Least-Squares Warped Distance for Adaptive Linear Image Interpolation

Presentation Outline• Introduction• Basic Concept of Interpolation• Conventional Interpolation• Previous Adaptive Linear Interpolation• Proposed Method• Example of Proposed Method• Simulation Results• Conclusions

Introduction• Image interpolation plays a key role in the

image processing literature– Image resizing/rotation/warping/morphing – Image/video compression– Mosaicking color filter array in DSC– De-interlacing in DTV– Lifting-based wavelet transform– Timing recovery in a digital modem– Sample rate converter

• Adaptive image interpolation can provide a substantial gain in image quality.– Especially, warped distance (WaDi) approach

Basic Concept of Interpolation• With given discrete samples f (xk),

generating continuous function as follows

• The ideal kernel is the sinc function

k

kk xxxfxf )()()(ˆ

)(ˆ xf

)( kxf)( kxx

Interpolation Kernel

Conventional Interpolation• Linear

elsewhere,0

10,1)(

xxx

10,where1,

1

1

sxxxxxsxxs

kk

kk

-1 1

)(xLinear

)(ˆ xf)( 1kxf

)( kxf

sxkx

)()()1()()()(ˆ1 kk

kkk xsfxfsxxxfxf

Conventional Interpolation• Keys’ Cubic Convolution Interpolation

1

)(xCubic

2-2 -1

2/))((

2/)43)((

2/)253()(

2/)2)(()(ˆ

232

231

230

231

ssxf

sssxf

ssxf

sssxfxf

k

k

k

k

21s

)()()()()(ˆ216

1116

9016

9116

1

kkkk xfxfxfxfxf

elsewhere,021,2410,1

)( 2253

21

2253

23

xxxxxxx

x

)(ˆ xf)( 1kxf

)( kxf

sxkx

Previous Adaptive Linear Interpolation• Warped Distance Linear Interpolation

• Definition of warped distance as follows

• Note that – The variable A is a pixel-based parameter– The variable k is an image-based parameter

(k = 8 fixed for the Lena image)

)()()1()(ˆ1 kk xsfxfsxf

)1(' skAsss

1)()()()( 211

Lxfxfxfxf

A kkkk

Proposed Method• New WaDi a for a given distance s

• Use distance s as a pixel-based parameter• Introduce a system to calculate s

– Including low pass filter and MMSE

)()()1()( 1 kk xafxfaxfc

Proposed Method• Generic diagram

Xlow(z)

LinearInterpolation

Xhigh(z,s,a)

LPFG(z,s)

s,a

22

-

D(z,s,a)

Find aLinearInterpolation

Xhigh(z)

Proposed Method• Systematic approach to employ the least-

squares technique

xLn

xHk(a,s)

xRn(a,s)

LPF + Sampling

xH2n+1(a)(1-s)xL

n-1+sxLn

… …

xRn(a,s) xR

n+1(a,s)

-dn(a,s)

To be interpolated

(1-s)xLn+1+sxL

n+2

s

,...322

12

,)1(,

,)1(),(

L1

L

L

1LL

H

nknknk

sxxsx

axxaasx

nn

n

nn

k

2/)1(

2/)1(2

H2/)1(

R ),()(),(M

MmmnMmn asxsgasx

),(),( RL saxxsad nnn

0])),([( 2 sadEa n

Example of Proposed Method• Low complexity version

• Apply 3-tap low pass filter gi = 1/2{s, 1, 1-s} • Define cost function as follows

2/)1(

2/)1(2

H2/)1(

R ),()(),(M

MmmnMmn asxsgasx

2)),(()),((),(

21

R1

L2RL saxxsaxxsaC nnnn

Example of Proposed Method• Get a to minimize the cost as follows

• We have

where

2)),(()),((),(

21

R1

L2RL saxxsaxxsaC nnnn

1,10,0

)()()()(

20

31

athenaifathenaif

scscscsc

a

))(1( L1

L21

0 nn xxsc ))(1( L1

L21

1 nn xxssc

)( L1

L21

2 nn xxsc

))1()2(( 2L

1LL

21

3 nnn xsxsxsc

Simulation• Two scenarios to evaluate the methods• One is a decimation-interpolation

simulation– Filtered followed by down-sampler with a factor

of two– Interpolate decimated image with a factor of

two• The other is a rotation test

– Fifteen rotations performed successively– Rotate by 24 degree for each rotation

Simulation Results for DI Test• PSNR resulting from the decimation-

interpolation testImages Linear CCI WaDi-

LinearWaDi-

CCILSWaDi-Linear

Lena 33.28 34.25 34.09 34.21 34.62Peppers 31.57 31.96 31.61 31.63 32.00Baboon 23.28 23.59 23.42 23.35 23.66Airplane 30.33 31.08 30.48 30.50 31.15Goldhill 31.01 31.49 31.45 31.37 31.64Barbara 25.25 25.40 25.34 25.31 25.38

Simulation Results (DI Test)

Simulation Results (Rotation Test)

Conclusions• A Pixel-based adaptive linear interpolation

has been presented• A generic system, formulation, and its low

complexity version have been proposed• Simulation results show that the proposed

method– Give better visual quality– Give better objective quality in terms of PSNR– than previous methods such as conventional

linear, cubic convolution, and previous warped distance-based methods