Discrete camera calibration from the information distance between pixel streams
Etienne Grossmann
Tyzx, Inc

Francesco Orabona
LIRA-Lab, U. of Genoa

José António Gaspar
ISR, Instituto Superior Técnicoetienne@tyzx.com

Francesco.Orabona@unige.it

jag@isr.ist.utl.pt

Abstract

form processing operations such as edge, extrema, cornerand feature localization on uncalibrated images, just as wellas on calibrated images.

We consider the problem of estimating the relative orientation of a number of individual photocells -or pixels- that
hold fixed relative positions. The photocells measure the intensity of light traveling on a pencil of lines. We assume thatthe light-field thus sampled is changing, e.g. as the result of
motion of the sensors and use the obtained measurements
to estimate the orientations of the photocells.
We explore an information-theoretic and geometric approach: based on real-world data, we build a non-parametricfunctional relation linking the information distance between
the data streams of two photocells, and the angular separation between the photocells. Then, given data streams
produced by arrays of pixels in similar conditions, we use
the functional relation to estimate the angles between pixels. Finally, we embed the estimated angles in the unit 3D
sphere to obtain the estimated layout of the array.

This paper departs from traditional computer vision on
this point: we do not assume that pixels (u, v) and (u + 1, v)
are neighbors. In fact, we are given pixels indexed by a single index n. Since such a camera is peculiar by the absenceof a-priori known topological information, it is only natural
to call it a discontinuous, or discrete camera.
The problem that we address may seem too general topertain to computer vision. Indeed, digital cameras are unlikely to ever produce discontinuous images1 . Yet we mustalso consider the computation that occur in biological systems, where it is unclear how much of the geometry of
the sensor is known beforehand, and how much is determined by processing visual stimuli. We must also consider
robots equipped with arrays of photocells, and new visualsensors [16] that may have variable geometry.
Our goal here is thus to determine the 3D direction pointed
by each pixel, under the assumption that pixels sample lightthat travels along rays that intersect in a unique center ofprojection. In the impossibility of performing local image
processing, traditional calibration techniques [20, 8] are outof the question.

1. Introduction
When we look around or read a conference paper, weonly perceive the part of the visual stimulus that is relevant
to us. Our human visual system performs an abstraction of
the underlying optical and biological processes, of which
we are mostly unaware.
Computer vision also strives to only “perceive” the relevant part, and performs an abstraction of the underlying
optical, electronic and computational processes. A successful example is camera calibration, that allows to transformdistorted images into canonical perspective (or other) images. These images are better fit for higher-level processing
stages, by making abstraction of lens distortion, raw image
size and imager geometry. In a calibrated image, each pixel(u, v) is mapped to a known 3D direction (x, y, z), and determining that mapping is the task of calibration.
Calibration is typically bootstrapped by using some basic properties of the imaging system. In particular, in all
the cases that we are aware of, the local topology is known:
pixel (u, v) is a neighbor of (u + 1, v); the triplet (u, v),(u + 1, v), (u + 2, v) is approximately aligned. Locally,and for many practical purposes, an uncalibrated image looks
like a calibrated image. In particular, it is possible to per-Less traditional non-parametric methods that assume asmooth image mapping and smooth motion [11, 7] can obviously not be applied either. By using controlled-light stimuli or known calibration, matches could be obtained, allowing to use match-based non-parametric techniques [14].
In [4], a non-central projection sensor samples the lightfield in a discontinuous way: the response at each pixelis the convolution of the lightfield with a possibly multimodal function; this “point-spread function” is only sufficiently estimated to reconstruct images on a fixed plane,
and calibration is done with controlled stimuli. In this study
however, we wish to exclude known calibration objects andother controlled stimuli.
It thus seems that we are left with statistical approaches.
The input data will consist in a stream of values x (i, t),where the integer index i denotes the photocell or pixel, andt, the time. Statistical properties of this stream will be exploited to determine the geometry of the sensor.

0 This work was partially supported by TYZX, Inc, by the Portuguese
FCT POS_C program that includes FEDER funds, and by the EU-project
URUS FP6-EU-IST-045062

1 Except in rare cases such as a SLR camera with a stereo adapter or
split mirror.

1

Figure 1. A discrete camera consists of a number of photocells(pixels) that measure the light traveling along pencil of lines. Moregenerally, one could consider that each photocell is characterized
by a point spread function defined on the 3D sphere.

1.1. Related work
Our approach to calibration is closely related to the workof Pierce and Kuipers [12], who measure the dissimilarity,
or distance, between sensor elements that are not necessarily light sensors. The elements are then embedded in a metric space using metric scaling [9], which also determines thedimension of the space. A relaxation method then improves
this embedding, so that the Euclidean distance between sensor elements better matches the dissimilarity between thesensor inputs. Getting very close to the problem addressedin the present paper, the authors use this method to reconstitute the geometry of a rectangular array of visual sensors
that scans a fronto-parallel image.
Going further, Olsson et al. [10] use the information metric of [1] as a more appropriate method to measure the distance between visual or other sensor elements. They also
show how visual sensors -the pixels of the camera of a mobile robot- can be mapped to a plane, either using the methodof [12], or their own, that embeds sensor elements specifically in a square grid.
The works of Olsson et al. and of Pierce and Kuipers
are very interesting to computer vision researchers, but they
only give a glimpse of what sensor input encoding could do
for computer vision. In particular, the geometry of the embedding space is either abstract or fixed to a grid, and lacks
a general connection to the geometry of the sensor. Filling
this gap is one of the motivations for the present work.
Since the present work exploits statistical properties ofthe light-field of the world surrounding a light sensor, thisarticle is also related to research on the statistical propertiesof real-world images. In that area, a model of image formation is used, but images, rather than sequences, are studied.
That research has put in evidence fundamental properties,
in terms of local, global and spectral statistics, of real-worldimages, and found ways to exploit these properties for computer vision tasks, such as classification [19], image restoration [5] and 3D inference [13].

Although these results are of great interest, they are notdirectly applicable in our case, first because we lack images
and because that research does not use temporal informationthat is crucial to our work.
Moreover, these statistics are about images formed on aplanar image plane, which is hindrance in our case: first, we
do not want to exclude the case of visual sensor elementsthat are separated by more than 180 degrees, such as theincreasingly popular omnidirectional cameras. Also, the local statistical properties of perspective images depend of theorientation of the image plane with respect to the scene, except in special constrained cases such as the fronto-parallel“leaf world” of Wu et al. [21]. Defining images on theunit sphere thus appears as a natural way to render image
statistics independent of the sensor orientation, at least withproper assumptions on the surrounding world and/or the
motion of the sensor.

1.2. Proposed approach
In the present work, we assume that the statistical properties of the data streams produced by pairs of sensor elements depends only the angular separation between the elements. This assumption, when one considers the images
as observations of a random field defined on the sphere,is equivalent [15] to assuming that the random field is homogeneous - we could say “anisotropic.” This assumption
clearly does not hold in an anisotropic world, unless the orientation of the sensor is uniformly distributed amongst allunitary transformations of the sphere. In terms of computer
vision and robotics, our assumption amounts to saying thatthe sensor is randomly oriented, so that each photocell isjust as likely to sample the light-field in any direction.
The great practical utility of this assumption is that, asa consequence, the statistical properties of a pair of data
streams generated by two photocells depend only on the angle separating them. In this situation, one can envision estimating the angular separation from a measure of dissimilarity (information distance, Sec. 2.1) between the streams,
and this is precisely what we do (Sec. 2.5). In order to estimate angles from dissimilarity measures, we empiricallyobserve (Sec. 2) the relation between these two quantities.
We may then (Sec. 3.1) embed these angular estimates using
simple techniques from distance geometry [2]. The wholeprocess is outlined in Algorithm 1.
Finally, Section 4 presents some conclusions and a long
list of questions to be addressed by future research.

2. The relation between angular distance andinformation distanceIn this section, we recall the definition of information
distance [1]. We then study how the information distance
between two pixel signals is related to the angle formed by
the pixels, assuming a discrete camera moving indoors.

Algorithm 1 Procedure for estimating the geometry of adiscrete camera.
Input: Signals produced by N rigidly connected light sensors at T time instants: x (i, t) , 1 ≤ i ≤ N , 1 ≤t ≤ T . The light sensors are rigidly connected andpoint towards unknown 3D directions X1 , . . . , XN ,Xi ∈ R3 , X ⊤ X = 1.
Output: Estimates of the directions X1 , . . . , XN .
Algorithm:
1. Estimate the information distance dij between
the temporal signals x (i, .) and x (j, .), for each1 ≤ i, j ≤ N .
2. Estimate the angular separations θij
cos Xi⊤ Xj .

=3. Embed the angular separation in the 3D sphere:
find X1 , . . . , XN s.t. Xi⊤ Xi = 1, Xi⊤ Xj =cos (θij ), 1 ≤ i, j ≤ N .

2.1. Information distance
Given two random variables x and y (in our case, thevalues produced by individual pixels of a discrete camera)
taking values in a discrete set {1, . . . , Q}, the normalized
information distance between x and y is [1]:d (x, y) = 2 − (H (y) + H (x)) /H (x, y) ,(1)

where H (x, y) is the Shannon entropy of the paired randomvariable (x, y), and H (x) and H (y) are the entropies of x
and y, respectively. It is easy to show that Eq. (1) defines
a distance over random variables and that it takes values in[0, 1].2
We now explain how we study the relation between information distance and angular separation. For this purpose,
we consider a discrete camera with pixels separated by angles ranging from 0.5 to 180 degrees.

2.2. Image sensor
It is convenient to take a discrete camera consisting of
pixels located on a plane, separated by geometrically increasing angles, spanning half a great circle (180◦) of the
3D sphere (Fig. 2, top). The smallest gap is half a degree.
We simulate a discrete camera with known Euclidean
geometry by sampling a calibrated panoramic image withunique projection center at fixed locations. Images are acquired by a VStone catadiopric camera consisting of a perspective camera fitted to a hyperbolic mirror. This system
2 The normalized distance is more convenient than the information dis-

tance d (x, y) = 2H (x, y) − H (y) − H (x), which is bounded bylog2 Q.

is modeled as single projection center camera [6] with a360◦ × 210◦ field of view with a ∼ 45◦ blind spot at thesouth pole. The mirror occupies a 453 × 453 pixel regionof the image. Image values at non-integer pixel locations
are obtained by bilinear interpolation. The angular separation between neighboring pixels in the panoramic is usuallyslightly smaller than 0.5◦ , so that, in the “tighter” part ofthe discrete camera layout, there exists a slight linear dependence between the values of consecutive pixels. Also,
some mild vignetting occurs3 , but, except for these minor
inconveniences, simulating a discrete camera by an omnidirectional camera presents many advantages: no other specialized hardware is needed and each omnidirectional image
can be used to simulate many discrete camera “images”, asin Fig. 2, bottom. With respect to perspective cameras, theavailable field of view allows to study very-wide-angle discrete cameras.

2.3. Data acquisition
The camera is hand-held and undergoes “random” general rotation and translation, according to the author´s whim,
while remaining near the middle of the room, at 1.0 to 1.8
meters from the ground. We acquired three sequences consecutively, in very similar conditions and joined them in asingle sequence totaling 1359 images, i.e. approximately 5
minutes of video at ˜4.5 frames per second.
To simulate the discrete camera, we randomly choose anorientation (i.e. half a great circle) such that all pixels of thediscrete camera fall in the field of view of the panoramic
camera. Figure 2 shows two such choices of orientations.
For each choice of orientation, we produce a sequence ofsamples x (i, t), 1 ≤ i ≤ 31, 1 ≤ t ≤ 1359, where eachx (i, t) ∈ {0, . . . , 255}. Choosing 100 different orientations, we obtain as 100 discrete sensors and 100 arrays ofdata xn (i, t), 1 ≤ n ≤ 100.
It is convenient to re-quantize the signals from 255 graylevels to a more parsimonious representation with Q < 255
bins. As in [10], we choose bins that maximize the entropy,
i.e. bins that contain equal numbers of values xn (i, t), 1 ≤i ≤ 31, 1 ≤ t ≤ 1359 and 1 ≤ n ≤ 100.
Considering any pair of pixel (indices) 1 ≤ i, j ≤ 31,
the angular separation θi,j is the main object to estimate byour method, and it is known in our test setup. We show inSec. 2.4 how to compute the information distance dn (i, j)between signals xn (i, t) and xn (j, t), 1 ≤ t ≤ 1359 and, asa result, get a dataset of angle-distance pairs (θi,j , dn (i, j)),1 ≤ n ≤ 100, 1 ≤ i, j ≤ 31, which we call D:
D = {(θij , dn (i, j)) | 1 ≤ i, j ≤ 31, 1 ≤ n ≤ 100} . (2)We will then use this dataset in Section 2.5 to build a
3 One may verify that the average and standard-deviation of pixels are

not exactly uniform in the image.

Mean Information Distance vs Angle

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Angular Separation (deg)Figure 2. Top: Geometry of a discrete camera consisting of a planar array of thirty one (31) pixels, spanning 180◦ in the plane. The
first two pixels are separated by 0.5◦ , the separation between consecutive photocells increases geometrically (ratio ≃ 1.14), so thatthe 31st photocell is antipodal with respect to the first.

Bottom: Two instances of the linear discrete camera, inserted in an omnidirectional image. Pixels locations are indicated by small crosses connected by white lines.

functional relation that allows to estimate the angular separation θij from the information distance d (i, j).

2.4. Estimating the information distanceCaution should be taken when estimating the information distance (1) from finite samples x (t), y (t), 1 ≤ t ≤T : it is relatively common knowledge that replacing unknown probabilities px (q) by sample frequencies pˆx (q) =|{t|x (t) = q}| /Q4 in the expressions of the entropy resultsin biased estimates Ĥ (x), with expectancy
P 1
 
n o
Q − 1 1 − q px (q)1
E Ĥ = H −
. (3)
+
O
+
2T
12T 2
T3
4 |.| denotes the set cardinal.

Figure 3. Dependency of the information distance d (i, j) on the
angular separation θij . The height of each plotted point is the average of 100 estimates of the information distance between signals
x (i, 1 ≤ t ≤ 1359) and x (j, 1 ≤ t ≤ 1359). The abscissa is theangular separations between pixels i and j of the discrete camera.
Each curve corresponds to a different number Q of quantizationbins. The quantization bins are all taken to have weight 1/Q.

Top: Information distances computed with the first-orderbias correction term of Eq. (3) removed. Bottom: Distancescomputed without correction.
This bias in turn causes a bias in the information distanceestimates in Eq. (1). While it is easy to correct for the firstbias term, (Q − 1) /2T , correcting for the other terms ismore delicate [17]. In the present work, we only correct thefirst bias term and we show below the benefits of doing so.
It should also be noted that the second term in Eq. (3) has
a variance that increases with the number Q of quantizationbins, so that it is also important to choose Q appropriately.
Figure 3 shows the average information distance Eq. (1)between the outputs of sensors with known angular separation. Each plot holds four curves, corresponding to Q = 2,
4, 16 and 64. The top plot uses bias correction, whereas thebottom plot does not.
The curves in the top plot are better grouped, showing
that, when bias reduction is applied, the information distance estimates depend less on Q. This is important if one
is interested in estimating properties of the information distance that depend on angular separation and on the ambient
light field, rather than on Q and T .

Mean Information Distance vs Angle
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Angular Separation (deg)Figure 4. When the signal sample is longer as here, the information distance estimates are better behaved: here, a signal of length
T = 135900 frames is obtained by concatenating the signals usedin the previous experiment, and the information distance is computed between pairs of signals. Here, the information distance iscomputed without bias reduction. The values obtained with bias
reduction are very similar.

Another way to reduce the importance of Q, when estimating information distance values, is to use longer signals.
The concatenation of the 100 signal arrays, where all sensors have the same geometry, can be seen as a pseudo-signalthat is 100 times longer, corresponding to approximately 90
minutes of data.
Figure 4 shows that, for long enough signals, the distanceestimates are much better behaved, as they vary less on Q.
However, one also sees that the estimates obtained for large
Q require still longer signals. For angles smaller than 20◦ ,the number of quantization bins, Q still influences to a large
extent the distance estimates.
The conclusion of these experiments is that adding thefirst correction term in Eq. (3) to entropy estimates in Eq. (1)is justified and beneficial, especially when only short sequences are available. One should note that correcting forbias in the entropy estimates is certainly less important in
the situation of Olsson et al. [10], where the distance measurements are embedded directly in an abstract space. Obtaining accurate information distance estimates is importantin our approach because our goal is to map information distance to angles. In the experiments below, we apply thebias-correction term to entropy estimates, and use Q = 4.

2.5. Estimating angular separation from information distance
Figures 3 and 4 also show one of the major issues thatappear when one wishes to estimate the angular separation between pixels from the information distance estimatedfrom their signals: for large angular separation, the information distance reaches a value close to one (corresponding
to nearly independent signals) and remains nearly constant.
In this section, we study the statistical properties of angle-

distance pairs, with the goal of estimating the angle fromthe information distance.
In the absence of a model linking the information distance to the angular separation, we take a nonparametricapproach. We estimate the probability density function of
the angular separation knowing the information distance,
P (θ | d) by applying standard Bayesian inference methods
to our dataset of angle-distance pairs D of Eq. (2).
We cover the space of angles [0, π] with 35 disjoint intervals I1 , . . . , I35 defined by angles5 0 < θ1 < . . . < θ34 <π, and similarly cover the space of information distances[0, 1] with 50 intervals J1 , . . . , J50 , defined by uniformly
spaced values 0.2 = d1 < . . . < d50 = 1. We then callC (p, q) the number of instances of angle-distance pairs ineach rectangle: for any 1 ≤ p ≤ 35 and 1 ≤ q ≤ 50:
C (p, q) = |{(n, i, j) | θij ∈ Ip , dn (i, j) ∈ Jq }| .

(4)

The expected value of C (p, q) is approximately
E (C (p, q)) ≃ P (d ∈ Jq | θ ∈ Ip ) Cθ (p) ,P
where Cθ (p) = q C (p, q) is the count of data points corresponding to θij ∈ Ip , and is known, since it is defined by
the geometry of our probe (shown in Fig. 2, top). Knowing
C (p, q) thus provides an estimate of P (d ∈ Jq | θ ∈ Ip ).
All that is now needed, to get an estimate of P (θ ∈ Ip |d ∈ Jq ), is a prior for P (θ), which would be chosen in accordance to the knowledge about the angles that will be estimated. In our case, we simply take P (θ ∈ Ip ) ∝ Cθ (p),which is appropriate to estimate angles between pixels of asensor similar to that of Fig. 2. We thus use the estimate
P (θ ∈ Ip | d ∈ Jq ) ≃C (p, q),
Cd (q)(5)

P
where Cd (q) = p C (p, q).
Figure 5 shows the estimates of P (θ ∈ Ip | d ∈ Jq ) graphically, as a graylevel image. The black curve shows the expected value of this conditional p.d.f., which we use as theestimate of θ knowing d.
In practice, for an information distance of (di−1 + di ) /2,
the estimated angle is taken to beθ̂ ((dq−1 + dq ) /2) = E (θ | d ∈ Jq ) ,and the estimated angle at intermediate values is obtainedby affine interpolation from the nearest values.
Experimental validation
We now assess the quality of this estimate, by applying it toa set of information distances and comparing the estimatedand true angles. In this experiment, the signals are obtainedin the same conditions -same image sequence, same sensor
shape, but different sensor orientations- as the dataset D.
5 Again, we choose angles with geometrically increasing spacing.

Angle knowing Information Distance1subsequently for estimating the whole sensor geometry.

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3.1. Embedding points in the sphere

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We now consider the problem:

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Problem 1) Spherical embedding problem: Given angle estimates θij , 1 ≤ i, j ≤ N , find points Xi on the unit
sphere, separated by angles approximately equal toθij , i.e. Xi⊤ Xj ≃ cos θij , for all i, j.

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Figure 5. Probability density of the angular separation, when theinformation distance is known. Each column of pixels represents
a conditional pdf P (θ ∈ Ip | d ∈ Jq ) of the angle θ, knowing theinformation distance d. The curve superposed to the image is the
expected value of the angle knowing the distance.
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Problem 2) Euclidean embedding problem: Given distanceestimates Dij , 1 ≤ i, j ≤ N , find points Yi in a metric vector space, such that, for all i, j, kYi − Yj k ≃DijIndeed, by defining
an extra point Y0 = (0, 0, 0), and disp
tances Dij = 2 − 2 cos θij , the mapping of the first problem to the second is immediate. Moreover, Schoenberg´sclassical algorithm for solving Problem 2, when applied toProblem 1, simplifies to the rank-3 approximation of thematrix C with terms Cij = cos θij . In order to solve Problem 1, we thus use the algorithm:
1. Compute the matrix C with terms Cij = cos θij , 1 ≤i, j ≤ N .

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2. Compute, using the SVD decomposition, the rank-3
approximation C̃ = U U ⊤ of C, where U is N × 3.

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3. Define Xi = (Ui1 , Ui2 , Ui3 ) / k(Ui1 , Ui2 , Ui3 )k.

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Figure 6. Statistical characterization of angles estimated from the(estimated) information distance between the signals produced by
the pixels of a discrete camera. The angles are estimated by linear
interpolation of the values of the non-parametric model describedabove. The first and third quartile define the top of each box. Theline in each box is located at the median estimated angle, and thewhiskers indicate the extrema.

Figure 6 shows that the estimated angles are fairly accurate for angular separations smaller than 5◦ , but degrades
sharply for greater values: around 10◦ , a positive bias andan increase in variability appears; starting at 30◦ , estimates“saturate” at the highest value in the non-parametric model.

3. Calibrating a discrete camera
Having seen how to estimate the angle between two pixels from the information distance between their respective
data streams, we now show how useful this method is for
estimating angles coming from different sensor shapes, andOne should note that this very simple algorithm is not optimal in many ways. In particular, it does not take into account that the error in the angles θij is greater in some cases
than in others.

3.2. Sensor calibration
We now evaluate the results of this very simple algorithm on data produced by the angle-estimating method ofSec. 2.5. For this purpose, we produce sequences of pixel
signals in the same conditions as in Sec. 2.3, except that thesensor shape is different. The information distance between
pixels is then estimated from these signals, the angular separation between the pixels is estimated, and the embedding
method of Sec. 3.1 is applied to these angular estimates.
Figure 7 shows the results of the reconstruction, when
the pixels lie on a triangular grid of the image plane, scaled
so that the angular diameter of the whole discrete camera is5◦ (left), 10◦ (middle) or 20◦ (right). The lines connecting
photocells have no purpose other than illustration.
These results are typical results of what researchers reproducing our method may encounter. It is easy to see that

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((Sensor diameter: 20 )Figure 7. True and estimated pixel layouts of a discrete camera. To estimate the layout, each pixel was sampled 1359 times. The information
distance between pairs of pixels was estimated from these values, and converted to angular estimates using our non-parametric model. The
angular distances were then embedded in the sphere. For visualization, they are aligned by the usual procrustes method, mapped to theplane by projective mapping with unit focal length, and line segments indicate the original pixel neighborhood relations.

for small-diameter discrete cameras, the estimated shape ispretty well determined by the information distance between
the pixel streams, whereas, when the angular diameter increases, the noise in the estimated angles gradually overwhelms the embedding algorithm.
Figure 8 shows the results of the reconstruction, when
the pixels lie on a 10 × 10 square grid of the image plane,scaled so that the angular diameter of the whole discrete
camera is 5◦ (left) or 10◦ (right). This last grid is less goodthan typical, the usual outcome having an aspect closer tothat of Fig. 7 left and middle.
These preliminary results -we have not performed extensive benchmarking of the complete calibration methodpoint towards the necessity of a more performant sphericalembedding procedure, in order to better estimate the geometry of wide-angle nonparametric cameras. It is likely thatan iterative refinement procedure would improve the final
results. In particular, an embedding method is needed thatcan cope with unequal errors in the input. Such methods exist [18, 3] for Euclidean embedding, and could be adapted
to the case of spherical embedding.

4. Discussion
We have presented the proof-of-concept for the estimation of the geometry of a discrete array of light-sensitiveelements. The presented method is based on the exploitation of properties of the ambient light-field in a man-madeenvironment. We raised some of the issues that are boundto be met by similar methods, in particular:
1. considering images defined on the sphere rather thanon the plane,
2. estimating information distance from finite samples,

3. building a functional relationship that links information distance to angular separation, and4. embedding angular estimates in a sphere.
On each of these points, some questions remain open forfurther work: first, what is the effect of individual photocells not having the same sensitivity, as is likely to happen
in practice? what are the effects of simulating a sensor withan omnidirectional camera?
Concerning the estimation of information distance, it may
be useful to estimate directly the bias in information distance estimates, rather than simply applying bias-correction
to entropy estimates. In addition, our method assumes thatconsecutive samples are i.i.d.; it would probably be beneficial to acknowledge the temporal correlation of the signals.
Concerning image statistics, what are useful statistics ofimages defined on the sphere? Modeling images as observations of homogeneous random processes, while the world is
not isotropic, is only correct if the orientation of the sensor
follows a uniform distribution. A more appropriate model
would probably acknowledge the existence of a dominantvertical (and perhaps forward) direction.
Concerning embedding, what is a practical way of embedding in the sphere angular estimates with greatly varying noise characteristics? Will one be able to adapt existing
methods, or will new methods be necessary?
The empirical observation of the strong relation between
angular separation and information distance is crucial to thepresented algorithm. Can one build a physical basis forthis relation, or at least link it to empirically known spectral properties of real-world images?
Can the present approach be generalized to a non-central
camera? This would require determining statistical properties of the lightfield as a five-dimensional random field, andEstimated and True Camera Shape
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0.04

(Sensor diameter: 5◦ )Estimated and True Camera Shape
0.1

0.05

0

−0.05

−0.1
True positions
Estimated positions
−0.15
−0.1

−0.05

0

0.05

0.1

0.15

(Sensor diameter: 10◦ )Figure 8. True and estimated pixel layouts of a discrete camera
consisting of photocells lying on a rectangular grid. Apart fromsensor shape, the experimental conditions are the same as in Fig. 7.

solving an altogether different embedding problem.
These questions are so basic that one can only guess at
the amount of work required to produce a mature method
for calibrating discrete cameras.

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