Stein Unbiased Risk Estimator

Michael Elad

The ObjectiveWe have a denoising algorithm of some sort, and we want to set

its parameters so as to extract the best out of it

Algorithmx +

2v ~ 0, I

y h y,

y

22

2 2ˆmin E y x min E h y, x

Derivation – 1 Lets open the norm into its ingredients:

Therefore, we will proceed with the second term and show that in fact it can be computed

2 2

2 2

T

2

2

E h y, x E h y,

2E x h y,

E x

Easy

Impossible?

No matter

Derivation – 2 Using the fact that

we get

Again, the first term is fine for us to compute, while the second seems hard (we do not know the noise vector!)

T T

T

E x h y, E y h y,

E v h y,

Easy

Impossible?

x y v

Derivation – 3 Using the definition of expectation

This may look ugly

BUT …..

Tk k

k

2k

k k k2k

E v h y, E v h y,

v1v h y, exp dv

22

Derivation – 4 We notice that the same integral can be written as

which should remind us of integration by parts:

2

2

22

2

vv h y, exp dv

2

d vh y, exp dv

dv 2

d df x g x dx f x g x f x g x dx

dx dx

Derivation – 5 Using this to our expression leads to

2

2

2 2

2 2

d vh y, exp dv

dv 2

v v dh y, exp exp h y, dv

2 2 dv

Assuming that the function

h is finite for all y, this term is zero

The derivative w.r.t. v can be replaced by a derivative w.r.t. y

Derivation – 6 One last step – the expression we got is in fact an expectation …

2

2

2

2

d vh y, exp dv

dv 2

v dexp h y, dv

2 dy

dE h y,

dy

Wrap Up (1)We got the following expression after all the above steps

2 2

2 2

T

2y

E h y, x E h y,

2E y h y,

2 E h y,

const.

The squared norm of the estimated image

An inner product between the noisy and the denoised images

Sum over the “sensitivity” of our algorithm to perturbations in the input vector

Our estimator is true up to an unknown constant

Wrap Up (2)Since we cannot compute the expectation, we will simply drop it

with the hope that the summation over all the image pixels is sufficient to provide the desired accuracy

If you want to set the parameters, , do this while minimizing the above expression

This implies that the algorithm should be differentiable w.r.t. the input.

2 2 T

2 2

2y

E h y, x h y, 2y h y,

2 h y,

Example – Thresholding

Algorithmx +

2v ~ 0, I

y

Lets come back to the global image denoising scheme by thresholding

1 T

T

y h y,

S y

DW W D

Example – Smoothing Lets make sure that our estimator is differentiable by smoothing

it (assume k is even)

k

k

T kk k

2k k

T2k

zz T

S z z zz T z

1T

z z(k 1)

dS z T Tdz z

1T

-2 -1 0 1 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

k=10k=20Hard-Thresholding

Example - SURELets make sure that our estimator is differentiable by smoothing

it (assume k is even)

2 2 T

2 2

2y

21 T

T2

T 1 TT

2 ' 1 T 1 TT

E h y, x h y, 2y h y,

2 h y,

S y

2y S y

2 tr S y

DW W D

DW W D

DW W D W D

Example - SUREWe can simplify the last term

' 1 T 1 T ' 1 T 1 TT T

' 1 T 1 TT

A Diagonal Matrix

' 1 T TT

' 1 T 2T

tr S y tr S y

tr S y

tr S y

tr S y

DW W D W D W W D W D D

W W D W D D

W D D D

W D W

11 2 1 2

tr tr

tr tr

tr tr diag( )

AB BA

WWW W

WR W R

Some Properties:

Example - SUREBottom line:

Does this work?

22 1 TT2 2

T 1 TT

2 ' 1 T 2T

ˆE y x S y

2y S y

2 tr S y

DW W D

DW W D

W D W