lecture8 sampling f11 - nyu tandon school of...
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Image Sampling andImage Sampling and Resizing
Yao WangPolytechnic Institute of NYU, Brooklyn, NY 11201
With contribution from Zhu LiuPartly based on A. K. Jain, Fundamentals of Digital Image Processing
Lecture Outline
• IntroductionN q ist sampling and interpolation• Nyquist sampling and interpolation theorem
• Common sampling and interpolation filters• Sampling rate conversion of discrete
images (image resizing)
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 2
Illustration of Image Sampling and InterpolationInterpolation
dx=dy=2mmy
16 mm
dx=dy=1mm
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 3
How to choose dx, dy to reach a good trade off between accuracy and cost of storage, transmission, processing?
Uniform Sampling• f(x,y) represents the original continuous
image, fs(m,n) the sampled image, andimage, fs(m,n) the sampled image, and the reconstructed image.
• Uniform sampling),(ˆ yxf
Uniform sampling),,(),( ynxmfnmfs
– ∆x and ∆y are vertical and horizontal
.1,...,0;1,...,0 NnMmy
sampling intervals. fs,x=1/∆x, fs,y=1/ ∆y are vertical and horizontal sampling frequencies.
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 4
Image Sampling as Product with Impulse TrainProduct with Impulse Train
• Periodic impulse sequence
1 1M N
1
0
1
0),(),(
M
m
N
nyx nymxyxp
1 1
1
0
1
0
~
)()(
),(),(),(),(),(
M N
M
m
N
nyxyxs
f
nymxnmfyxpyxfyxf
0 0
),(),(m n
yxs nymxnmf
1m = 0
∆x1
…p(x,y)
M - 1
x y∆y
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 5
n= 0 1 N-12x y
Fourier Transform of Impulse Train• 1D
nfut
uPtnttp s
)(1)()()(
tfwhere
t
s
nnm
1
,
• 2D
nfvmfuyx
vuPynyxmxyxpnm
ysxsnm
),(1),(),(),(,
,,,
yf
xfwhere ysxs
1,1
,,
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 6
Frequency Domain Interpretation of SamplingSampling
• Sampling is equivalent to multiplication of the original signal with a sampling pulsethe original signal with a sampling pulse sequence.
s yxpyxfyxf ),(),(),(
I f d i
nm
ynyxmxyxpwhere,
),(),(
• In frequency domains vuPvuFvuF
11),(),(),(
nm
ysxss
ysxs
nmysxs nfvmfuF
yxvuF
ffwhere
nfvmfuyx
vuP,
,,,
,, ),(1),(
1,1
),(1),(
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 7
ysxs yf
xf ,, ,
Frequency Domain Interpretation of Sampling in 1D
Original signal
p p g
Sampling impulse train
The spectrum of the sampled signal includes the original
S l d i l
includes the original spectrum and its aliases (copies) shifted to k fs , k=+/- 1,2,3,…
Sampled signalfs > 2fm
s
When fs< 2fm , aliases overlap with the original spectrum >Sampled signal
fs < 2fm(Aliasing effect)
original spectrum -> aliasing artifact
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 8
Sampling of 1D Sinusoid Signals
Sampling aboveNyquist rateyqs=3m>s0
Reconstructedi i l=original
Sampling underNyquist rates=1.5m<s0
ReconstructedReconstructed!= original
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 9
Aliasing: The reconstructed sinusoid has a lower frequency than the original!
Frequency Domain Interpretation of Sampling in 2DSampling in 2D
• The sampled signal contains replicas of the original spectrum shifted by multiplesthe original spectrum shifted by multiples of sampling frequencies.
u uu u
fm x
fs,x
fs,x>2fm,x
vv
fm,x
fm,y fs,y
fs,y>2fm,y
Original spectrum F(u v) Sampled spectrum F (u v)
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 10
Original spectrum F(u,v) Sampled spectrum Fs(u,v)
Illustration of Aliasing Phenomenon
u u
fm,x fs,xfs x<2fm x
v vfm,yfs,y
s,x fm,xfs,y<2fm,y
Original spectrum F(u,v) Sampled spectrum Fs(u,v)
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 11
p p s( )
Nyquist Sampling Rate
• Nyquist sampling rateTo be able to preserve the original signal in– To be able to preserve the original signal in the sampled signals, these aliasing components should not overlap with thecomponents should not overlap with the original one. This requires that the sampling frequency fs,x, fs,y must be at least twice of the , ,yhighest frequency of the signal, known as Nyquist sampling rate.
• Interpolation– Remove all the aliasing components
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 12
Nyquist Sampling and Reconstruction TheoremTheorem
• A band-limited image with highest frequencies at fm,x, fm,ycan be reconstructed perfectly from its samples, provided that the sampling frequencies satisfy: fs,x >2fm,x, fs,y>2fm,y
• The reconstruction can be accomplished by the ideal• The reconstruction can be accomplished by the ideal low-pass filter with cutoff frequency at fc,x = fs,x/2, fc,y = fs,y/2, with magnitude ∆x∆y.
yfyf
xfxf
yxhotherwise
fv
fuyxvuH
ys
ys
xs
xsysxs
,
,
,
,,, sinsin
),(0
2||,
2||),(
• The interpolated image
)(sin)(sin),(),(ˆ ,, ymyfxmxf
nmfyxf ysxss
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 13
)()(),(),(
,, ymyfxmxfnmfyxf
ysxsm ns
Applying Nyquist Theorem• Two issues
– The signals are not bandlimited.The signals are not bandlimited.• A bandlimiting filter with cutoff frequency fc=fs/2 needs to be
applied before sampling. This is called prefilter or sampling filterfilter.
– The sinc filter is not realizable.• Shorter, finite length filters are usually used in practice for
b th filt d i t l ti filtboth prefilter and interpolation filter.
• A general paradigm
Prefilter Interpolation(postfilter)A D
B C
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 14
Sampling pulse fs
Non-ideal Sampling and InterpolationInterpolation
AliasedComponent
AliasedComponent
Pre-filteredSignal
-fs fs0
Non-idealInterpolationfilter
Aliasing Imaging
filter
Non ideal prefiltering causes AliasingNon ideal interpolation filter causes Imaging
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 15
p g g
Sampling a Sinusoidal Signal )1,2()1,2(
21),()24cos(),( vuvuvuFyxyxf
S l d t ∆ ∆ 1/3 f f 3Sampled at ∆x=∆y=1/3 fs,x=fs,y=3
v vOriginal Spectrum Sampled Spectrum
3(-2,1)
3(-2,1)
u
3
-3 3
(2,-1)
u
3
-3 3(2,-1) Ideal-3 (2, 1) -3
Original pulse Replicated pulse
interpolation filter
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 16
Replication center)22cos(),(ˆ yxyxf
Sampling in 2D: Sampling a 2D Sinusoidal PatternSampling a 2D Sinusoidal Pattern
f(x,y)=sin(2*π*(3x+y))Sampling: dx=0 01 dy=0 01 f(x,y)=sin(2*π*(3x+y))Sampling: dx=0.01,dy=0.01Satisfying Nyquist ratefx,max=3, fy,max=1fs,x=100>6, fs,y=100>2
Sampling: dx=0.2,dy=0.2(Displayed with pixel replication)Sampling at a rate lower than Nyquist rate
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 17
A Simple Prefilter – Averaging Filter• Each sampling value is the mean value of the
original continuous function in a rectangularoriginal continuous function in a rectangular region of dimension ∆x and ∆y, i.e:
dxdyyxfnmf )(1)(
n
nmDyx
s dxdyyxfyx
nmf,),(
),(),(
where
m
The equivalent prefilter is
ynyynxmxxmD nm )2/1()2/1(,)2/1()2/1(,
yvyv
xuxuyxvuH
otherwise
yyxxyxyxh
sinsin),(
0
2/||,2/||1),(
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 18
otherwise 0
How good is the averaging Filter?
• Look at its frequency response, how far is it from the ideal low pass filterit from the ideal low pass filter
Ideal low pass filter with cutoff at fs/2p fs
Averaging filter
f 2f 3f-f2f3f
Averaging filter
fs 2fs 3fsfs-2fs-3fs
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 19
Reconstruction of continuous image from Samplesfrom Samples
• The interpolation problem• Interpretation as weighted average of sample• Interpretation as weighted average of sample
values• Interpretation as filteringInterpretation as filtering• What filter should we use?
– Ideal interpolation filter: remove the replicatedIdeal interpolation filter: remove the replicated spectrum using ideal low-pass filter with cutoff frequency at fs/2 (the sinc function)
)()(sin
)()(sin
),(),(ˆ,
,
,
,
ynyfynyf
xmxfxmxf
nmfyxfys
ys
xs
xs
m ns
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 20
Interpretation as a weighted average of sample valuesof sample values
• The value of a function at arbitrary point (x, y) is estimated from a weighted sum of its sample values in the neighborhood of ([x/∆x],weighted sum of its sample values in the neighborhood of ([x/∆x], [y/∆y]):– Let h(x,y; m,n) specifies the weight assigned to sample m,n, when
determining the image value at x,y
nm
s nmfnmyxhyxf,
),(),;,(),(ˆ
H(x;8)
H(x;9)H(x;7)
x
H(x;10)
H(x;9)
H(x;6)
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 21
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 n
H(x;5)
Desirable Properties of the Weight FunctionFunction
• The weighting function h(x,y;m,n) should depend only on distance between (x,y) and the spatial location of (m,n), i ei.e.
Should be a decreasing function of the distance
).,(),;,( ynyxmxhnmyxh
• Should be a decreasing function of the distance– Higher weight for nearby samples
• Should be an even function of the distanceL ft i hb d i ht i hb f di t h th– Left neighbor and right neighbor of same distance have the same weight
– h1(x)=h1(-x)• Generally Separable:• Generally Separable:
– h2(x,y)=h1(x) h1(y) • To retain the original sample values, should have
h(0 0)=1 h(mx ny)=0
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 22
– h(0,0)=1, h(mx,ny)=0
Interpretation as Filtering• The weighted
average operation is ),(),(),(ˆ
~
,nmfynyxmxhyxf s
nm
equivalent to filtering fs(x,y) with h(x,y)
• Usually h(x y) is ),(),(
),(),(),(
~
,
yxfyxh
ynyxmxnmfyxf
s
nmss
• Usually, h(x,y) is separable h(x,y) = hx(x)hy(y) ),(),(),(
),(),(~
ddynxmnmfyxh
ddfyxh
s
s
• To retain the original sample values, should have )()(
),(),(),(,
,
hf
ddynxmyxhnmfnm
s
nm
should have– h(0,0)=1,
h(mx,ny)=0Nyquist filter
),(),(),(ˆ
),(),(
~
,
yxfyxhyxf
ynyxmxhnmf
s
nms
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 23
– Nyquist filter
A Simple Interpolation Filter: Sample-And-Hold (pixel replication)Sample And Hold (pixel replication)
• The interpolated value at a point is obtained from that of its nearest sampleobtained from that of its nearest sample
ynyynxmxxmnmfyxf s )2/1()2/1(,)2/1()2/1(),(),(ˆ
• Corresponding interpolation filter is
yyyxxx
yxh2/2/,2/2/1
),( n otherwise
y0
),(
m(m-1/2)∆x
m(m+1/2)∆x
(n 1/2)∆y
0th order interpolation filter
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 24
(n-1/2)∆y(n+1/2)∆y
Bilinear Filter
f0fa
f1(m,n) (m,n+1)
b
0 1aa
b
q1 q 2)1,1()1,()1()(
),1(),()1()(:1
2
1
Snmafnmfaqf
nmafnmfaqfStep
10)1( affafa
(m+1,n) (m+1,n+1)
pq1 q2
)()()1()(:2
21 qbfqfbpfStep
1D Linear interpolation
2D bilinear interpolation
Corresponding interpolation filter
yyyxxx
yy
xx
yxh ,||1||1),(
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 25
otherwise0
Which filter is better?
• Recall what happened in the frequency domain when we sample an imagedomain when we sample an image
• Ideal filter: ½ band ideal low pass filter• Quantitatively we can evaluate how far is
the filter from the ideal filter• But we should also look at visual artifacts
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 26
Frequency Domain Interpretation of Sampling in 2DSampling in 2D
• The sampled signal contains replicas of the original spectrum shifted by multiplesthe original spectrum shifted by multiples of sampling frequencies.
u uu u
fm x
fs,x
fs,x>2fm,x
vv
fm,x
fm,y fs,y
fs,y>2fm,y
Original spectrum F(u v) Sampled spectrum F (u v)
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 27
Original spectrum F(u,v) Sampled spectrum Fs(u,v)
Ideal Interpolation Filter• The ideal interpolation filter should be a low-pass filter
with cutoff frequency at fc,x = fs,x/2, fc,y = fs,y/2, with y ymagnitude ∆x∆y
yfxfyxh
fv
fuyxvuH ysxs
ysxs,,
,, sinsin)(2
||,2
||)(
yfxf
yxhotherwise
yvuHysxs ,,
),(0
2||
2||),(
The sinc filter
• The interpolated image
The sinc filter
Weight function h(x,y;m,n)
)()(sin
)()(sin
),(),(ˆ,
,
,
,
ynyfynyf
xmxfxmxf
nmfyxfys
ys
xs
xs
m ns
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 28
Comparison of Different Interpolation FiltersFilters
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 29
Image Resizing• Image resizing:
– Enlarge or reduce the image size (number of pixels)– Equivalent to
• First reconstruct the continuous image from samples• Then Resample the image at a different sampling ratep g p g
– Can be done w/o reconstructing the continuous image explicitly
• Image down-sampling (resample at a lower rate)Image down-sampling (resample at a lower rate)– Spatial domain view– Frequency domain view: need for prefilter
• Image up-sampling (resample at a higher rate)– Spatial domain view– Different interpolation filters
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 30
Different interpolation filters• Nearest neighbor, Bilinear, Bicubic
Image Down-Sampling
• Example: reduce a 512x512 image to 256x256 = factor– reduce a 512x512 image to 256x256 = factor of 2 downsampling in both horizontal and vertical directionsvertical directions
– In general, we can down-sample by an arbitrary factor in the horizontal and vertical ydirections
• How should we obtain the smaller image ?How should we obtain the smaller image ?
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 31
Down Sampling by a Factor of Two
8x8 Image 4x4 Image
• Without Pre-filtering (simple approach)Without Pre filtering (simple approach)
Averaging Filter
)2,2(),( nmfnmfd
• Averaging Filter4/)]12,12()2,12()12,2()2,2([),( nmfnmfnmfnmfnmfd
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 32
Problem of Simple Approach• Aliasing if the effective sampling rate is below
the Nyquist sample rate = 2 * highest frequencythe Nyquist sample rate 2 highest frequency in the original continuous signal
• We need to prefilter the signal before down-p gsampling
• Ideally the prefilter should be a low-pass filter with a cut-off frequency half of the new sampling rate.– In digital frequency of the original sampled image, the
cutoff frequency is ¼.• In practice we may use simple averaging filter
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 33
• In practice, we may use simple averaging filter
Down Sampling by a Factor of K
)(f̂Hs ↓K
f(m,n) fd(m,n)),( nmf
Pre-filtering Down-sampling
),(ˆ),( KnKmfnmfd
For factor of K down sampling, the prefilter should be low pass filter with cutoff at fs/(2K), if fs is the original sampling frequency
I t f di it l f th t ff h ld b 1/(2K)
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 34
In terms of digital frequency, the cutoff should be 1/(2K)
Example: Image Down-Sample
Without prefiltering
With prefiltering (no aliasing, but blurring!)
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 35
Down-Sampling Using Matlab• Without prefiltering
– If f(,) is an MxN image, down-sampling by a factor of K can be done simply by
>> g=f(1:K:M,1:K:N)
• With prefilteringp g– First convolve the image with a desired filter
• Low pass filter with digital cutoff frequency 1/(2K)– In matlab, 1/2 is normalized to 1In matlab, 1/2 is normalized to 1
– Then subsample>> h=fir1(N, 1/K)
%design a lowpass filter with cutoff at 1/K%design a lowpass filter with cutoff at 1/K.>> fp=conv2(f,h)>> g=fp(1:K:M,1:K:N)
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 36
Image Up-Sampling• Produce a larger image from a smaller one
– Eg. 512x512 -> 1024x1024M ll l b bit f t L– More generally we may up-sample by an arbitrary factor L
• Questions: – How should we generate a larger image?– Does the enlarged image carry more information?
• Connection with interpolation of a continuous image from discrete image– First interpolate to continuous image, then sampling at a higher sampling
rate, L*fs– Can be realized with the same interpolation filter, but only evaluate at
x=mx’ y=ny’ x’=x/L y’=y/Lx=mx , y=ny , x =x/L, y =y/L– Ideally using the sinc filter!
)(sin)(sin)()(ˆ ,, ymyfxmxf
ff ysxs
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 37
)()(),(),(
,
,
,
,
ymyfxmxfnmfyxf
ys
ys
xs
xs
m ns
Example: Factor of 2 Up-Sampling
(m,n+1)(m,n) (2m,2n) (2m,2n+1)
(2m+1 2n) (2m+1 2n+1)
(m+1,n+1)(m+1,n)
(2m+1,2n) (2m+1,2n+1)
Green samples are retained in the interpolated image;Orange samples are estimated from surrounding green samples.
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 38
Nearest Neighbor Interpolation(pixel replication)(pixel replication)
(m n+1)(m n)
a
b
(m,n+1)
(m’/M,n’/M)
(m,n)
a
(m+1,n+1)(m+1,n)
O[m’,n’] (the resized image) takes the value of the sample nearest to (m’/M,n’/M) in I[m,n] (the original image):
/Mn'=n/Mm'=m0 5)]+(n(int)0 5)+(mI[(int)=]n'O[m'
Also known as pixel replication: each original pixel is replaced by MxM pixels of
/M.nn/M,mm,0.5)]+(n(int)0.5),+(mI[(int) ]n,O[m
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 39
Also known as pixel replication: each original pixel is replaced by MxM pixels of the sample valueEquivalent to using the sample-and-hold interpolation filter.
Special Case: M=2
(m,n+1)(m,n) (2m,2n) (2m,2n+1)
(2m+1 2n) (2m+1 2n+1)
(m+1,n+1)(m+1,n)
(2m+1,2n) (2m+1,2n+1)
Nearest Neighbor:O[2m,2n]=I[m,n]O[2m,2n+1]= I[m,n]O[2m+1,2n]= I[m,n]
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 40
O[2m+1,2n+1]= I[m,n]
Bilinear Interpolation
(m,n) (m,n+1)
a
b
(m’/M,n’/M)
• O(m’,n’) takes a weighted average of 4 samples nearest to (m’/M,n’/M) in I(m,n).
(m+1,n) (m+1,n+1)
O(m ,n ) takes a weighted average of 4 samples nearest to (m /M,n /M) in I(m,n).
• Direct interpolation: each new sample takes 4 multiplications:O[m’,n’]=(1-a)*(1-b)*I[m,n]+a*(1-b)*I[m,n+1]+(1-a)*b*I[m+1,n]+a*b*I[m+1,n+1]
• Separable interpolation:i) interpolate along each row y: F[m,n’]=(1-a)*I[m,n]+a*I[m,n+1]ii) interpolate along each column x’: O[m’,n’]=(1-b)*F[m’,n]+b*F[m’+1,n]
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 41
Special Case: M=2
(m,n+1)(m,n) (2m,2n) (2m,2n+1)
(2m+1 2n) (2m+1 2n+1)
(m+1,n+1)(m+1,n)
(2m+1,2n) (2m+1,2n+1)
Bilinear Interpolation:O[2m,2n]=I[m,n]O[2m,2n+1]=(I[m,n]+I[m,n+1])/2O[2m+1,2n]=(I[m,n]+I[m+1,n])/2
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 42
O[2m+1,2n+1]=(I[m,n]+I[m,n+1]+I[m+1,n]+I[m+1,n+1])/4
Bicubic Interpolation
(m-1,n+1)(m-1,n)
b
(m,n) (m,n+1)(m,n-1) (m,n+2)
a
(m+1,n+1)(m+1,n)
(m’/M,n’/M)
(m+1,n-1) (m+1,n+2)
(m+2,n) (m+2,n+1)
O( ’ ’) i i t l t d f 16 l t t ( ’/M ’/M) i I( )• O(m’,n’) is interpolated from 16 samples nearest to (m’/M,n’/M) in I(m,n).• Direct interpolation: each new sample takes 16 multiplications• Separable interpolation:
i) interpolate along each row y: I[m,n]->F[m,n’] (from 4 samples)ii) i t l t l h l ’ F[ ’] O[ ’ ’] (f 4 l )
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 43
ii) interpolate along each column x’: F[m,n’]-> O[m’,n’] (from 4 samples)
Interpolation Formula
(m n) (m n+1)(m n 1) (m n+2)
(m-1,n+1)(m-1,n)(m-1,n-1) (m-1,n+2)
b
(m,n) (m,n+1)(m,n-1) (m,n+2)
F(m’,n-1) F(m’ n) F(m’ n+1) F(m’ n+2)a
(m+1,n+1)(m+1,n)
(m’/M,n’/M)
(m+1,n-1) (m+1,n+2)
(m+2,n) (m+2,n+1)
( , ) F(m ,n) F(m ,n+1) F(m ,n+2)
( , ) ( , )
(m+2,n-1) (m+2,n+2)
IbbIbbbIbbIbbF ]2[)1(]1[)1(][)21(]1[)1(]'[ 22322
mMmb
Mmmwhere
nmIbbnmIbbbnmIbbnmIbbnmF
','(int)
],,2[)1(],1[)1(],[)21(],1[)1(],'[ 22322
nnnmFaanmFaaanmFaanmFaanmO
''],2,'[)1(]1,'[)1(],'[)21(]1,'[)1(]','[ 22322
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 44
nMna
Mnnwhere ,(int)
Special Case: M=2
(2m,2n) (2m,2n+1)(m-1,n+1)(m-1,n)
(2m+1 2n) (2m+1 2n+1)
(m,n) (m,n+1)(m,n-1) (m,n+2)
(2m,2n+1) (2m+1,2n) (2m+1,2n+1)
(m+1,n+1)(m+1,n)(m+1,n-1) (m+1,n+2)
Bicubic interpolation in Horizontal direction
(m+2,n) (m+2,n+1)
Bicubic interpolation in Horizontal direction
F[2m,2n]=I[m,n]F[2m,2n+1]= -(1/8)I[m,n-1]+(5/8)I[m,n]+(5/8)I[m,n+1]-(1/8)I(m,n+2)
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 45
Same operation then repeats in vertical direction
Comparison of Interpolation MethodsInterpolation Methods
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 46
Resize_peak.m
Up-Sampled from w/o Prefiltering
NearestneighborOriginal
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 47
Bilinear Bicubic
Up-Sampled from with Prefiltering
NearestneighborOriginal
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 48
Bilinear Bicubic
Matlab for Image Resizing[img]=imread('fruit.jpg','jpg');
%downsampling without prefiltering
i 1 i i (i 0 5 ' ')img1=imresize(img,0.5,'nearest');
%upsampling with different filters:
img2rep=imresize(img1,2,'nearest');
i 2li i i (i 1 2 'bili ')img2lin=imresize(img1,2,'bilinear');
img2cubic=imresize(img1,2,'bicubic');
%down sampling with filtering%down sampling with filtering
img1=imresize(img,0.5,'bilinear',11);
%upsampling with different filters
img2rep=imresize(img1 2 'nearest');img2rep=imresize(img1,2, nearest );
img2lin=imresize(img1,2,'bilinear');
img2cubic=imresize(img1,2,'bicubic');
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 49
Filtering View: Up Sampling by a Factor of Kof K
)(~fHi↑K
f(m,n) fu(m,n)),( nmf
Zero-padding Post-filtering
Kfl lfKKf )//(
otherwise
KofmultiplearenmifKnKmfnmf
0,)/,/(
),(~
lk
u lnkmflkhnmf,
),(~),(),(
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 50
Ideally H should be a low pass filter with cutoff at 1/2K in digital frequency, or fs/2K in continuous frequency
Homework (1)1. Consider a 1D signal f(t) = sin(4πt). Illustrate the original and the sampled
signal f(n) obtained with a sampling interval ∆t = 1/3. Draw on the same figure the interpolated signal from the sampled one using the sample-and-hold and the linear interpolation filter respectively Explain theand-hold and the linear interpolation filter, respectively. Explain the observed phenomenon based on both the Nyquist sampling theorem as well as physical interpretation. What is the largest sampling interval that can be used to avoid aliasing?
2. Consider a function f(x, y) = cos2π(4x + 2y) sampled with a sampling2. Consider a function f(x, y) cos2π(4x 2y) sampled with a sampling period of ∆x = ∆y = ∆ = 1/6 or sampling frequency fs = 1/∆ = 6.
a) Assume that it is reconstructed with an ideal low-pass filter with cut-off frequency fcx = fcy = 1/2fs. Illustrate the spectra of the original, sampled, and reconstructed signals. Give the spatial domain function representation of the reconstructed signal Is the result as expected?reconstructed signal. Is the result as expected?
b) If the reconstruction filter has the following impulse response:
otherwise
yxyxh
02/,2/1
),(
Illustrate the spectra of the reconstructed signal in the range -fs ≤ u,v ≤ fs. Give a spatial domain function representation of the reconstructed signal if the reconstruction filter is band-limited to (-fs ≤ u,v ≤ fs). (i.e., this filter remains the
otherwise0
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 51
( s , s) ( ,same for the frequency range -fs ≤ u,v ≤ fs, and is set to 0 outsize this range.)
Homework(2)3. (Computer Assignment) Write your own program or programs
which can: a) Down sample an image by a factor of 2, with and without using the averaging filter; b) Up-sample the previously d l d i b f t f 2 i th i l li tidown-sampled images by a factor of 2, using the pixel replication and bilinear interpolation methods, respectively. You should have a total of 4 interpolated images, with different combination of down-sampling and interpolation methods. Your program could ith di tl di l th d i d ieither directly display on screen the processed images during
program execution, or save the processed images as computer files for display after program execution. Run your program with the image Barbara. Comment on the quality of the down/up
l d i bt i d ith diff t th dsampled images obtained with different methods.Note: you should not use the ”imresize” function in Matlab to do this assignment. But you are encouraged to compare results of your program with ”resize”.y p g
Yao Wang, NYU-Poly EL5123: Sampling and Resizing 52