n sip 01 matrix gain talk
TRANSCRIPT
8/12/2019 n Sip 01 Matrix Gain Talk
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OutlineOutline
• Digital halftoning– Modeling grayscale error diffusion halftoning– Modeling color error diffusion halftoning
• Matrix gain model for color error diffusion– Validating the signal shaping predicted by the model– Validating the noise shaping predicted by the model
• Conclusions
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Grayscale Error DiffusionGrayscale Error Diffusion• Shape quantization noise into high frequencies
• Two-dimensional sigma-delta modulation
• Design of error filter is key to high quality
current pixel
weights
3/16
7/16
5/16 1/16
+ _
_+
e(m )
b (m ) x(m )
difference threshold
computeerror
shape error
u (m )
)(mh
Error Diffusion
Spectrum
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Modeling Grayscale Error DiffusionModeling Grayscale Error Diffusion
• Sharpening is caused by a correlated error image [Knox, 1992]
Floyd-Steinberg
Jarvis
Error images Halftones
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Modeling Grayscale Error DiffusionModeling Grayscale Error Diffusion
• Apply sigma-delta modulation analysis to two dimensions– Linear gain model for quantizer in 1-D [Ardalan and Paulos, 1988]
– Linear gain model for grayscale image [Kite, Evans, Bovik, 2000]
– Signal transfer function (STF) and noise transfer function (NTF)– 1 – H( z) is highpass so H( z) is lowpass
K s
u s(m )
Signal Path
K s u s(m )
K n
u n (m )
+
n (m )
K n u n (m ) + n (m)
Noise Path, K n =1
( ))(1
)(z
zz
H N
B NTF n
−==
( )( ) ( )zz
z H K
K
X
BSTF s
ss
11)( −+==
Q(.)u (m ) b (m ) {
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Vector Color Error DiffusionVector Color Error Diffusion
• Error filter has matrix-valued coefficients
• Algorithm for adapting matrix coefficients[Akarun, Yardimci, Cetin 1997]
( ) ( )ÿ
( )vectormatrix
kmekhmtk
−= ÿ℘∈
+ _
_+
e(m )
b (m )x(m )
difference threshold
computeerror
shape error
u (m )
)(mh
t(m )
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The Matrix Gain ModelThe Matrix Gain Model
• Replace scalar gain with a matrix
– Noise is uncorrelated with signal component of quantizer input– Convolution becomes matrix–vector multiplication in frequencydomain
( ) ( ) 12
minarg −
=−=uubuA
CCmuAmbK E s
IK =n
( ) ( ) ( )zNzHIzB −=n
( ) ( )( )( ) ( )zXIKzHIKzB 1−
−+=s
Noise component of output
Signal component of output
u (m ) quantizer inputb (m ) quantizer output
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How to Construct an Undistorted HalftoneHow to Construct an Undistorted Halftone
• Pre-filter with inverse of signal transfer function to obtain
undistorted halftone
• Pre-filtering is equivalent to the following when
( ) ( )( )[ ] 1−−+= KIKzHIzG
IKL −= −1
Modified error diffusion
+ _
_+
e(m )
b (m )x(m )u (m )
)(mh
t(m )
L
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Validation #1 by Constructing Undistorted HalftoneValidation #1 by Constructing Undistorted Halftone
• Generate linearly undistorted halftone
• Subtract original image from halftone
• Since halftone should be “undistorted”, the residual should
be uncorrelated with the original
=
0027.00011.00058.0
0020.00023.00054.0
0040.00009.00052.0
rxC
Correlation matrix of residualimage (undistorted halftone minusinput image) with the input image
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Validation #2 by Knox’s ConjectureValidation #2 by Knox’s Conjecture
=
1836.02952.02063.0
1605.03295.02787.0
0999.02989.03204.0
exC
Correlation matrix for anerror image and input imagefor an error diffused halftone
=
0150.00142.00428.0
0164.00144.00493.0
0122.00235.00455.0
exC
Correlation matrix for anerror image and input imagefor an undistorted halftone
( ) 0=zE s
( ) ( )zNzE =n
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Validation #3 by Distorting Original ImageValidation #3 by Distorting Original Image• Validation by constructing a
linearly distorted original
– Pass original image through
error diffusion with matrix
gain substituted for quantizer
– Subtract resulting color
image from color halftone
– Residual should be shaped
uncorrelated noise
Correlation matrix of residual
image (halftone minus distortedinput image) with the input image
=
0062.00040.00082.0
0049.00039.00065.0
0051.00007.00067.0
rxC
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Validation #4 by Noise ShapingValidation #4 by Noise Shaping
• Noise process is error image for an undistorted halftone
• Use model noise transfer function to compute noise spectrum
• Subtract actual halftone from modeled halftone and compute
actual noise spectrum
0 0.5 1 1.510
15
20
25
30
35
40
radial frequency
2 0 * l o
g 1 0 (
e r r o r
)
red error spectra
observedpredicted
0 0.5 1 1.515
20
25
30
35
40
radial frequency
2 0
* l o
g 1 0 (
e r r o r
)
green error spectra
observedpredicted
0 0.5 1 1.515
20
25
30
35
radial frequency
2 0 * l o
g 1 0 (
e r r o r
)
blue error spectra
observedpredicted
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ConclusionsConclusions
• Modeling of color error diffusion in the frequency domainusing a coupled matrix gain and noise injection approach
• Linearizes error diffusion
• Predicts linear distortion and noise shaping effects
• Signal frequency distortion may be ‘‘cancelled’’
• Filters may be designed for optimum noise shaping