Motion Blur Estimation at Corners

Computer Vision Group

Motion Blur Estimation at CornersGiacomo Boracchi and Vincenzo Caglioti

[email protected]


Motion Blurred Image




















Preliminary Remarks

Dealing with blurred images it is complicated (lack of information)

Blur is often assumed uniform, but this is restrictive

We propose to analyze blur on some image regions

We focus on regions containing a corners

We consider only motion blur• Blur is approximated as parametric – direction and length -• Estimation of corner linear displacement

Any other blurring phenomena are neglected (e.g out of focus blur).

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Outline

Image Model Corner Model Problem Solution Robust Solution Experiments Concluding Remarks


Blurred Image Model

A blurred noisy image given the original image

and the blur operator

( ) ( )( ) ( )I x K y x x (0, ),N x X

K

( )( ) ( , ) ( )X

K y x k x y d

we assume that blur locally is constant

point spread function

0

0 0( )( ) ( ) ( )xU

K y x v x y d v y x

0v

I

00 , xx X U X and

y


Blur Assumptions

Point spread function has 1D support,

Constant value (Uniform Speed)

Parametric approach, estimate and

we call the corner displacement , the vector having direction and length

These assumptions hold only locally…

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0 ( ) ( )lv R s x 1 21/(2 1) , 0

0l

l l x l xs

else

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v

1x

2x

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Presentation Outline



Why Corners?

No Aperture Problem, when blurred.

Easy to Detect (Harris, Hessian)

Easy to Model

Meaningful for scene understanding


1

2

1

2

The Corner Model

The Corner has to be binary in the considered region D

Not every displacement can be managed.

/ 2, / 2 Exclude “self intersecting” corners

D





Vector Relation

Consider an Image Region D containing a blurred corner

in the noise free case,

v

0( ) ,v K y x x D

0 , ( ) 0D x K y x D

the aperture problem holds

However both at corners blurred edges may be used to solve this ambiguity

0 , ( ) , 0D x K y x T D T


Least Square Solution

A “good” image region should containpixels from both blurred edges

Several pixels have to be considered, for example x

wi ;¡ n < i < n weights

~vx =argminv°°°A(x) v ¡ ¢ [w¡ n ; :::;w0; :::;wn]

T°°°2

A(x) ~vx =

2

66664

w¡ n r I (x¡ n)T

:::w0 r I (x)T

:::wnr I (xn)T

3

77775~vx = ¢ [w¡ n ; :::;w0; :::;wn]

T


Drawbacks

Solution is not robust in presence of outliers and noise

Whenever image assumptions are not met (e.g. textured or shaded corners, smoothed contours, other image artifacts) solution is seriously corrupted.

Compute the solution on every pixel : method is slow

Requires a filtering procedure as every estimate depends on

Then, better look for a vector that satisfy the basic equation for a significant number of pixels, disregarding how far from the solution is for few pixels

x

v

v

~vx =argminv°°°A(x) v ¡ ¢ [w¡ n ; :::;w0; :::;wn]

T°°°2

x





v

Robust Solution

Considering only two gradient vectors it would be enough if appropriately chosen

axbx

( )aN x

( )bN x( )aN x

( )bN x

v

N(x) = r I (x)jjr I (x)jj2 ¢


The Hough Transform

For each input data determine theset of possible solutions.

The solutions are represented in the parameter space

A vote (1) is assigned to all parameters that are compatible with a given data

being , the parameters,the coordinates of end point (in pixels)

Evaluate all inputs and sum the votes

The most voted pair in the parameter space are taken as solution,as they represent the parameters satisfying most of data

1 2( , )u uuv

( )N x


The votes in parameter space

Consider also parameters close to the solutions• Assign them a fraction of vote (<1)• Assign a full vote to exact solutions

Being a tuning parameter and noise standard deviation

For every data , votesare assigned by this vote function opportunely rotated and translated

k¾́

N (x)

x̀(u1;u2) = R( ¼2 ¡ µ)¡`¢([u1;u2]T ¡ N (x))

x̀(u1;u2)

(̀u1;u2) = exph¡³

u21+kju1 j¾r ´

´2i


Robust Solution- Votes sum

Sum of votes





Experiment on Synthetic Images

Synthetic images constructed according to

I (x) = K¡y+»

¢(x) +´(x) ; x = (x1;x2)

»(x) » N (0;¾»)

´(x) » N (0;¾́) where represents electronic noise

represents differences between the binary corner and the synthetic image


Experiments on Synthetic images

Point Spread Function of 10° degrees and 20 pixels extents


Experiments on Synthetic images

Point Spread Function of 70° degrees and 30 pixels extents


Experiment on a Test Image

5 regions containing a corner have been selected on house image


Results on a Test Image

house image has been artificially blurred by motion blur psf having• direction 30 degrees• length 25 pixels

Error in pixels : 2.07






















Experiments on camera images

Triplets of images have been taken according to the following scheme• still image (A)• Blurred image moving the camera on a rack (B)• still image (C)

Motion has been estimated in selected image regions in B and compared with the ground truth obtained by matching the feature in the corresponding regions in A and in C.


A - still image at initial camera position


B – the Blurred Image


C – still image at final camera position


Five selected Image regions










Results from camera images




Image Model Corner Model Problem Solution Robust Solution Experiments Ongoing Works


Conclusions

Fourier based methods usually fail at corners and on small regions

Method to estimate motion blur parameters at corners from a single blurred image

We handle space varying blur as every image region is considered separately


Ongoing Work

Manage all possible displacement, also the self – intersecting case

Detect Blurred Corners

Adaptively select region for motion estimation

Extend the algorithm to psf having 1D support, and non-uniform density.

Measure the Goodness of our estimate

Fusing estimates coming from different corners

Documents

Motion Blur Estimation at Corners