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,where
1,
1
1
sxxx
xxsxxs
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,0
21,24
10,1
)(2
253
21
2
253
23
xxxx
xxx
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
L
xfxfxfxfA 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 aLinear
Interpolation
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
,...32
2
12
,)1(
,
,)1(
),(L
1L
L
1LL
H
nk
nk
nk
sxxs
x
axxa
asx
nn
n
nn
k
2/)1(
2/)1(
2H
2/)1(R ),()(),(
M
Mm
mnMmn 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(
2H
2/)1(R ),()(),(
M
Mm
mnMmn 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
1,1
0,0
)()(
)()(
20
31
athenaif
athenaif
scsc
scsca
))(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 test
Images Linear CCIWaDi-Linear
WaDi-CCI
LSWaDi-Linear
Lena 33.28 34.25 34.09 34.21 34.62
Peppers 31.57 31.96 31.61 31.63 32.00
Baboon 23.28 23.59 23.42 23.35 23.66
Airplane 30.33 31.08 30.48 30.50 31.15
Goldhill 31.01 31.49 31.45 31.37 31.64
Barbara 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