ESTIMATING CAMERA ROTATION PARAMETERS FROM A

BLURRED IMAGE

Giacomo Boracchi

a,b

, Vincenzo Caglioti

and Alberto Danese

Dipartimento di Elettronica e Informazione

Politecnico di Milano, Via Ponzio, 34/5 20133 Milano, Italy

Tampere International Center for Signal Processing,

∗

Tampere University of Technology, P.O. Box 553, 33101 Tampere, Finland

Keywords:

Blurred Image Analysis, Camera Rotation, Camera Ego-motion, Blur Estimation, Space Variant Blur.

Abstract:

A fast rotation of the camera during the image acquisition results in a blurred image, which typically shows

curved smears. We propose a novel algorithm for estimating both the camera rotation axis and the camera

angular speed from a single blurred image. The algorithm is based on local analysis of the blur smears.

Contrary to the existing methods, we treat the more general case where the rotation axis can be not orthogonal

to the image plane, taking into account the perspective effects that in such case affect the smears.

The algorithm is validated in experiments with synthetic and real blurred images, providing accurate estimates.

1 INTRODUCTION

This paper concerns images corrupted by blur due to

a camera rotation or to a rotating object in the scene.

When the camera or the captured object are purely ro-

tating, the image blur is determined by only two fac-

tors: the camera rotation axis a and its angular speed

ω. We present a novel algorithm for estimating both

a and ω, by analyzing the blur in a single image.

When the camera rotation axis and the angular

speed are known, the rotationally blurred image can

be restored by image coordinates transformation and

blur inversion. In broad terms, the image is trans-

formed from Cartesian to polar coordinates so that the

blur becomes space invariant and can be inverted us-

ing a deconvolution based algorithm. Estimating cor-

rectly the camera rotation axis and its angular speed

is therefore crucial for restoring these images as small

errors in the polar transformation are ampliﬁed by

blur inversion.

On the other hand, estimating a and ω from a sin-

gle image can be also of interest for robotic applica-

tion as these describe the camera ego-motion.

Figure 1 shows an image acquired during camera

rotation. The shapes of the blur smears show that the

∗

This work was partially supported by CIMO, the

Finnish Centre for International Mobility (fellowship num-

ber TM-07-4952)

Figure 1: A rotationally blurred image.

blur is space variant. Typically, these are assumed

arcs of circumferences, all having the same center.

However, this approach neglects the perspective ef-

fects that occur when the rotation axis is not orthog-

onal to the image plane. The proposed algorithm es-

timates the camera rotation axis in the most general

case when it is not necessarily orthogonal to the im-

age plane. To the best of our knowledge this issue has

never been correctly addressed before.

Besides the early works concerning rotational blur

modeling and restoration, Hong and Zhang (Hong and

Zhang, 2003) addressed the issue of both rotational

blur estimation and removal. Their method is based

on an image segmentation along circumferences to

389

Boracchi G., Caglioti V. and Danese A. (2008).

ESTIMATING CAMERA ROTATION PARAMETERS FROM A BLURRED IMAGE.

In Proceedings of the Third International Conference on Computer Vision Theory and Applications, pages 389-395

DOI: 10.5220/0001085403890395

 SciTePress

C ≡P

Figure 2: (a) Blurred image formation, a ⊥ π. Blurring

paths are circumferences. (b) Blurred image formation, a

is not orthogonal to image plane π and V ∈ a. Blurring

paths are conic sections on the image plane, while they are

circular when projected on an ideal spherical sensor and on

a plane perpendicular to the rotation axis.

estimate the blur and restore the image separately in

these subsets. Recently, an algorithm for estimating

the camera rotation from a single blurred image has

been proposed (Klein and Drummond, 2005). The al-

gorithm is meant as a visual gyroscope and it is tar-

geted to an efﬁcient implementation. In particular,

this algorithm requires edges in the scene.

All the existing methods, concerning both image

restoration and blur estimation, assume that the blur

smears are arcs of circumferences having the same

center. Therefore these methods are accurate only on

images where the rotation axis is orthogonal to the

image plane.

We present an algorithm for estimating the camera

rotation axis and angular speed in the most general

case, where the rotation axis is not orthogonal to the

image plane. The proposed algorithm is mostly tar-

geted to accuracy rather than efﬁciency and does not

require the presence of edges in the scene.

2 PROBLEM FORMULATION

We propose an algorithm for estimating the camera

rotation axis a and its angular speed ω by analyzing a

single blurred image acquired during camera rotation.

We assume that the camera is calibrated, the rotation

axis a passes through its viewpoint V, i.e. V ∈ a, and

w is constant. Figure 2.a illustrates the situation typi-

cally considered in literature, where the rotation axis

is perpendicular to image plane π. The principal point

P and the intersection between the image plane and

the rotation axis C = π∩a then coincide. Analogous

blur is obtained when a ⊥π and V /∈a, but the capture

scene is planar and parallel to π (Ribaric et al., 2000).

In this work we consider the most general situa-

tion, illustrated in Figure 2.b, where a is not orthogo-

nal to π and the camera viewpoint V ∈ a.

2.1 Image Blur

A blurring path is deﬁned as the set of image pixels

that a viewing ray intersects during a camera rotation

of 2π around axis a. Figure 2 illustrates examples

of blurring paths. In rotationally blurred images ev-

ery pixel is merged with neighboring pixels from the

same blurring path, see Figure 1. The blur is there-

fore space variant and can not be represented as a lin-

ear shift invariant system. We therefore model the ro-

tational blur by an operator K on the original image

y (Bertero and Boccacci, 1998) so that the observed

(blurred and noisy) image z is

z(x) = K





(x) + η(x) x = (x

) ∈ X , (1)

x being the coordinates in the discrete image domain

X and η ∼ N(0,σ

) is white Gaussian noise. The blur

operator K can be written as





(x) =

k(x,s)y(s)ds. (2)

where k(x, •) is a kernel

k(x,•) = A

θ,e

(•), (3)

and A

θ,e

corresponds to the point spread function

(PSF) at x. A

θ,e

being an arc the blurring path at x, i.e.

is an arc of conic section having tangent line with di-

rection θ and arc length e. The parameters θ,e varies

between image pixels according to the rotation axis a.

Other blurring effects, such as the out of focus blur,

lenses aberrations and camera shake, are not consid-

ered.

3 THE ALGORITHM

The proposed algorithm is based on three steps: in the

ﬁrst step the lines tangent to blurring paths at some

image pixels are estimated (Section 3.1). In the sec-

ond step, these lines are used in a voting procedure for

estimating the rotation axis a (Sections 3.2 and 3.3).

The third step consists of the angular speed ω estima-

tion (Section 3.4).

3.1 Blur Tangent Direction Estimation

Image blur is analyzed within N image regions taken

around selected pixels {x

}

i=1,..,N

. There are no

particular requirements in selecting {x

}

i=1,..,N

, but

avoiding smooth areas while covering uniformly the

image. Therefore we take the local maxima of Harris

corner measure (Harris and Stephens, 1988), or when-

ever these do not cover uniformly the image, we take

}

i=1,..,N

on a regular grid.

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

390

Figure 3: Rotationally blurred image and plots of direc-

tional derivatives energy in four regions.

Blur is analyzed using the approach proposed by

Yitzhaky et al (Yitzhaky and Kopeika, 1996) for esti-

mating the direction of blur “smears” by means of di-

rectional derivative ﬁlters. This method, proposed for

space invariant blurs with PSF having vectorial sup-

port, assumes the image isotropic. The blur direction

θ is estimated as the direction of the derivative ﬁlter

having minimum energy response

θ = arg min

θ∈[0,π]



||(d

⊛ z)||



, (4)

where ⊛ denotes the convolution and

||(d

⊛ z)||

∑

x∈X

|(d

⊛ z)(x)| the ℓ

norm.

Equation (4) is motivated by the fact that the blur

removes all the details and attenuates edges of y along

blur direction. Therefore the blur direction can be

determined by the directional derivative ﬁlter having

minimum energy. This method can not be directly

applied to rotationally blurred images, as the blur is

not space invariant because in every pixel the circum-

ference approximating the blurring path (i.e the PSF)

changes.

At x

, the center of each regionU

, we estimate the

direction θ

of the line l

tangent to the blurring path

in x

, as

= arg min

θ∈[0,π]

∑

∈U



⊛ z)(x

)



. (5)

being w a window function rotationally symmetric

with respect to the center. By using Gaussian dis-

tributed weights, it is possible to reduce the inﬂuence

of pixels in Equation (5) with the distance from x

. We

adopted the 3 tap derivative ﬁlters presented in (Farid

and Simoncelli, 2004) for blur analysis in Equation

(5). These ﬁlters have been selected as they provide

good accuracy and as they are separable. Experimen-

tally the ℓ

norm gave better results than the ℓ

norm.

Figures 3 shows

∑

∈U



⊛ z)(x

)



as a

function of θ ∈[0,π] within regions of the blurred im-

age containing isotropic textures or edges. Regions

Figure 4: Weight function used for the votes spread.

containing edges, as pointed out in (Klein and Drum-

mond, 2005), can be exploited for estimating the cam-

era rotation: in z only edges tangent to the blurring

paths are preserved. Formula (5) gives accurate re-

sults also when U

contains a blurred edge, as the di-

rection minimizing the derivatives energy is the edge

direction, i.e the blur tangent direction.

3.2 Voting Procedure for Circular

Blurring Paths

When the camera optical axis and the rotation axis a

coincide, the blurring paths are circumferences cen-

tered in C = π ∩a, see Figure 2.a. Circular blurring

paths are obtained also when a is parallel to the op-

tical axis and the scene is planar and parallel to the

image plane (Ribaric et al., 2000; Hong and Zhang,

2003). In this case C can be determined by a Gener-

alized Hough Transform (Ballard, 1987).

The Generalized Hough Transform is a procedure

for computing robust solution to a problem, given

some input data. The procedure is developed by

means of a parameters space P, which is the set of

all the possible solutions. A vote is assigned to ev-

ery parameter that satisfy a datum and then summed

to the votes coming from the other data. After having

considered all the data, the parameter that receivedthe

highest vote is taken as a solution.

In our case P is a discrete grid of all the pos-

sible location for C ∈ π and data are the pairs

,θ

) i = 1, ..,N. Note that C could be outside of

the image grid X. Every data (x

,θ

) identiﬁes a line

, the line tangent to the blurring path at x

. The set

of all the possible rotation centers C, given the line l

is the line perpendicular to l

and passing through x

It is worth to take into account the root mean

square error of each θ

i.e.

E[(θ

−θ

∗

)

] (6)

where θ

∗

represents the true tangent blur direction

at x

and E[•] the mathematical expectation. Since

ESTIMATING CAMERA ROTATION PARAMETERS FROM A BLURRED IMAGE

391

we can not directly compute σ

, we approximate it

with an indirect measurement: for example consid-

ering the amplitude of the area near θ

in the energy

function minimized in (5) or considering σ

propor-

tional to σ

(1). Noise standard deviation is estimated

using (Donoho and Johnstone, 1994). Given a datum

,θ

), we assign a full vote to all the exact solutions

and we spread smaller votes to the neighboring pa-

rameters, according to the errors in θ

Let now p = (p

, p

) represent a coordinate sys-

tem in the parameters space and assume θ

= 0 and

= p

= (0, 0). Let now model the vote spread

assuming that along the line p

= 1 the errors are

distributed as σ

√

2π ·N(0,σ

). We model the vote

spread so that along line p

= k, the votes are still

Gaussian distributed with a full vote at the exact so-

lution (k,0) and for neighboring parameters the votes

depend only on the angular distance from θ

, see Fig-

ure 4. Therefore the following weight function is used

for distributing the votes in the parameter space (when

= p

= (0,0) and θ

= 0),

, p

) = e

−

1+p

, (7)

The votes weight function v

, associated to other data

,θ

), correspond to Equation (7) opportunely ro-

tated and translated. When all pairs (x

,θ

) i = 1, .., N

have been considered, the parameter that received the

highest vote is taken as the solution, i.e.

p = argmax

p∈P

V (p), being V (p) =

∑

i=1

(p). (8)

The coordinates of C = π∩a are determined from

3.3 Conic Section Blurring Paths

Assuming circular blurring paths reduces the com-

plexity load but gives inaccurate solutions whenever

a is not perpendicular to π. We present an algorithm

for estimating a and ω when V ∈ a and a is in a gen-

eral position w.r.t. π. In particular, if we call π

plane perpendicular to a, π

is obtained by two rota-

tions of α and β from π. We do not consider V /∈a as

in this case the blur would depend on the scene depth.

Votes in the parameters space show at a glance

what happens assuming circular blurring paths when

a is not orthogonal to π. Figure 5.a shows a blurred

image produced when the plane orthogonal to a forms

angles α

∗

= 45

◦

and β

∗

= 0

◦

with π. If we treat the

blurring paths as circumferences, the votes in the pa-

rameters space do not point out a clear solution, as

shown in Figure 5.b and 5.c.

Directions θ

obtained from (5) represent the blur-

ring paths tangent direction, even when the blurring

paths are conic sections. But the blurring paths them-

selves are not circumferences, thus lines perpendic-

ular to these tangent lines do not cross at the same

point.

From basic 3D geometry considerations, and as

pointed out in (Klein and Drummond, 2005), it fol-

lows that the blurring paths are circumferences on

an ideal spherical sensor S, Figure 2.b. Then, if we

project the image from π on S surface, the blurring

paths become circumferences. Each of these circum-

ferences belongs to a plane and all these planes have

the same normal: the rotation axis a. Let now con-

sider one of these planes, π

, tangent to the sphere.

The projections of the blurring paths on π

are cir-

cumferences, Figure 2.b.

The plane π and the plane π

are related by a pro-

jective transformation determined by two parameters,

namely (α,β), the angles between the two planes. De-

ﬁne the map M

α,β

: π 7→ π

α,β

as the projection from V

between π and π

α,β

, which is the plane tangent to S,

forming angles (α,β) with π (Rothwell et al., 1992).

We search for (α,β) that project the blurring paths

into circumferences, by modifying the voting proce-

dure of Section 3.2.

There is no need to transform the whole image

with M

α,β

as each l

, the line tangent to the blur-

ring path at x

, can be directly mapped via M

α,β

Let v

α,β

be the weight function (7) associated to data

,θ

) i = 1,..,N mapped via M

α,β

. The parameters

pair identifying the plane π

is estimated as

(

α,

β) = argmax

α,β

(

α,β

), (9)

being

α,β

= argmax

p∈P

α,β

(p), V

α,β

(p) =

∑

i=1

α,β

(p).

(10)

Figure 5.d and 5.e represent the votes in case the

data have been transformed according to the correctly

estimated parameters

α = 45

◦

β = 0

◦

. These votes

are much more concentrated than votes in Figure 5.b

and 5.c.

Once

α and

β have been estimated, the cam-

era rotation axis a is determined and it is possible

to map the image z to M

α,

(z). As said before, in

(z) the blurring paths are circumferences cen-

tered at M

α,

∩a and it is therefore possible

to transform M

(z) in polar coordinates for estimat-

ing the angular speed.

3.4 Angular Speed Estimation

Once C has been determined, it is possible to trans-

form M

α,

(z) (the image projected on π

) on a polar

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

392

Figure 5: (a) Rotationally blurred image with rotation axis α

∗

= 45

◦

,β

∗

= 0

◦

. (b) Votes assuming circular blurring paths, (c)

votes contours. (d) Votes obtained transforming the data with

α = 45

◦

β = 0

◦

, (e) votes contours. The maximum vote in (d)

is 33% higher than the maximum vote in (b). This is due to the fact that transforming the data with M

45,0

the blurring paths

become circumferences having the same center.

Figure 6: Still and rotationally blurred synthetic images.

First row, left to right: Boat (α

∗

= 20

◦

,β

∗

= 0

◦

), Man-

drill (α

∗

= −20

◦

,β

∗

= 20

◦

) and Lena (α

∗

= 0

◦

,β

∗

= −20

◦

Second row: Boat, Mandrill and Lena, rotationally blurred

with an angular speed of 6, 8 and 6 deg/s, respectively, as-

suming 1 second of exposure time. Intersection between

image plane and rotation axis is marked with a red circle.

lattice (ρ,θ) w.r.t to M

α,

On the polar lattice, the blur is space invariant with the

PSF directed along lines ρ = const. We estimate the

PSF extent using the method proposed by Yitzhaky

(Yitzhaky and Kopeika, 1996) as this can be applied

to a restricted image area, avoiding lines which con-

tain several pixels of padding introduced by the polar

transformation. The PSF extent, opportunely scaled

by the factor due to the polar lattice resolution, di-

vided by the exposure time gives the camera angular

speed.

4 EXPERIMENTS

The algorithm has been validated both on synthetic

and camera images. Synthetic images of Figure

6 have been generated with a raytracer software

(http://www.povray.org/,) rotating the camera in front

of planar tiles of test images. Blurred images are ob-

Table 1: Boat. Highest votes corresponding to (α,β) in the

parameters space, expressed as a percentage with respect to

the maximum vote.

α β -40 -20 0 20 40

-40 33 52 72 53 48

-20 39 52 83 63 44

0 34 57 83 63 43

20 40 55 100 55 44

40 35 42 62 39 34

Table 2: Reﬁnement around (

α,

β) from Table 1.

α β -10 0 10

10 71 79 80

20 63 100 74

30 57 82 61

tained averaging all the rendered frames, according to

Equation (1). Ten frames (512x512 pixels, grayscale

0-255) are rendered per each rotation degree. The

blurring paths tangent directions are estimated in 121

equally spaced regions having a 10 pixel radius, using

formula (5).

Table 1 shows V

α,β

(

α,β

) (the value of the maxi-

mum vote obtained with (α,β)) as a percentage w.r.t

α,

(

α,

) (the maximum vote obtained with (

α,

β)).

Here (

α,

β) coincides with (α

∗

,β

∗

), the ground truth.

Table 2 shows the results at a second iteration consid-

ering a reﬁnement around (

α,

β)).

Table 3 shows results obtained on synthetic im-

ages of Figure 6. Each of them has been tested

adding white gaussian noise with standard devia-

tion 0, 0.5 and 1 and considering α and β in

A = B = {−40

◦

,−20

◦

,20

◦

,40

◦

}. Algorithm per-

formances are evaluated with ∆(α)=|

α − α

∗

| and

∆(β)=|

β − β

∗

|. ∆(

C) and ∆(

ω) represent the abso-

lute error between the ground truth and the estimated

values of C = π∩a and ω, respectively.

ESTIMATING CAMERA ROTATION PARAMETERS FROM A BLURRED IMAGE

393

Figure 7: Boat, Mandrill and Lena rectiﬁed with the corre-

sponding estimated (

α,

β). Intersection between the image

plane and the rotation axis is marked with a red circle.

The effectiveness of our algorithm is evaluated as

adv =

α,

(

) −V

,β

(

,β

)

,β

(

,β

)

, (11)

being V

,β

(

,β

) the maximum vote obtained

among other parameters (α,β). The higher this ratio,

the better. Finally, ∆(C

0,0

) and ∆(ω

0,0

) are the corre-

sponding errors obtained assuming circular blurring

paths. Results for noisy images represent the average

over ten different noise realizations.

Results reported in Table 3 show that our algo-

rithm can cope with a reasonable amount of noise,

obtaining regularly better results than the circular blur

assumption. This is more evident in the estimation

of the angular speed, which lacks physical meaning

when the rotation axis is not correctly identiﬁed. Fig-

ure 7 shows blurred images of Figure 6 transformed

with the corresponding M

α,

Camera images have been captured rotating a

Canon EOS 400D camera on a tripod, assuring that

a is orthogonal to the ﬂoor. The ground truth α

∗

and

∗

, can be then computed rectifying still images of a

checkerboard on the ﬂoor. Figures 8.a and 10.a show

the downsampled RAW converted in grayscale used

to test our algorithm. The blurring path tangent direc-

tions are estimated on 187 uniformly spaced regions,

having 10 pixel radius.

Tables 4 and 5 show the results of the execution

of two iterations of the algorithm on Figure 8.a. The

solution obtained is

α = −30

◦

and

β = 0

◦

, which

is acceptable as the ground truth, obtained from the

checkerboard, is (−27

◦

). Figure 9 points out the

differences between the blurring paths estimated with

the circular approximation (in red) and the conic sec-

tion paths estimated by our method (in green). As

clearly seen from the detail, the blur is correctly in-

terpeted by the green blurring paths. Figure 10 shows

results on another camera image, having α

∗

= −20

◦

and β

∗

= 0

◦

. After two iterations, the algorithm con-

verges exactly to the correct solution. Figures 8 and

10 show the blurred images and the checkerboard im-

ages rectiﬁed with the estimated (

α,

β).

Figure 8: Blurred camera image. (a) Blurred image

(α

∗

= −27

◦

, β

∗

= 0

◦

), (b) rectiﬁed image with estimated

α = −30

◦

β = 0

◦

, (c) checkerboard with the same cam-

era inclination, (d) checkerboard rectiﬁed with

α = −30

◦

β = 0

◦

Figure 9: Comparison between circular (red) and conic sec-

tion (green) blurring paths on 8.a. Green blurring paths de-

scribe more accurately the image blur.

Figure 10: Blurred camera image. (a) Blurred image

(α

∗

= −20

◦

, β

∗

= 0

◦

), (b) rectiﬁed image with estimated

α = −20

◦

β = 0

◦

, (c) checkerboard with the same cam-

era inclination, (d) checkerboard rectiﬁed with

α = −20

◦

β = 0

◦

5 CONCLUSIONS

We described a novel algorithm for estimating the

camera rotation axis and the angular speed from a sin-

gle blurred image. The algorithm provides accurate

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

394

Table 3: Algorithm performances on synthetic images. When σ

> 0, averages over 10 noise realizations.

Image σ

∆(α) ∆(β) ∆(

C) ∆(

ω) adv(%) ∆(C

0,0

) ∆(ω

0,0

)

Boat 0 0 0 2.20 0.23 20.44 33.06 4.83

Boat 0.5 0 0 5.46 0.24 20.23 21.27 114.55

Boat 1 0 0 8.84 0.19 8.84 19.25 71.98

Mandrill 0 0 0 1.00 0.09 5.66 7.07 0.96

Mandrill 0.5 2 2 1.48 0.11 6.13 4.81 2.85

Mandrill 1 4 4 1.17 0.26 5.25 4.41 2.29

Lena 0 0 0 3.00 0.08 11.01 12.08 0.60

Lena 0.5 0 0 3.88 0.20 14.06 33.64 64.94

Lena 1 0 4 5.23 0.48 6.00 29.43 62.58

Table 4: Camera Image. Highest votes corresponding to

(α,β) in the parameters space, expressed as a percentage

with respect to the maximum vote.

α β -40 -20 0 20 40

-40 63 88 100 88 52

-20 62 77 78 70 54

0 67 74 80 70 62

20 62 81 85 65 63

40 71 73 82 78 77

Table 5: Camera Image. Reﬁnement around (

α,

β) from

Table 4.

α β -10 0 10

-50 82 76 80

-40 81 96 72

-30 85 100 83

estimates also in the most challenging cases, when

the rotation axis is not orthogonal to the image plane.

To the best of the authors’ knowledge, none of the

existing methods handles these cases correctly since

said methods assume circular blurring paths. We have

shown how this assumption produces inaccurate esti-

mates when the rotation axis is not orthogonal to the

image plane, while our algorithm is more accurate.

The algorithm is targeted to accuracy rather than

efﬁciency. Accuracy in the estimation of these pa-

rameters is a primary issue in restoring such images

as the deblurring is typically based on a coordinate

transformation and a deconvolution, which are highly

sensitive to errors.

Ongoing works concern the design of a more

noise-robust method for blur analysis on image re-

gions and the implementation of a faster voting pro-

cedure.

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