The Nature of Motion Blur

Abstract
The removal of motion blur induced into images is currently an active eld of research. The analysis
of motion de-blur algorithms has shown, that they perform dierently on synthetic and real motion blur.
Therefore we have build an apperatus which allows us to produce pictures with dened motion blur induced,
so we can study these possible dierences. In this paper we focus on clearly visible lines in the spectrum of
synthetically blurred images. We investigate for such lines in real blur images and are able to show their
existence with both blur types. Furthermore we give theoretical reasoning for these lines to appear. Hence
this which is a strong indice that the basic blur model used to generate synthetic blur incoperates some
truth.

1 Purpose
Motion Blur is a phenomena encountered in many
application. Not only photographers, but also as-
tronomers and operators of surveillance cameras are
struggling with distorted images due to motion blur
[5]. Several techniques to restore the image already
exists, but none of the existing algorithms allows
perfect restoration [5]. The suboptimal results are
partly due to dierences between the motion blur Figure 1:
modeled and the motion blur encountered in real Original and synthetically blurred image
world images. Revealing the similarities and dif-
ferences hopefully leads to a better understanding The upper row represents the spectrum of the un-
of the motion blur phenomena, thus allows to de- blurred, original image whereas the lower row show
sign better algorithms. Therefore a certain aspect the spectrum of the blurred image.
of synthetically blurred image is considered here,
namely the appearance of of lines in the spectrum The upper spectra look like random noise and there
of the blurred image. seems to be no additional information included in
the picture. Whereas in the spectra of the blurred
image clearly some lines get visible. These lines are
2 Experimental Observations visible at best in the magnitude spectrum, but also
can be seen in the real or imaginary only spectrum.
Let's take a look at a picture (Figure 1) which got If we compute the average value of a vertical line
blurred (synthetically with a known horizontal dis- and plot the result (Figure 3), we get a graph where
placement, in this case 10 pixels). the peaks represent the vertical lines in the magni-
tude spectrum.

Now the spectrum of both images is generated via

a FFT and both spectra are compared (Figure 2).

1
 Figure 4:
Spectra of blurred images with dierent
displacements
Figure 2:
Spectra of blurred and non-blurred image about the blur parameter, but dier to a little ex-
tend. This is possible due to the little amplitude
near the ends.
Another important issue is, before questioning why
these lines occur is, if our ndings are not only of
theoretical nature but are also visible in blurred pic-
tures taken with a camera. For this purpose we have
build an experimental setup, which allows to pro-
duce blurred images with a dened displacement.
Furthermore the blurred images should be compa-
rable to the synthetically created ones in terms of
linear blur and horizontal-only blur.
Figure 3:
Average magnitude along the horizontal axis of
blurred image

Counting the number of peaks, we nd 9 peaks,

which is about the value of the blur parameter used
to generate this image. If one now considers that
the peak in the middle is about twice as wide as
all the other peaks, we can count it as a double
peak and therefore get 10 peaks, which is corre-
spond to the blur parameter used earlier. We now
assume as a thesis that the number peaks in the
intensity graph of the spectrum of an blurred im- Figure 5:
age corresponds to blur parameter used during its Experimental setup
generation.
This unit is shown in Figure 5 and comprises a
camera unit, a guiding rail and a stepper motor.
The camera carriage is accelerated to a constant
Let's see if this holds true for other blur parame- speed and the camera takes a photo with a medium
ters. Figure 4 show the intensity graph of pictures exposure time (around 100ms) to allow signicant
blurred using dierent blur parameters. Counting motion blur appear in the picture. As a motif a
the number of peaks, we get values which are round checkerboard structure has been chosen, since this

2
allows an easy method for estimating the associated no motion in the vertical direction (otherwise the
blur parameters. graph must look like shown in Figure 7, where the
decay is linear and spread about a signicant num-
ber of pixels).

Figure 6:
Estimating the PSF with real motion blur

An image captured with our experimental setup is Figure 8: Verication of horizontal motion
shown in Figure 6. From the way we have set up
the capturing process we assume the motion path Measuring the blur area in Figure 7 allows us to
of the camera to be linear and uniform in the hor- directly infer the length of the PSF (the length has
izontal direction. The rst assumption is veried been visualized in Figure 6).
by looking at the plot shown in Figure 7. The plot
Another picture taken by this unit is shown in Fig-
depicts the luminance of the picture, which taken
ure 9, which is used to see if your assumption of the
across the blurred zone (marked with (a) in Figure
characteristic spectrum holds true.
6). The luminance curve is very close to linearly
decreasing.

Figure 9:
Figure 7: Real blur of a checker board structure
Verication of linearity

The second assumption that the motion is only in

the horizontal direction is veried with a second Using the same technique as for the synthetic im-
luminance graph (of region (b)), shown in Figure age, we can compute the intensity graph (Figure
8. The sharp decay of luminance at the border be- 10). Clearly the graph shows much more noise than
tween the two boxes proves that there is almost the one of the synthetic blurred images. In this case

3
we calculate the period of the peaks, since it is con- It is furthermore known, that a convolution with a
stant in the graphs above and we need only a few Dirac is a shifts by the amount the Dirac's argument
peaks to calculate the total number. The marked of the original function. Assuming no noise (n = 0),
data points show a period between 5 and 6. This the blurred image can be written as
gives us 21 to 23 peaks which is about the same as
the blur parameter determined earlier. a
P
b=h?f = δ(k) ? f =
k=0
a
P
f (x, y)+f (x+1, y)+...+f (x+a, y) = f (x+k, y)
k=0

Clearly the resulting image increased in intensity,

which has to be compensated by a factor c = a+1 1
,
which has been omitted here fore simplication pur-
poses.
As we are interested in the spectrum of the blurred
Figure 10:
image, we will use the Fourier transformation (FT)
Spectrum of checker board structure
F{} to achieve this. Since the FT is a linear oper-
ation, we can move the FT-operation into the sum.

3 Theoretical Reasoning F{b} = F{

a
P
f (x + k, y)}
linearity
=
k=0
a
Commonly a linear, non-recursive (FIR) is used to
P
F{f (x + k, y)}
model the degradation of digital (sampled) images k=0

caused by motion blur. Let's consider the origi-

nal, blur-free M × N -image f to be convolved with Now let's look at a single, displaced image. The
a convolution kernel h, referred to as the Point rule of circular displacement allows us, to rewrite
Spread Function (PSF). Additionally, some noise the expression in that manner, that it only contains
is introduced during the capturing process, which the original image.
is modeled with the additive noise term n. Hence,
2πk
the blurred M × N image b, as it is captured by the F{f (x + k, y)} = e−j N · F{f (x, y)}
moving camera, is modeled as
where N denotes the total Length of the image.
b=h?f +n (1) Now we make again use of linearity and factor out
the spectrum of the original image:
where the symbol ? represents the convolution op-
erator. a 2πk
e−j
P
F{b} = N · F{f (x, y)}
The most simple form of the PSF consists of two k=0
Heaviside functions, dening a rectangular lter:
Assuming that the spectrum of the motion blur free
h = u(0) − u(a) image consists only of random noise, we can neglect
its inuence on the blurred image spectrum. Plot-
ting the sum of exponential function yields a graph
with a being the blur parameter or respectivly the (Figure 11) which is somewhat similar in its peak
length of the blur zone. structure to the one of the real blur image. The plot
If we consider discrete values, the PSF consists of has been shifted by half its width to correspond to
a series of Diracs: MATLAB's way of computing the spectrum. Fur-
thermore this graph has been translated into an im-
a
P age, with the characteristic line structure. Here it is
h= δ(k)
k=0
harder to see the corresponding lines in the real blur

4
spectrum, but they can be seen even though they [2] M. Ben-Ezra and S.K. Nayar. Motion-Based
are dominated by non-equally distributed noise. We Motion Deblurring. IEEE Transactions on
can infere, since lines are present in both synthetic Pattern Analysis and Machine Intelligence,
and real spectrum, convolution is a valid method to 26(6):689699, 2004.
model motion blur.
[3] MM Chang, AM Tekalp, and AT Erdem. Blur
identication using the bispectrum. Signal
Processing, IEEE Transactions on [see also
Acoustics, Speech, and Signal Processing, IEEE
Transactions on], 39(10):23232325, 1991.

[4] R. Fergus, B. Singh, A. Hertzmann, S.T.

Roweis, and W.T. Freeman. Removing camera
shake from a single photograph. ACM Transac-
tions on Graphics (TOG), 25(3):787794, 2006.

Figure 11: [5] S. Schuon and K. Diepold. Comparison of Mo-

Series of exponential functions tion Deblur Algorithms and Real World Deploy-
ment. Paper on IAC 2006, 2006.

4 Conclusion
We have seen that synthetic and real blur seems
to be quite similar in terms of lines in the spec-
tra. Thus this can not be a reason for restoration
algorithms to fail miserably on real world pictures.
Taking a look at the spectrum of the real blur image
gives a indication that noise might be the reason.
In comparison to the synthetic spectrum it com-
prises more noise. Furthermore comparison tests
have shown increasing restoration problems when
noise is added to the synthetic blur [5]. Hence more
research into the noise issue is required.
Furthermore one might be tempted to use the lines
appearing in the spectrum to determine the blur pa-
rameters [3]. Quick tests on real pictures taken by
cameras with long exposure time reveal that the rel-
evant motion is much more complex and can there-
fore not be identied by just looking at the lines in
the spectra. But only recently new ndings [1, 2, 4]
oer promising ways to acquire the blur parameters.

References

[1] M. Ben-Ezra and SK Nayar. Motion deblurring

using hybrid imaging. Computer Vision and
Pattern Recognition, 2003. Proceedings. 2003
IEEE Computer Society Conference on, 1, 2003.