at time t moves to a new position at time t + dt. In theabsence of knowledge of camera motion, when we are looking ata viewpoint-independent edge in an image through an aperture,all we can say about the evolving image position of an indistin-guishable point along the edges is that this position continues tolie somewhere along the evolving image of the edge. In the limit,we can locally determine only that component of the imagemotion of an image-intensity edge that is orthogonal to theedge—the normal flow.

* Trademark of Texas Instruments Inc..

vision system has two CCD monochromatic video cameraswith motorized lenses (allowing for the control of the iris,focus and zoom position) and five step motors that conferan equal number of degrees of freedom for the vergence ofeach camera, baseline shifting, eyes tilt and neck pan. TheMaster Processing Unit is responsible for the control ofthese degrees of freedom, using step motor controllers.

The Master Unit is also responsible for the platform’strajectory. This unit sends commands to the platform byusing a RS-232 serial port that is the physical link of thecommunication system. The mobile platform has a multi-tasking operating system running on a 68020 CPU that isresponsible for generating and controlling the commandsreceived from the Master Unit. The supervision andmanagement of the whole system is done by an externalcomputer, connected to the Master Processing Unit andusing a serial link with a wireless modem.

This system is used to perform the pursuit of moving tar-gets so that the distance from the mobile robot to the targetis kept approximately constant. Normal optical flow is usedto enable visual fixation. After fixation is achieved the dis-

tance and heading of the target can be estimated and pursuitperformed.

Computing normal flow on binocular log-polar images—the C40 solution

The log-polar sampling and the normal flow algorithm areimplemented in software for DSP TMS320C40 from TexasInstruments. This software runs on a special board basedon C40s (DSP TMS320C40). The board corresponds toone of the possible configurations of a modular hardwaresystem based on these processors that explores the tech-nical and communications capabilities of these devices.The modular nature of the system enables a scalable sys-tem if more processing power is required. With thisarchitecture it is also possible to explore parallel process-ing implementations. The carrier board provides themodules with power, an interface for the ISA bus of theMaster Unit and the infrastructure that allows data com-munication between modules. The processor TMS320C40provides six communications ports that are bidirectionaland can carry data at up to 20 MB/s. A six-channel DMAco-processor allows for concurrent I/O and CPU operation,maximizing sustained CPU performance. The CPU, run-ning at 50 MHz, is capable of 275 million operations/s and50 M FLOPS. It utilizes a 40/32 bit floating/fixed pointmultiplier/ALU and has hardware support for divide andinverse square root operations. The addressing modesinclude special addressing as linear, circular and bit-reversed addressing. A single-cycle barrel shifter for 0–31bits right/left ensures fast bit manipulation. TheTMS320C40 has two identical l00 MB/s buses for dataand addresses (local and global).

The board used on our implementation comprises twomodules, with their own processor C40 at 40 Hz, local andglobal memory and fast communications ports (see Figure11). One of the modules is a frame grabber for imageacquisition and processing. The board has 4 MB of localmemory and 1024 3 1024 3 24 bits of video memory. The


Mobile platform

Motor controlsignals

Seriallink

Active visionsystem

Right video systemLeft video system

Masterprocessing

unit

Wirelessseriallink

SystemsupervisorDSP

Figure 9. System architecture.

Figure 10. The active vision system and the mobile robot.

Personal Computer BUS

Frame grabber

C40 C40

Figure 11. Block diagram of the image processing architecturebased on DSPs (Digital Signal Processors).

board is able to acquire color images with 1024 3 1024pixels by sampling the R, G and B signals, codifying eachsample using 8 bits. The frame grabber is also able to cap-ture images from different cameras and into a private videoarray buffer, without the intervention of the local C40. Theother module has 1 MB of local fast RAM and 4 MB ofglobal RAM. The processors are programmed to co-operateand to achieve the minimum execution time during the pro-cessing.

For real-time operation, we developed routines that areable to compute the several operations in the least amountof time possible. In order to apply a ‘divide and conquer’technique, each time an image is captured the C40 is usedto convert it into its log-polar equivalent, thus freeing theprivate memory of the grabber for the next image, and leav-ing the resulting image in its local memory for furtherprocessing. The additional processing is, in this case, thecomputation of the normal flow.

The algorithm to convert an image to the log-polar plane,and the method used to compute the normal flow, wereimplemented taking advantage of the set of special instruc-tions provided by the C40 (parallel, block repeat, etc.).Care should be taken when using those instructions, parti-cularly the parallel instructions (refer to [29] for detaileddescriptions of the instructions).

Implementation of Log-polar Mapping

To implement the conversion of a common video arrayimage into a log-polar image, the cortical pixel valuesshould be computed as averages of the receptive fields, asdescribed previously. Several implementations described inthe literature use this principle [22, 30–32]. However, inour case we decided to use just sampling (without averag-ing) to speed up the whole computational process. For thatpurpose we need to sample the video image at the correctcoordinates. One way of doing it is to program the mathe-matical equations needed to convert (j, g) into (x, y), andthen use them for every needed value of (j, g). Althoughsimple, that method would represent a big load to the pro-cessor, since it needs to perform multiplications andcompute logarithms to do it. Also, after the (x, y) valueswere obtained from the desired (j, g) coordinates, the pro-cessor would need to convert them into the correctdisplacement on the video array image, adding one moremultiplication to the method for each coordinate.

The procedure to convert a single (j, g) pixel into its cor-responding displacement in the video array image can be

speeded up if a conversion table is built. That table givesthe displacement in the video array image for all the pos-sible values of (j, g) and needs only to be initiated atstart-up. However, like all the other methods involvingtables it is quite fast, but needs large memory resources.The process of table access must be also reversible, andonce the video image has been captured, we want to con-vert it completely into its log-polar equivalent. So we cantake advantage of the C40 repeat instructions that allow usto repeat one or more instructions using no cycles to branchback to the start of the loop (they are thus called zero-over-head looping instructions). Also, we are using a processorthat is able to perform three bus accesses per cycle: oneinstruction read and two operand loads (reads) or stores(writes). In fact, the C40 takes advantage of this facilitythrough parallel instructions that allow two loads, twostores, or one load and one store to be performed in oneinstruction.

Unfortunately, each conversion using the table involvesone load for getting a displacement from the table, thenanother load when using the displacement in the videoarray to get a pixel, and one store for saving on a memorybuffer the pixel taken from the video array. So we cannotperform the conversion with a single parallel instruction,but we can still take advantage of them. Using C language-style expressions we can write two cycles of the conversionprocess like:

video_array_address_plus_displacemen_1 = *table++;pixel_1 = *video_array_address_plus_displacement_1;*buffer++ = pixel_1;video_array_address_plus_displacement _2 = *table++;pixel_2 = *video_array_address_plus_displacement_2;*buffer++ = pixel_2;

Although there are several ways to implement two cyclesof the conversion process, the suggested expressions werewritten based on several aspects that should be noted:

(i) The conversion table contains the displacements addedto the base address of the video array, instead of justthe displacements.

(ii) Although the parallel instructions can perform oneoperand load and one operand store with the sameinstruction, they cannot load a value and store the samevalue in the same instruction. So an expression like *a= *b could not be converted directly into a parallelinstruction. The loaded value can only be used by thenext instruction, and the stored value must have beenloaded with the previous instructions.


(iii) The C40 assembly code, and the parallel instructions inparticular, must follow certain rules, like the need touse auxiliary registers when pointers are needed, andthe impossibility to load those registers with parallelinstructions. So the expressions that load thevideo_array_address_plus_displacement_N pointerscannot be performed by parallel instructions.

Based on this, we can re-arrange the conversion process as:

video_array_address_plus_displacement_1 = *table++;video_array_address_plus_displacement_2 = *table++;pixel_1 = *video_array_address_plus_displacement_1;pixel_2 = *video_array_address_plus_displacement_2;*buffer++ = pixel_1;*buffer++ = pixel_2;

The first two expressions must be implemented by twoseparate load instructions (loading of auxiliary registers),but the last four expressions may be performed by twoparallel instructions. The first parallel instruction will exe-cute a double load, and the second a double store. Definedin that block, we need only to repeat it as many times ashalf the number of entries in the table (since the block willread two of them). In the end, our buffer will contain thelog-polar equivalent of the video array image.

Notice that the C40 uses words with 32 bits. So eachentry in the table will be a 32 bit pointer, and the table ele-ments must be consecutive in memory. The above routineis executed each time an image is captured. The destinationbuffer is located in the C40 local memory, allowing furtherprocessing without interfering with the acquisition process.Each entry in the destination buffer will also be 32 bits,although only the least significant 8 bits are important.

Computing the Normal Flow Using DSPs

Assuming that we have captured two images in two consec-utive instants of time, and converted them into theirlog-polar equivalent, this section will show a fast way tocompute the normal flow between them. We start bysmoothing the images, but we will not describe it, since it isa simple operation. Extracting the normal flow is, basically,a problem of computing the partial derivatives of the imageintensity on a certain pixel, with respect to j, g and t (time).To do that, we use the first differences of the pixel valueson a cube (see Figure 12). Let Ij,g,t be the intensity of thepixel located at (j, g) in the image taken at time t, then thepartial derivatives Dj, Dg and Dt can be obtained as fol-lows:

(19)

To apply Eqn (19) directly, we need to perform 21 addi-tions/subtractions to reach the expected results. In anattempt to reduce that number, let us start by defining thefollowing relations:

(20)

We can now simplify the partial derivatives using theabove relations, resulting in only 11 additions/subtractionsto compute:

(21)

D a b c d

D a b c d e f

D a b c d e ft

ξ

γ

≈ − − − −

≈ − + + − = −

≈ + − − = +

1

41

4

1

41

4

1

4

( )

( ) ( )

( ) ( ).

a I I

b I I

c I I

d I I

e b d

f a c

t t

t t

t t

t t

= −

= −

= −

= −

= −= −

+ + +

+ + +

+ + +

+ + +

ξ γ ξ γ

ξ γ ξ γ

ξ γ ξ γ

ξ γ ξ γ

, , , ,

, , , ,

, , , ,

, , , ,

.

1 1 1

1 1 1

1 1 1

1 1 1

D I I I I

I I I I

D I I

t t t t

t t t t

t t

ξ ξ γ ξ γ ξ γ ξ γ

ξ γ ξ γ ξ γ ξ γ

γ ξ γ ξ γ

≈ + + + −

− + + +

≈ + +

+ + + + + + + +

+ + + +

+ + +

1

41

41

4

1 1 1 1 1 1 1 1

1 1 1 1

1 1 1

( )

( )

(

, , , , , , , ,

, , , , , , , ,

, , , , II I

I I I I

D I I I I

t t

t t t t

t t t t t

ξ γ ξ γ



+ + + + +

+ + + +

+ + + + + + + +

+ −

− + + +

≈ + + + −

1 1 1 1 1

1 1 1 1

1 1 1 1 1 1 1 1

1

41

4

, , , ,

, , , , , , , ,

, , , , , , , ,

)

( )

( )

−− + + ++ + + +1

4 1 1 1 1( )., , , , , , , ,I I I It t t tξ γ ξ γ ξ γ ξ γ


t + 1

t

γ

γ + 1

ξ + 1ξ

Figure 12. The 8 pixels involved in the computation of the par-tial derivatives on the shaded pixel.

Another problem we must face is reading the pixels inthe two images. Each time the partial derivatives are com-puted, we need to read 4 pixels from each image. Sinceaccessing memory is always a slow task, we should try toreduce the number of accesses needed. That is not a hardgoal to accomplish. In fact, if we compute the derivativessequentially, we only need to access the memory four timesfor each new set of partial derivatives, since 4 of the pixelsused in the last computation will be used again. Hence, wecan store those pixels in the machine registers, and just readthe next 4 pixels.

After computing the partial derivatives, how will we usethem? The answer depends on the problem at hand, and onwhat we intend to do with the values. As an example, let ussuppose we want to compute the angle of the vector withcoordinates (Dj, Dg). Notice that this angle would be com-puted in the j 2 g space, and therefore one should becareful in relating this value with the orientation in imagespace. As was the case with the sampling of the log-polarimage, the inverse tangent is not going to be computed eachtime the angle is needed. The fastest way is to build a tablewith the inverse tangent of several pairs of values of Dj andDg. How big will the table be? Since the pixel intensity is avalue between 0 and 255, the values of Dj and Dg arealways between 2255 and 255. Hence, a table with all thepossible combinations of values of Dj and Dg would have511 3 511 entries. This is obviously an enormous table. Ingeneral, we do not need a table with that many entries.What we should do is just ignore one or more of the leastsignificant bits of the values, and use them as indexes to thetable.

Timing Considerations for Binocular Real-time Operation

The system throughput depends very much on the algorithmsimplemented, on the size of the images we are working on,and on our needs and hardware solutions. Figure 13 shows avery fast processing scheme, taking full advantage of thehardware available. Looking back at Figure 11, we see thatonly the C40 supporting the frame grabber can communicatewith the PC host. For that reason, that C40 is called the mas-ter, and all the others (just one in the case described) arecalled the slaves. The frame grabber has multiple videoinputs, but can only capture from one of them at a time.Since the images captured are interlaced PAL video signals,and since we decided not to use the full resolution of theimages, we only capture one field from each camera. Figure13 shows that both video signals are synchronized. For thatto happen, the cameras must have external synchronizationsignals. If they are not synchronized, there will be a delay

between the ending of a capture and the beginning of thenext, which will reduce the global processing speed.

The timing diagram in Figure 13 describes the startingstages of the processing scheme. The rest of the processingwill be a simple repetition of the final steps shown. Belowwe enumerate all the steps:

(i) While the even field of the right camera is beingacquired, both C40s are in busy wait.

(ii) When the field ends, the master C40 switches theframe grabber to acquire the odd field of the leftcamera, and starts the log-polar sampling of the rightimage, using the method described above. After thelog-polar image is complete, it will be downloaded tothe slave C40 for processing.

(iii) The slave C40 receives the log-polar image, andapplies the normal-flow algorithm to it. The slave willstore that image until another image from the rightcamera is received, since the algorithm must be usedwith two images.

(iv) When the odd field from the left camera ends, theframe grabber is switched back to the right camera,and the left image will be converted and downloaded.When the time comes to download the left log-polarimage, the results from the slave C40 are available, andthe master C40 will upload them.

(v) The master C40 will then process the results, andinform the PC host of the actions to take (in our case,the control of an active vision system).

With this scheme we are able to reach a processingcycle of 50 Hz (the frequency of one field of the video sig-nal), with a maximum processing delay of 40 ms. Since weare interested on the results from both images, the accumu-lated processing reduces the frequency to 25 Hz. What ifthe algorithms are too slow to fit in that diagram? Thenadditional C40s should be used, and instead of the resultsfrom one slave being uploaded by the master, they wouldbe sent to another slave, and so on. The master wouldreceive the final results from the last slave in the chain,


VideoRight

VideoLeft

MasterC40

SlaveC40

EvenEven Even

Odd Odd

T T TFlow Flow

S T S T S T PP

Odd Odd

Even Even

No operation

Not used[S] Sampling[T] Transferring[P] Processing

Figure 13. The timing diagram. The video signal from bothcameras is shown, and also the processing stages of both C40s.

adding a processing delay of 20 ms per slave, but keepingthe same processing frequency. The amount of time spentin transferring between the C40s is negligible. Indeed,DMA is used, allowing the C40s to process while receiv-ing and sending data.

Results obtained by the application of this normal flowalgorithm are illustrated in Figure 14.

After the computation of the log-polar normal opticalflow, additional processing has to be done in order toenable the active vision system to perform fixation. Thebasic idea is to use the “amount” of motion taking place inthe image as an indication of the region that should be fix-ated, i.e., that should be in the foveal area of the image. Asa measure of the amount of motion, the length of the flowvectors is used. Let vn represent a normal flow vector and??vn?? its magnitude. Two functions of the magnitude of theflow vectors are computed: the integral along the g-axis foreach one of the j values and the integral along the j-axisfor each one of the g values. Because we are dealing withdiscrete values, these functions are

(22)

and

(23)

where ji and gi represent the discrete values of the vari-ables. Figure 15 displays the values of these two functionscomputed for the sequence of images of Figure 14. Afterthese two functions are computed, the centers of mass ofboth distributions are used as the location of the movingtarget. During this process several thresholds are applied,namely to the magnitude of the vectors and to the valuesand extent of the functions. These thresholds prevent thesystem from reacting to noisy and small motions. The con-trol system (running in the Master Unit) takes these valuesas estimates of the locations and directs both cameras sothat fixation is performed. The control structure includespredictive filters. Further processing may be required toconfirm that both cameras are fixating the same object.Fixation of the object enables the robot to track it.

Discussion and Conclusions

This paper describes a computational solution for normaloptical flow estimation using space-variant image sampling.

The article describes some of the most important propertiesof the log-polar structure for fast and real-time processing.These include the high data compressing rates by maintain-ing a high resolution area—the fovea—and a space-variantresolution—the retinal periphery. This sampling scheme isuseful for the implementation of selective processing, char-acteristic of attentive vision and active vision svstems.

The implementation of normal optical flow detection wasdeveloped around digital signal processors (DSP) TMSC40. The normal optical flow computation is performed inreal time (video frame rates) for both the left and rightimages. The use of the log-polar sampling scheme was cru-cial for this type of performance.

The computation of the normal optical flow is part of arobotic system that uses binocular visual fixation in order totrack moving objects so that the distance between the targetand the mobile robot is kept approximately constant.Binocular visual fixation enables easy computation of thetarget distance and heading. Despite the fact that globallythe image is much smaller, the higher resolution in the cen-tral region of the image enables a more robust operation ofthe control structure of the active vision system while per-forming visual fixation.

To fixate the target, its location in both images is com-puted from the distribution of the normal flow. Furtherprediction filtering is required to enable smooth fixationcontrol. The system described demonstrates the use ofnormal optical flow computed in log-polar sampled imagesto perform binocular visual fixation in real time with anactive vision system integrated in a mobile robot.

In the future we plan to implement more complex visualbehaviors that will enable the robot to perform obstacleavoidance as well as other reactive behaviors. The goal isto equip the mobile system with behaviors that enable thedevelopment of co-operation with other robots.

References

1. Papanikopoulos, N., Khosla, P. & Kanade, T. (1993) Visualtracking of a moving target by a camera mounted on a robot: acombination of control and vision. IEEE Trans. on Roboticsand Automation, 9(1): 14–34

2. Allen, P., Timcenko, A., Yoshimi, B. & Michelman, P. (1993)Automated tracking and grasping of a moving object with arobotic hand-eye system. IEEE Trans. Robotics andAutomation, 9(2): 152–165.

3. Coombs, D. (1992) Real-Time Gaze Holding in BinocularRobot Vision, PhD Dissertation, University of Rochester,USA, June.

H vj

i i

( ) ,γξ ξ

= ∑

G vj

i i

( )ξγ γ

= ∑


4. Grosso, E. & Ballard, D. (1992) Head-Centered OrientationStrategies in Animate Vision, Technical Report 442, Comp.Science Dept., University of Rochester, USA, October.

5. Cipolla, R. & Hollinghurst, N. (1994) Visual robot guidancefrom uncalibrated stereo. In: Brown, C. & Terzopoulos, D.,Real-time Computer Vision, University Press, pp. 169–187.

6. Brown, C. (1990) Gaze controls with interaction and delays.IEEE Transactions on Systems, Man and Cybernetics, IEEETSMC 20, May, pp. 518–527.

7. Coombs, D. & Brown, C. (1992) Real-time smooth pursuittracking for a moving binocular robot. In: Proc. 1992 IEEEComp. Soc. Conf. on Computer Vision and PatternRecognition, pp. 23–28, Champaign, USA.

8. Pahlavan, K., Uhlin, T. & Eklund, J.O. (1992) Integratingprimary ocular processes. In: Proc. Second EuropeanConference on Computer Vision, Santa Margherita, Italy:Springer-Verlag, pp. 526–541.

9. Pahlavan, K. (1993) Active Robot Vision and Primary OcularProcesses, PhD thesis, Royal Institute of Technology,Sweden.

10. Uhlin, T., Pahlavan, K., Maki, A. & Eklund, J.O. (1995)Towards an active visual observer. In: Prof. FifthInternational Conference on Computer Vision, pp. 679–686,Cambridge, MA, June 20–23.

11. Burt, P., Bergen, J., Hingorani, R., Kolcynski, R., Lee, W.,Leung, A., Lubin, J. & Shvaytser, H. (1989) Object tracking


Figure 14. Two different image sequences showing the computation of the normal optical flow. For each sequence the top row displaysthe original images and the second row displays the computed normal flow vectors on log-polar. In the log-polar images the horizontalaxis corresponds to the coordinate j and the vertical axis corresponds to the coordinate g . The origin of each log-polar image is locatedat the bottom left corner.

with a moving camera. In: Proc. IEEE Workshop VisualMotion, Irvine.

12. Murray, D., McLaughlin, P., Read, I. & Sharkey, P. (1993)Reactions to peripheral image motion using a head/eye plat-form. In: Proc. 4th International Conference on ComputerVision, pp. 403–411.

13. Manzotti, R., Tiso, R., Grosso, E. & Sandini, G. (1994)Primary ocular movements revisited. Technical Report 7/94,LIRA Lab, University of Genoa, Italy, November.

14. Panerai, F., Capurro, C. & Sandini, G. (1995) Space variantvision for an active camera mount. Technical Report TR 1/95,LIRA Lab, University of Genova, Italy.

15. Carpenter, H. (1988) Movements of the eyes, 2nd edn,London: Pion Limited.

16. Hacisalihzade, S., Stark, L. & Allen, J. (1992) Visual percep-tion and sequences of eye movement fixations: a stochasticmodeling approach. IEEE Trans. on Systems, Man andCybernetics, 22(3): 474–481.

17. Dias, J., Paredes, C., Fonseca, I., Batista, J., Araújo, H. & deAlmeida, A. (1995) Simulating Pursuit with Machines –Experiments with Robots and Artificial Vision. In: 1995 IEEEInt. Conf. on Robotics and Automation, Nagoya, Japan, May.

18. Sandini, G. & Tagliasco, V. (1980) An anthropomorphicretina-like structure for scene analysis. Computer, Graphicsand Image Process., 14: 365–372.

19. Schwartz, E. (1984) Anatomical and physiological correlatesof visual computation from striate to infero-temporal cortex.IEEE Trans. on Systems, Man, and Cybernetics, SMC-14(2):257–271.

20. Massone, L., Sandini, G. & Tagliasco, V. (1985) Form-invari-ant topological mapping strategy for 2D shape recognition.

Computer Vision, Graphics and Image Processing, 30:169–189.

21. Weiman, C.F. (1988) Exponential sensor array geometry andsimulation. In: Proc. SPIE Conf. on Pattern Recognition andSignal Processing, 938, pp. 129–137, Orlando, FL.

22. der Spiegel, J.V., Kreider, G., Claeys, C., Debusschere, I.,Sandini, G., Belutti, P. & Soconi, G. (1989) A foveatedretina-like sensor using CCD technology. In: Mead, C. (ed.)Analog VLSI implementations of neural systems, Boston, MA:Kluwer.

23. Wallace, R.S., Ong, P.-W., Bederson, B.B. & Schwartz, E.L.(1994) Space variant image processing. Int. J. ComputerVision, 13(1): 71–90.

24. Weiman, C.F. (1994) Log-polar binocular vision system –final report, SBIR Phase II Contract #NAS 9–18637, T.R.C.,Danbury, December.

25. Bederson, B.B., Wallace, R.S. & Schwartz, E.L. (1995) Aminiaturized space-variant active vision system: cortex-I.Machine Vision and Applications, 8: 101–109.

25. Reitboeck, H. & Altmann, J. (1984) A model for size- androtation-invariant pattern processing in the visual system.Biol. Cybern., 51: 113–121.

26. Batista, J., Dias, J., Araújo, H. & de Almeida, A. (1993)Monoplanar camera calibration, British Machine VisionConference, 21–23 September, Surrey, UK, 2: 476–488.

27. Horn, B. & Schunk, B. (1981) Determining optical flow.Artificial Intelligence, 17: 185–204.

28. Fermuller, C. & Aloimonos, Y. (1995) Qualitative egomotion.Int. J. Computer Vision, 15: 7–29.

29. TMS320C4x User’s Guide, Texas Instruments (editor), 1992.30. Bailey, J.G. & Messner, R.A. (1990) Docking target design


5

Σθ

Σ

3

Σ

Σ

1 log(ρ)

Σ

Σ

6

Σθ

Σ

4

Σ

Σ

2

Σ

Σθ θ

θ θ

log(ρ)

log(ρ)log(ρ)

log(ρ)log(ρ)

Figure 15. Computation of the integrals of the normal flow vector magnitudes.

and spacecraft tracking system stability. SPIE – IntelligentRobots Computer Vision VIII: Algorithms and Techniques,1192: 820–831.

31. Weiman, C.F. (1990) Video compression via log-polar map-ping. In: Proc. SPIE Conf. on Real Time Image Processing II,1295: 266–277, Orlando, FL.

32. Bederson, B.B., Wallace, R.S. & Schwartz, E.L. (1993) Aminiaturized space-variant active vision system: cortex-I. In:Mammone, R.J. (ed.), Artificial Neural Networks for Speechand Vision, Chapman and Hall, pp. 429–456.

33. Aloimonos, Y., Weiss, Y. & Bandopadhay, A. (1988) Activevision. Int. J. Computer Vision, 7: 333–356.

34. Aloimonos, Y. (1990) Purposive and qualitative active vision.Proc. Image Understanding Workshop, 816–828.

35. Bajcsy, R. (1988) Active perception. Proceedings of theIEEE, 76: 996–1005.

36. Ballard, D. (1991) Animate vision. Artificial Intelligence, 48:57–86.

37. Bederson, B.B., Wallace, R.S. & Schwartz, E.L. (1992) A mini-aturized active vision system. In: Proc. 11th IAPR InternationalConference on Pattern Recognition Vol. IV, Conference D:Architectures for Vision and Pattern Recognition, pp. 58–61.

38. Bergen, J., Burt, P., Hingorani, R. & Peleg, S. (1990)Computing two motions from three frames. Technical Reportfrom David Sarnoff Research Center, subsidiary of SRIInternational, Princeton, NJ, 08543–5300.

39. Blackman, S. (1986) Multiple-target tracking with radarapplications. Artech House, Inc., pp. 19–25.

40. Burt, P. (1981) Fast filter transforms for image processing.Computer Vision, Graphics, and Image Processing, 16:16–20.

41. Burt, P. & Adelson, E. (1984) The pyramid as a structure forefficient computation. In: Rosenfeld, A. (ed.), Multiresolutionimage processing and analysis, Spring Series in InformationSciences, Vol. 12, New York: Springer.

42. Burt, P., Anderson, C., Sinniger, J. & van der Wal, G. (1986)A pipelined pyramid machine, NATO ASI Series, Vol. F 25.

43. Crowley, J. (1981) A Representation for Visual Information,PhD Thesis, Carnegie-Mellon University, Robotics Institute,Pittsburgh, PA.

44. Dias, J., Batista, J., Simplício, C., Araújo, H. & de Almeida,A. (1993) Implementation of an Active Vision System. In:Proceedings of International Workshop on MechatronicalComputer Systems for Perception and Action, HalmstadUniversity, Sweden, June 1–3.

45. Meer, P., Baugher, E. & Rosenfeld, A. (1987) Frequencydomain analysis and synthesis of image pyramid generatingkernels. IEEE Transactions on Pattern Analysis and MachineIntelligence, PAMI-9(4): 512–522.

46. Paul, R. (1984) Robot manipulators: mathematics, program-ming, and control. Cambridge, MA: The MIT Press, pp. 9–11.


Optical Normal Flow Estimation on Log-polar Images. A Solution for Real-Time Binocular Vision

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