image interpolation · introduction 3 what is image interpolation? an image f(x,y) tells us the...
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IMAGE INTERPOLATION Francesca Pizzorni Ferrarese
1
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Image Interpolation 2
Introduction What is image interpolation? (D-A conversion) Why do we need it?
Interpolation Techniques 1D zero-order, first-order, third-order 2D = two sequential 1D (divide-and-conquer) Directional(Adaptive) interpolation*
Interpolation Applications Digital zooming (resolution enhancement) Image inpainting (error concealment) Geometric transformations (where your
imagination can fly)
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Introduction 3
What is image interpolation? An image f(x,y) tells us the intensity values
at the integral lattice locations, i.e., when x and y are both integers
Image interpolation refers to the “guess” of intensity values at missing locations, i.e., x and y can be arbitrary
Note that it is just a guess (Note that all sensors have finite sampling distance)
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A Sentimental Comment 4
Haven’t we just learned from discrete sampling (A-D conversion)?
Yes, image interpolation is about D-A conversion
Recall the gap between biological vision and artificial vision systems Digital: camera + computer Analog: retina + brain
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Engineering Motivations 5
Why do we need image interpolation? We want BIG images
When we see a video clip on a PC, we like to see it in the full screen mode
We want GOOD images If some block of an image gets damaged during
the transmission, we want to repair it We want COOL images
Manipulate images digitally can render fancy artistic effects as we often see in movies
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Scenario I: Resolution Enhancement
6
Low-Res.
High-Res.
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Scenario II: Image Inpainting 7
Non-damaged Damaged
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Scenario III: Image Warping 8
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Image Interpolation 9
• Introduction – What is image interpolation? – Why do we need it?
• Interpolation Techniques – 1D linear interpolation (elementary algebra) – 2D = 2 sequential 1D (divide-and-conquer) – Directional(adaptive) interpolation*
• Interpolation Applications – Digital zooming (resolution enhancement) – Image inpainting (error concealment) – Geometric transformations
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Upsampling
This image is too small for this screen: How can we make it 10 9mes as big? Simplest approach: repeat each row and column 10 9mes
(“Nearest neighbor interpola9on”)
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Image interpola9on
Recall how a digital image is formed
• It is a discrete point-‐sampling of a con9nuous func9on • If we could somehow reconstruct the original func9on, any new
image could be generated, at any resolu9on and scale
1 2 3 4 5
Adapted from: S. Seitz
d = 1 in this example
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Image interpola9on
1 2 3 4 5
d = 1 in this example
Recall how a digital image is formed
• It is a discrete point-‐sampling of a con9nuous func9on • If we could somehow reconstruct the original func9on, any new
image could be generated, at any resolu9on and scale
Adapted from: S. Seitz
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Image interpola9on
1 2 3 4 5 2.5
1
• Convert to a con9nuous func9on:
• Reconstruct by convolu9on with a reconstruc)on filter, h
• What if we don’t know ? • Guess an approxima9on: • Can be done in a principled way: filtering
d = 1 in this example
Adapted from: S. Seitz
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“Ideal” reconstruc9on
Nearest-‐neighbor interpola9on
Linear interpola9on
Gaussian reconstruc9on
Source: B. Curless
Image interpolation
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Ideal reconstruction 15
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Ideal reconstruction 16
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Ideal reconstruction 17
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Nearest-‐neighbor interpola9on Bilinear interpola9on Bicubic interpola9on
Original image: x 10
Image interpolation
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1D Zero-order (Replication) 19
n f(n)
x
f(x)
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1D First-order Interpolation (Linear) 20
n f(n)
x
f(x)
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Linear Interpolation Formula 21
a 1-a
f(n)
f(n+1) f(n+a)
f(n+a)=(1-a)×f(n)+a×f(n+1), 0<a<1
Heuristic: the closer to a pixel, the higher weight is assigned Principle: line fitting to polynomial fitting (analytical formula)
Note: when a=0.5, we simply have the average of two
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Numerical Examples 22
f(n)=[0,120,180,120,0]
f(x)=[0,60,120,150,180,150,120,60,0], x=n/2
f(x)=[0,20,40,60,80,100,120,130,140,150,160,170,180,…], x=n/6
Interpolate at 1/2-pixel
Interpolate at 1/3-pixel
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1D Third-order Interpolation (Cubic)* 23
n
x
f(n)
f(x)
Cubic spline fitting
http://en.wikipedia.org/wiki/Spline_interpolation
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From 1D to 2D 24
• Engineers’ wisdom: divide and conquer • 2D interpolation can be decomposed into two sequential
1D interpolations. • The ordering does not matter (row-column = column-row) • Such separable implementation is not optimal but enjoys low
computational complexity “If you don’t know how to solve a problem, there must be a related but easier problem you know how to solve. See if you can reduce the problem to the easier one.” - rephrased from G. Polya’s “How to Solve It”
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Graphical Interpretation of Interpolation at Half-pel
25
row column
f(m,n) g(m,n)
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Numerical Examples 26
a b c d
a a b b a a b b c c d d c c d d
zero-order
first-order
a (a+b)/2 b (a+c)/2 (a+b+c+d)/4 (b+d)/2 c (c+d)/2 d
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Numerical Examples (Con’t) 27
X(m,n) row m
row m+1
Col n Col n+1
X(m,n+1)
X(m+1,n+1) X(m+1,n)
Y a 1-a
b
1-b
Q: what is the interpolated value at Y? Ans.: (1-a)(1-b)X(m,n)+(1-a)bX(m+1,n) +a(1-b)X(m,n+1)+abX(m+1,n+1)
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Bicubic Interpolation* 28
http://en.wikipedia.org/wiki/Bicubic_interpolation
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Limitation with bilinear/bicubic 29
Edge blurring Jagged artifacts
X Z
Jagged artifacts
X
Z
Edge blurring
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Edge-Sensitive Interpolation 30
Step 1: interpolate the missing pixels along the diagonal
black or white?
Step 2: interpolate the other half missing pixels
a b
c d
Since |a-c|=|b-d| x
x has equal probability of being black or white
a
b
c
d Since |a-c|>|b-d|
x=(b+d)/2=black
x
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Image Interpolation 31
• Introduction • Interpolation Techniques
– 1D zero-order, first-order, third-order – 2D zero-order, first-order, third-order – Directional interpolation*
• Interpolation Applications – Digital zooming (resolution enhancement) – Image inpainting (error concealment) – Geometric transformations (where your
imagination can fly)
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Pixel Replication 32
low-resolution image (100×100)
high-resolution image (400×400)
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Bilinear Interpolation 33
low-resolution image (100×100)
high-resolution image (400×400)
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Bicubic Interpolation 34
low-resolution image (100×100)
high-resolution image (400×400)
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Edge-Directed Interpolation (Li&Orchard’2000)
35
low-resolution image (100×100)
high-resolution image (400×400)
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Image Demosaicing (Color-Filter-Array Interpolation)
36
Bayer Pattern
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Image Example 37
Ad-hoc CFA Interpolation Advanced CFA Interpolation
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Error Concealment* 38
damaged interpolated
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Image Inpainting* 39
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Image Mosaicing* 40
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Geometric Transformation 41
MATLAB functions: griddata, interp2, maketform, imtransform
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Basic Principle 42
(x,y) → (x’,y’) is a geometric transformation
We are given pixel values at (x,y) and want to interpolate the unknown values at (x’,y’)
Usually (x’,y’) are not integers and therefore we can use linear interpolation to guess their values
MATLAB implementation: z’=interp2(x,y,z,x’,y’,method);
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Rotation 43
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡
−=⎥
⎦
⎤⎢⎣
⎡
yx
yx
θθ
θθ
cossinsincos
''
x
y
x’
y’
θ
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MATLAB Example 44
% original coordinates [x,y]=meshgrid(1:256,1:256);
z=imread('cameraman.tif');
% new coordinates a=2; for i=1:256;for j=1:256; x1(i,j)=a*x(i,j); y1(i,j=y(i,j)/a; end;end % Do the interpolation z1=interp2(x,y,z,x1,y1,'cubic');
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Rotation Example 45
θ=3o
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Scale 46
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
yx
aa
yx
/100
''
a=1/2
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Affine Transform 47
⎥⎦
⎤⎢⎣
⎡+⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
y
x
dd
yx
aaaa
yx
2221
1211
''
square parallelogram
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Affine Transform Example 48
⎥⎦
⎤⎢⎣
⎡+⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡
−=⎥
⎦
⎤⎢⎣
⎡
10
25.15.
''
yx
yx
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Shear 49
⎥⎦
⎤⎢⎣
⎡+⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
y
x
dd
yx
syx
101
''
square parallelogram
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Shear Example 50
⎥⎦
⎤⎢⎣
⎡+⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
10
15.01
''
yx
yx
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Projective Transform 51
1'
87
321
++
++=
yaxaayaxa
x
1'
87
654
++
++=
yaxaayaxay
quadrilateral square
A B
C D
A’
B’
C’
D’
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Projective Transform Example 52
[ 0 0; 1 0; 1 1; 0 1] [-4 2; -8 -3; -3 -5; 6 3]
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Polar Transform 53
22 yxr +=
xy1tan−=θ
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Iris Image Unwrapping 54
r
θ
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Use Your Imagination 55
r -> sqrt(r)
http://astronomy.swin.edu.au/~pbourke/projection/imagewarp/ �
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Free Form Deformation 56
Seung-Yong Lee et al., “Image Metamorphosis Using Snakes and Free-Form Deformations,”SIGGRAPH’1985, Pages 439-448
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Application into Image Metamorphosis
57
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Summary of Image Interpolation
58
A fundamental tool in digital processing of images: bridging the continuous world and the discrete world
Wide applications from consumer electronics to biomedical imaging
Remains a hot topic after the IT bubbles break