Removing Non-Uniform Motion Blur from Images∗
Sunghyun Cho
POSTECH
Pohang, Korea

Yasuyuki Matsushita
Microsoft Research Asia
Beijing, China

Seungyong Lee
POSTECH
Pohang, Korea

sodomau@postech.ac.kr

yasumat@microsoft.com

leesy@postech.ac.kr

Abstract
We propose a method for removing non-uniform motion
blur from multiple blurry images. Traditional methods focus on estimating a single motion blur kernel for the entire
image. In contrast, we aim to restore images blurred by unknown, spatially varying motion blur kernels caused by different relative motions between the camera and the scene.
Our algorithm simultaneously estimates multiple motions,motion blur kernels, and the associated image segments. We
formulate the problem as a regularized energy function andsolve it using an alternating optimization technique. Realworld experiments demonstrate the effectiveness of the proposed method.

1. Introduction
Motion blur caused by a relative motion between a camera and a scene is inevitable due to the nature of a camera
sensor that accumulates incoming light over a certain periodof time. Many computer vision algorithms rely on the assumption that a scene is captured without such motion blur.
However, this assumption generally does not hold unless thescene and the camera are both static. It is therefore important to correctly remove motion blur from images so that thesubsequent algorithms can neglect the effect of motion blur.
Motion deblurring has been studied by many researchers. Most methods solve the problem under an assumption that there is only a single motion blur kernel
for the entire image. However, in real-world cases, photographed images often have spatially-varying motion blurs
due to multiple relative motions caused by moving objects
or depth variations from the camera (Fig. 1).
This paper proposes an algorithm for removing spatiallyvarying motion blurs from images. The removal of
spatially-varying motion blurs involves three different problems: estimation of motions, segmentation into regions ofhomogeneous motions, and estimation of motion blur ker∗ Part of this work was done while the ﬁrst author was visiting Microsoft
Research Asia as an intern.

Image 1

CameraImage 2

Image 1

Camera A Camera B

Image 2

Figure 1. Different motions are introduced to the foreground andbackground objects due to a moving object (left) and a depth difference (right).

nels. Solutions of these problems affect to each other reciprocally. Therefore, we jointly solve these problems by anenergy minimization approach. Our algorithm uses a regularized form of the energy function and minimizes it by analternating optimization technique. From two or more input
images, images are restored by removing spatially-varying
motion blurs.
The primary contributions of our work are twofold. First,
it proposes a new approach to restoring images that arecontaminated by spatially-varying motion blurs. Second,in addition to the restoration, associated motion blur kernels, segmentation, and motions are simultaneously obtained. The proposed method has wider applicability compared with the prior approaches that only assume a single
motion blur kernel.

1.1. Related work
The major difﬁculty of motion deblurring is accurate estimation of motion blur kernels, in other words, motion
point spread functions (PSFs). Most prior approaches extensively focus on a single motion PSF caused by a camera, but not spatially-varying motion PSFs caused by multiple objects’ motions. Yitzhakey et al. propose a method
for motion deblurring by identifying the motion PSF using
an isotropic image assumption [2]. Assuming an 1D directional motion blur in their model, they extract the motion

InputInitial motion
estimation

Estimation

Restoration

OutputBlurry images

Rough estimation ofmultiple motions

Motion estimation, segmentation, andPSF estimation

PSF refinement andImage deblurring

Deblurred images

PSF
Backgroundmask

Motion 2

BG PSF

FG PSFPSF

BG PSF

FG PSF

Foreground
mask

PSF

Motion 1

PSF

Backgroundmask

Foregroundmask

Figure 2. Overview of the proposed method. Given blurry images as input, the algorithm ﬁnds the initial guess for the multiple motion
estimation using Bhat et al.’s method [1]. In the estimation step, accurate motion estimation, segmentation, and motion PSF estimation are
performed. With these estimates, the restoration step deblurs input images with reﬁnement of the motion PSF estimates.

component from the image in the frequency domain using
autocorrelation. Afterwards, they extend their method to account for a high frequency motion blur caused by vibrationthat is observed along with the major 1D directional motion blur [3]. Recently, Jia proposes a deblurring methodthat uses the translucency information of blurry regions between two opaque regions [4]. The information is acquiredby image matting and used to estimate a single motion PSF.
In the multiple image framework, Chen et al. propose amethod for recovering the original image from two consecutively blurred images [5]. The method adopts an energy
minimization technique that computes the best translationof the camera from the two consecutive images. Similarly,
Rav-Acha and Peleg [6] use two blurry images (one is horizontally and the other is vertically blurred) for deblurring
by simultaneously estimating parameters of image displacements using iterative energy minimization. Bascle et al.
propose a method to remove motion blur and generate highresolution images from an image sequence by ﬁnding pixelwise motion [7]. Bar et al. developed a uniﬁed frameworkof segmentation and blind restoration by treating the image segmentation and restoration as a coupled problem [8].
While they use a classical color-based image segmentation
without using the motion blur kernel for the segmentation,
convincing results are obtained with the proposed method.
Yuan et al. use a blurred/noisy image pair to handle uniform
motion blur [9]. They also propose a novel deconvolution
method that reduces ringing artifacts.
Recently, a few works for removing spatially-variant motion blur have been proposed. Levin proposes a method torestore a blurry image that contains one motion blurred object and a sharp background [10]. The algorithm uses thestatistical distributions of image gradient to estimate motion blurs and to locate a blurry foreground object. Bardsley et al. use a phase diversity method to ﬁnd motion PSFs

in which segmentation is performed by splitting an image
into regions that have homogeneous motion PSFs [11].
Apart from these post-processing approaches, a few
hardware approaches have been proposed as well. Ben-Ezra
and Nayar develop a hybrid imaging system [12] that consists of two cameras: one is high-resolution with a long exposure time, the other is low-resolution but can capture multiple frames with a shorter exposure time. Using the lowresolution camera for estimating the camera motion, their
method estimates PSF for the high resolution camera. It is
known that hardware approaches can handle more generalcamera motions than software approaches. However, such
hardware systems are still expensive, and unfortunately they
are not commonly available.

1.2. Overview
Our goal is to restore motion blurred images which contain spatially-varying motion blurs. Figure 2 illustrates theoverview of our method.
Our algorithm takes multiple blurry images of roughly
the same scene as input, e.g., consecutive frames from a
video. To simplify the explanation, we describe our algorithm in the case of two input images although the algorithm
can naturally work with more than two input images. Given
the input images, our method ﬁrst ﬁnds an initial guess ofmultiple different motions that appear in the images. For
this step, we use Bhat et al.’s motion ﬁtting method [1].
Since the accurate motion estimation is performed in thefollowing step, the initial estimation result need not necessarily be precise.
Once the initial estimation of motion parameters is obtained, the ﬁrst stage of the proposed method simultaneously reﬁnes the motion estimates, performs image segmentation into regions of homogeneous motions, and estimatesthe corresponding motion PSFs. The estimated PSFs arefurther reﬁned in the following restoration stage, where we
perform the reﬁnement and image restoration at the sametime, using a method similar to Jin et al.’s work [13].
A few assumptions are used in the proposed method.
First, a piecewise afﬁne motion is assumed between image segments. Second, we assume that a motion PSF
is dominated by a linear component. These assumptions
are commonly used in many prior deblurring methods(e.g., [2], [3], [6], [7]) and reasonably hold with image sequences captured by a video camera.

2. Formulation
Section 2.1 describes the motion blur model used in thispaper. Section 2.2 describes the algorithm of simultaneous
estimation for obtaining multiple relative motions, segmentation, and motion PSFs. Section 2.3 explains the algorithm
for image restoration and motion PSFs reﬁnement.

2.1. Motion blur model
Let I be an image that contains a motion blur and Is
be the corresponding sharp image in which the motion blurdoes not exist. These two images I and Is are related by thefollowing equation ∞2
1√ e−t /2 Is (x + vt)dt, (1)I(x) = kv ∗ Is (x) =2π−∞where v is a 2D motion vector corresponding to the relative
motion between a camera and the scene, and kv is a motion
blur kernel parameterized by the motion vector v.
Since an image is composed of a collection of discrete
pixels, Equation (1) can be expressed in a discrete form;u = Kv us ,(2)

in which the column vectors u and us embed pixels of theimage I and Is , respectively, and Kv is a matrix representation of convolution with the motion blur kernel kv .
In a captured image, multiple different motions andspatially-varying motion blurs may exist. To represent n
different relative motions and motion blurs, we use the subscript i, such as vi and Kvi , where 0 ≤ i < n, to indicate
the i-th motion. We also introduce a segmentation mask
vector mi and its matrix representation Mi , which is a diagonal matrix whose diagonal elements are equivalent to
the elements of mi , to indicate the i-th image region that
corresponds to the i-th relative motion. In the vector mi ,we use ﬂoating-point numbers instead of binary numbers,
and all elements are set in the range [0, 1]. For example,
if mi [l] = 1, the l-th pixel belongs to the i-th motion. Asmi [l] decreases, the l-th pixel is less likely to belong to thei-th motion. Using this notation, the motion blur caused by
the i-th motion can be described as
Mi u = Mi Kvi us .

(3)

Assume that there are two sharp images us0 and us1 ofroughly the same scene. Image segments of these two images are associated by piecewise afﬁne motions asi
Mi us0 = Mi uA
s1 ,

(4)

i
where uAs0 is the vectorized image us0 transformed by the
afﬁne transform Ai , and Mi is a segmentation mask deﬁned in the coordinates of us0 . When these image segments
are blurred by spatially-varying motion PSFs, Equation (4)changes to

Mi u0
i
Mi uA
1

= Mi Kv0i us0 , andAii= Mi KAv1i us1 ,(5)iwhere u0 and u1 are blurry images, and KA
v1i is a matrix
representation of convolution with the i-th blur kernel paAi
rameterized by a vector v1i. v0i represents the i-th motion
blur parameter in image u0 . Likewise, v1i denotes the i-th
motion blur parameter of image u1 . In the following, weAiiuse K0i and KA
1i to denote Kv0i and Kv1i , respectively.
A
For a 2D motion vector v, v indicates a vector warped bya transform A without the translation component.
From Equations (4) and (5), we obtain
i
Mi uA
1

i
= Mi KA
1i us0 .

(6)

Because of occlusions or image boundaries, we may
have image areas that do not have corresponding regionsin the other image. For these image areas, segments cannot
be deﬁned. We call such image areas occluded regions.

2.2. Estimation step
Since motion blurs are commutative, applying motion
Aii
PSFs K0i and KA
1i to the i-th segments of u1 and u0 respectively will produce the same result, which can be written as
Ai
i
Mi KA
1i u0 = Mi K1i K0i us
Aii= Mi K0i KA
1i us = Mi K0i u1 ,(7)

where us = us0 . In the following, to simplify the notation,
we will use us to represent us0 . From Equation (7), thefollowing equation is obtained;
dTi Mi di = 0,
(8)
i
Ai
i
where di = K0i uA
1 − K1i u0 . Therefore, we expect thatminimizing the left term of Equation (8) yields the solutionsof K and A, if mask vectors mi are properly set. However, due to the non-linearity of the equation, the solutionthat minimizes the term cannot be determined uniquely. To
assure the uniqueness of the optimal solution, we use additional constraints similar to Jin et al.’s method [13], whichpenalize long motion blurs. For mask vectors mi , setting
all the elements to zero can produce a numerically optimal result when minimizing the left term of Equation (8),however, this approach should be avoided. In addition, it
is better to preserve edges in mask images while smoothtransition of mask values should be observed in blurry image regions. Combining all these constraints, we design aregularized form of the energy function as follows.
E (v, A, m) =dTi Mi di
(9)

+

α




i





error term
Ai T
Ai
T
[v0i
v0i + (v1i
) (v1i
)]

i





β(1 −

i


+

γ






mi )T (1 −




Ai
i
where d0i = u0 − K0i(b) us and d1i = uA
1 − K1i(b) us .

mi )

i

i
K0i(b) and KA
1i(b) are matrices parameterized by motion

occlusion term[(Dx mi )T G(Dx mi ) + (Dy mi )T G(Dy mi )],iwhere b is an unknown scalar. From these relations, theimage u0 can be deblurred to yield a sharp image us , except for the occluded regions, by minimizing the following
function:
Er1 (us , b) =(dT0i Mi d0i + dT1i Mi d1i ) + ρ(us ), (12)imotion blur term+

PSFs that associate a sharp image with the blurry image theoptimal PSFs to distinguish from the relative PSFs.
As described by Jin et al. [13], the optimal PSFs and therelative PSFs are related asAiv̂0i =
b2 + 1v0i − bv1i
, and
(10)

Ai
Ai
v̂1i
= −bv0i + b2 + 1v1i
,
(11)



edge term

where α, β, and γ are weighting factors.
In Equation (9), the error term computes the weighted
differences between two segments. This is the core termin the energy function for estimating the multiple motions,motion PSFs, and segmentation. The motion blur term
assures the uniqueness of the motion PSFs by penalizing excessively long motion blur vectors. The occlusion
term accounts
 for the occluded pixels that are measuredby (1 − i mi ), in which 1 is a vector whose elements
are all one. The edge term makes mask estimates rapidlychange around edges, but slowly change in the smooth regions. To evaluate transitions of mask values, spatial gradients of mask values are computed. Dx mi and Dy mi are
the spatial gradients, where Dx and Dy are matrix representations of convolution with vertical and horizontal gradientkernels, i.e., [0 − 1 1] and [0 − 1 1]T . G is a diagonal
matrix whose elements are 1/(gj2 + δ), where gj is the gradient magnitude of the j-th pixel of u0 . δ is a small positive
constant used to prevent
 division by zero. The mask vectors
are constrained by i mij ≤ 1 ∧ mij ≥ 0 ∀j, where mij
is the j-th element of mi .
We optimize the energy function in Equation (9) using analternating optimization approach and gradient-based minimization techniques. The implementation details about the
optimization are described in Section 3.

2.3. Restoration step
The estimated PSFs in the previous stage are relative
PSFs among images which are not necessarily optimal for
generating sharp images via deconvolution. We call theAicomputed by Equations (10)blur vectors v̂0i and v̂1iand (11). ρ(us ) is a regularization function. We used aregularization
 function introduced by Geman et al. [14],ρ(us ) = ζ j gj2 /(1 + gj2 ), where ζ is a weighting factor
for the regularization term. gj is the spatial gradient magnitude of the j-th pixel of us . ζ should be relatively small to
avoid strong bias to ρ(us ).
In the proposed method, occluded regions need special
handling because no relative motions can be deﬁned forthem. To handle occluded regions, we assume that suchregions have the most dominant motion in the image. This
assumption works for many images, e.g., where the mostdominant motion is observed in the background of the scene
by a camera motion. Using this assumption, we modify themask matrix M to assign new mask values in the occluded
regions. With the modiﬁed mask matrix Mo , the energyfunction Er1 is slightly modiﬁed so that the occluded regions are correctly handled;Er2 (us , b) =(dT0i Moi d0i + dT1i Mi d1i ) + ρ(us ). (13)i

By minimizing Equation (13), the deblurred image us corresponding to u0 and optimal PSFs K are obtained.

3. Implementation
This section describes minimization methods for the energy functions in Equations (9) and (13).

3.1. Computation of estimation stepIn the estimation step, motions A, image segments m,and non-uniform motion PSFs K (which are parameterizedby v) are estimated using Equation (9). To jointly optimize these parameters, we use an alternating optimization
approach for three sets of parameters [mi ], [Ki ] and [Ai ].

One of these parameter sets is updated at one step, andthe others are updated in the following steps in sequence.
This iteration continues until the energy function converges.
We use a gradient-based unconstrained non-linear optimization method, fminunc of Matlab optimization toolbox,
to optimize motion PSFs [Ki ], and a gradient projection
method [15] to optimize segmentation masks [mi ]. For
estimating motions [Ai ], we use a registration method described by Baker et al. [16].

pyramidal approach for the computation in a coarse-to-ﬁnemanner. To assign initial values for the estimation in a ﬁner
level, segmentation masks are enlarged by bilinear interpolation. The estimated motion blur parameter at a pixel ismagniﬁed twice and propagated to the corresponding four
pixels at the next level of the pyramid. Likewise, the estimated motion parameters are propagated to the next levelof the pyramid with the translation parameters multipliedby two.

Segmentation masks Segmentation masks mi are estimated by optimizing the terms of Equation (9) except the
motion blur term. The gradient of mi is computed from
Equation (9) as

3.2. Computation of restoration step

∂E 
∂mi

= dTi di − 2β(1 −



mj )

(14)

j

+

2γ[DTx GDx + DTy GDy ]mi .

Motion PSFs Motion PSFs can be estimated by optimizing the error term and the motion blur term of Equation (9).
By computing gradients of motion blur parameters v0i andAi
v1i
in a similar way to Jin et al. [13], we can obtain
∂E
∂v0i
∂E
Ai
∂v1i

=

2

i
Mi K̇0i ∇uA1

i= −2 Mi K̇A
1i ∇u0T

di + αv0i , and (15)T

Aidi + αv1i,

The image restoration and PSF reﬁnement are also performed by a gradient-based optimization. We use a gradient projection method to force pixel values of us to bewithin the valid range, and an unconstrained gradient descent method to optimize the parameter b of Equation (13).
We describe the details of optimizing Equation (13) for obtaining the deblurred image us and optimal PSFs.
Image deblurring From Equation (13), the gradient of
the deblurred image us can be written as∂Er2= −2KT0i(b) Moi d0i∂usi− 2 KA1i(b)T

Mi d1i +dρ(us )
.
dus

(18)

(16)

Motion PSFs To obtain the optimal PSFs, the gradient ofparameter b in Equation (13) can be computed aswhere ∇u0 = [Dx u0 Dy u0 ] is a matrix consisting of twocolumn vectors of image gradients, and K̇v is a matrix representation of convolution with a kernel k̇v deﬁned as ∞2t√ e−t /2 I(x + vt)dt.
(17)k̇v ∗ I = −
2π
−∞


∂Er2
Ai
)
= −2dT0i Moi (K̇0i(b) ∇us )(qv0i − v1i∂bAAi− 2dT1i Mi (K̇1i(b)∇us )(qv1ii − v0i ),(19)√
where q = b/ b2 + 1.

4. Experiments
Motion parameters Motion parameters Ai can be computed by optimizing the error term of Equation (9). To opAiitimize the error term, K0i uA
1 and K1i u0 should be computed at each iteration. However, if we assume that current motion parameters Ai are close enough to the solution,
we can approximate the minimization by image registraAi
ition between w1Ai = K0i uA
1 and w0 = K1i u0 . Consequently, motion parameters can be computed by LucasKanade based registration method between w0 and w1Ai
with weighting factors Mi . For this computation, weused a weighted inverse-compositional method describedby Baker et al. [16].
Hierarchical estimation For large images, estimation ofthese parameters is computationally intensive, and obtaining an optimal solution becomes difﬁcult due to the numberof unknown parameters. To avoid this problem, we used a

To evaluate the proposed algorithm, we used both synthetic and real-world images. The input images are ﬁrst
converted into grayscale images to perform the estimation
step. Once the parameters are obtained, the restoration stepis performed for each color band to produce the ﬁnal result.
Throughout our experiments, we set parameters in Equation (9) as (α, β, γ, δ) = (10−2 , 10−3 , 2.0 × 10−5 , 10−4 ),and the weighting factor of ρ in Equation (13) as ζ = 10−5 .
The image intensity is normalized in the range [0, 1].
We ran our algorithm over ﬁve synthetically-blurred images. Root mean squared errors (RMSE) from the groundtruth (original sharp image) were measured in normalized
intensity (Fig. 3). From left to right for each dataset, theﬁrst column shows RMSE of the blurry image, and the second and third columns show RMSEs computed from the restored images using the foreground PSF and the background0.09

Blurry image

0.08

Deblurred (foreground PSF)

0.07

Deblurred (background PSF)

RMSE

0.06

Our result

0.05

0.04
0.03
0.02
0.01
0
Dataset #1

Dataset #2

Dataset #3

Dataset #4

Dataset #5

Figure 3. Comparison of RMSEs from the ground truth using synthetic datasets that contain background and foreground motion
PSFs. From left to right bars in each dataset, RMSE of the input blurry image, the deblurred image using the background PSF,
the deblurred image using the foreground PSF, and our result are
shown.

PSF respectively. The right-most column shows the RMSE
of our result. Images restored by only foreground PSFs
show errors even larger than that of blurred images. Also,
the images restored only by background PSFs show slightly
larger errors than our results. This is because the area ofthe background region is usually large enough to make the
RMSE numerically small. However, perceptual differences
are large due to the large error in the foreground region. In
these experiments, all the image sizes are 640 × 480 pixels.
The average computation time is about six minutes for theestimation step and 40 seconds for the restoration step.
Four different real-world scenes that contain spatiallyvarying motion blurs were deblurred by our method (Fig. 4).
The ﬁrst and second rows show the blurry input images.
The third and fourth rows show the segmentation masks that
correspond to the background and foreground, respectively.
Brighter pixels indicate the higher conﬁdence in being thepart of background/foreground. Pixels that do not belong toeither the foreground or the background are estimated to bein the occluded regions. In the ﬁfth row, estimated PSFs areshown in a tabular form, e.g., 1-A corresponds to the PSF of
Input 1, segment A (background). The following two rows
show the deblurred results restored from input 1 and 2. For
the deblurring, the estimated non-uniform PSFs are used forthe corresponding image regions. For better visualization,
the bottom row shows magniﬁed image portions.
Input images containing three different motions were
tested to show that our method is able to handle more than
two different motions (Fig. 5). These three motions correspond to the background, a pencil vase, and a cigarette
pack, respectively, resulting from depth differences among
them. The cigarette pack is the nearest object to the camera,
and the pencil vase lies between the cigarette pack and thebackground. Our method successfully segments the scene,estimates the motion PSFs, and deblurs the input images.

The bottom row of the ﬁgure shows the magniﬁed views of
the image portions for a better comparison.
Figure 6 depicts the power of handling spatially-varying
motion PSFs. The foreground object is blurred by its own
motion while background is also blurred by a camera motion. The image deblurred only with the background PSF
shows severe artifacts on the foreground object due to theinconsistency with the foreground PSF (Fig. 6b). On theother hand, the image deblurred only with the foreground
PSF yields a blurry background (Fig. 6c). Our method isable to avoid this problem because it uses non-uniform motion PSFs for image restoration (Fig. 6d).

5. Discussion
In this paper, we proposed a new method for removing non-uniform motion blurs from images. To achieve
this goal, the problem is formulated as simultaneous estimation of multiple motions, segmentation, and spatiallyvarying motion PSFs. The problem is solved by optimizing a regularized form of the energy function. Furthermore,
the estimated PSFs are reﬁned to achieve the restoration ofimages. We have evaluated the proposed method using avariety of synthetic and real-world images and validated theeffectiveness of the algorithm.
Although we used only two images as input, the algorithm can naturally take more than two images as well. In
this case, the estimation step is performed for each pair ofimages, and the restoration step is performed by simultaneously optimizing the energy functions of all pairs of images.
Limitations Our algorithm has a few limitations. One isthat it shows some artifacts around the boundaries of different motions in restored images. Blurry regions on boundaries of the foreground object still remain (Fig. 6). This
artifact is inevitable due to missing information of hidden
pixels behind the foreground objects. Second, like existing segmentation algorithms, our segmentation is not performed well on textureless regions because it is difﬁcult todetermine the motion in such regions. Therefore, the proposed method works better if the input images are more textured.

Acknowledgements
We would like to thank Kyuman Jeong for the help
in taking pictures used for our experiments, and Derek J.
Lactin for his help on English editing. This research was
supported in part by the ITRC support program.

References[1] P. Bhat, K. Zheng, N. Snavely, A. Agarwala, M. Agrawala, M. Cohen, and B. Curless, “Piecewise image registration in the presenceInput #1
Input #22
Seggment A
Segment B

2

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A

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ComparisonDebblurred #2 Deblurreed #1

PSFs

1

Figure 4. Results of our method. Among the four scenes, the input images for the left-most column were blurred synthetically, while
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(a) One of two input images (b) Deblurred by background PSF (c) Deblurred by foreground PSF (d) Our result (non-uniform PSF)
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