Tallin, November 2007
Introduction to Selected Concepts
of Image Processing
Jiří Jan
Department of Biomedical Engineering
Brno University of Technology
Czech Republic
Tallin, November 2007
Overview of the lectures
I.
Images as multidimensional signals
•
•
II.
analogue nD signals and systems, 2D FT
digital 2D images and operators, 2D DFT
Image enhancement
•
•
•
III.
contrast transforms
sharpening
noise smoothing
Reconstruction of images from tomographic projections
•
parallel projections
–
–
–
•
algebraic methods
slice theorem based reconstruction
filtered back projection
fan projections
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I. Images as multidimensional signals
• continuous representation of images (both in space and in values)
= analogue 2D, 3D or 4D image
• discrete representation of images (both in space and in values)
= digital image data (2D, 3D, 4D)
Analogue (continuous) images
Images as signals
2D image (with scalar values)
f ( x, y )
x ∈ − xmax , xmax ,
y ∈ − ymax, ymax
– different
representations
– profiles
2
Images with scalar pixel values:
f ( x, y )
- grey-scale
- black-and-white (binary)
Images with vector pixel values, e.g. :
f ( x, y ) = f re (x, y ) + j f im ( x, y )
– 2D image with complex values
– colour image in e.g. RGB components
f ( x, y ) = [ f R ( x, y ), f G (x, y ), f B ( x, y )]
T
f (x )
Multidimensional images
x = [x,y,z]
x = [x,y,t ]
x = [x,y,z,t ]
e.g. 3D
or 4D
Some important 2D elementary signals
2D Dirac impulse (df.)
while
δ ( x, y ) =
∞
∞ pro x = 0 ∧ y = 0
0 pro x ≠ 0 ∨ y = 0
∞
∫ ∫ δ (x, y )dxdy = 1
−∞ −∞
Approximation by a function sequence
δ m (x, y ) = m 2 rect(mx, my )
rect(x,y ) =
1 for x ≤ 0.5, y ≤ 0.5
0
otherwise
lim δ m ( x, y ) = δ ( x, y )
m →∞
3
The most important properties of the 2D Dirac impulse (δ-distribution):
• filtering property of the δ-distribution
∞
∫ ∫
∞
−∞ −∞
f ( x, y )δ ( x − ξ , y − η ) dxdy = f (ξ ,η )
• shift to arbitrary position
δ ( x − ξ , y − η)
• expression via the 2D Fourier transform
δ ( x, y ) =
Harmonic 2D signal (e.g. image)
∞
1
4π
2
∫ ∫
∞
−∞ −∞
e − j (ux + vy )du dv
f ( x, y ) = A cos(u 0 x + v 0 y + ϕ )
- frequencies in directions x, or y:
u, or v
- absolute spatial frequency
w0 = u 02 + v 02
- spatial period
P = 2π
u 02 + v 02
- direction of the „wave“
⎛ v0
⎝ u0
ϑ = arctan⎜⎜
⎞
⎟⎟
⎠
- phase shift of the „wave“
ϕ
- spatial shift of the first ridge
d=P
ϕ
2π
=
ϕ
w
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2D systems – description in the original domain
Definition (input-output df., i.e. as a mapping, described by an operator):
g ( x, y ) = P{ f (x, y )}
This can easily be generalised to multidimensional case (3D, 4D)
Classification of 2D systems according to different criteria:
linearity
- linear systems – principle of superposition applies
⎧
⎫
P ⎨∑ ai f i ( x, y )⎬ = ∑ aiP{ f i (x, y )}
⎩í
⎭ i
- nonlinear systems – principle of superposition does not apply
spatial extent of the operator influence
- point-wise operators
- local operators
- global operators
variability of the operator in space:
- spatially invariant operators - isoplanar
- spatially variable operators (e.g. adaptive) - anisoplanar
isotropy:
- isotropic operators
- anisotropic operators (e.g. possibly directional)
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2D linear systems – mathematical description in the original domain
f ( x, y ) =
∞
∫ ∫
∞
f (s, t )δ (x − s, y − t ) ds dt
−∞ −∞
g ( x, y ) = P
g ( x, y ) = ∫
{∫
∞
∫
∞
−∞ −∞
∞
f (s, t )δ ( x − s, y − t ) ds dt
}
∞
∫ P{ f (s, t )δ (x − s, y − t ) }ds dt
−∞ −∞
=∫
∞
∞
∫ f (s, t )P{δ (x − s, y − t ) }ds dt
−∞ −∞
impulse response (point-spread-function PSF)
(generally space-variable)
h( x, y, s, t ) = P{δ ( x − s, y − t ) }
output of the system – the superposition integral
g ( x, y ) = ∫
∞
∞
∫ h(x, y, s, t ) f (s, t )ds dt
−∞ −∞
physical interpretation of the superposition integral
example: an optical system
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special case: the
isoplanar (space-invariant) system
P{δ ( x − s, y − t ) } = h( x − s, y − t )
output: the convolutional integral
g ( x, y ) = ∫
∞
∞
∫ h(x − s, y − t ) f (s, t )ds dt
−∞ −∞
= h ( x, y ) * f ( x , y ) = h * f ( x, y )
2D Fourier transform
Definition of the forward transform:
F (u , v ) = FT2D { f (x, y )} = ∫
∞
∫
∞
−∞ −∞
f ( x, y ) e − j (ux + vy )dx dy
as a generalisation of the 1D FT:
F1 (u , y ) = ∫
∞
−∞
f ( x, y ) e − jux dx
∞
F2 (u, v ) = ∫ F1 (u, y ) e − jvy dx =
−∞
∫ ⎡⎢⎣∫
∞
∞
−∞
−∞
f ( x, y ) e − jux dx ⎤ e − jvy dy = FT2D { f ( x, y )}
⎥⎦
sketch of coordinates in the original and spectral domain
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Examples: - spectrum of a square
F (u , v ) = FT2D {rect(x, y )} =
sin u / 2 sin v / 2
= sinc(u / 2, v / 2)
u/2
v/2
- spectrum of a rectangle
.
Example 2: - spectrum of a natural
image
original image
its spectrum
amplitude spectrum
phase spectrum
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Example 3: - spectrum of the 2D Dirac impulse
non-existent but can be defined as the limit case of the sequence:
Fm (u, v ) = m 2
thus
1
sinc(u/ 2m, v / 2m )
m2
FT2D {δ (x, y )} = lim Fm (u, v ) = 1
m →∞
Definition of the 2D inverse Fourier transform
f ( x, y ) = FT -1{F (u , v )} = FT -1{FT{ f ( x, y )}}
It can be derived that:
f ( x, y ) = FT -1{F (u, v )} =
∞
1
4π
2
∫ ∫
∞
−∞ −∞
F (u, v ) e j (ux + vy )du dv
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Physical interpretation of the 2D FT
Explanation on the image formed by a single concrete harmonic component:
[
f ( x, y ) = A cos(u 0 x + v0 y + ϕ ) =
1
A e j (u0 x +v0 y +ϕ ) + e − j (u0 x+v0 y +ϕ )
2
]
for the first term, the spectrum is given by a shifted δ-impulse, as visible from the inverse FT:
FT −1{F1 (u , v )} =
∞ ∞
1
4π
ϕ
(
∫ ∫ [2 Aπ δ (u − u , v − v )e ] e
j ux + vy )
j
2
2
0
0
dudv =
− ∞− ∞
A j (u0 x + v0 y +ϕ )
e
2
(similarly for the second term)
thus the spectrum of the complete harmonic signal is
F (u, v ) = FT{A cos(u 0 x + v 0 y + ϕ )} = F1 (u, v ) + F2 (u , v ) =
[
2 Aπ 2 δ (u − u 0 , v − v 0 )e jϕ + δ (u + u 0 , v + v 0 )e − jϕ
original image (cropped)
]
and its spectrum (cropped)
schematic sketch (magnified)
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The spectrum of a single real harmonic component is symmetrical complex conjugate.
As a natural image can be considered an infinite set of harmonic components, its
spectrum can be understood as an infinite set of such pairs of impulses.
Consequences:
• to a rotation of an image, the same rotation of its spectrum corresponds
• the spectrum must have the same symmetry as its components, should the image
be real-valued:
F (u , v ) = F * (− u ,−v )
Basic properties of the 2D FT
- linearity
1 ⎛u v⎞
F⎜ , ⎟
ab ⎝ a b ⎠
- change-of-scale rule
FT{ f (ax, by )} =
- shift rule
FT{ f (x − a, y − b )} = F (u, v )e − j (ua + vb )
{
}
FT f (x, y )e − j (u0 x +v0 y ) = F (u − u0 , v − v0 )
- convolutional property
FT{ f ( x, y )* g ( x, y )} = F (u, v )G(u, v )
FT{ f ( x, y ) g ( x, y )} =
- Parseval’s theorem
∞ ∞
∫∫
1
4π 2
f ( x, y ) g * ( x, y ) dxdy =
− ∞− ∞
F (u, v )* G (u, v )
∞ ∞
∫ ∫ F (u, v ) G (u, v )dudv
*
−∞−∞
→ total energy preservation law
∞ ∞
∫∫
− ∞− ∞
f ( x, y ) dxdy =
2
∞ ∞
∫ ∫ F (u, v )
2
dudv
− ∞− ∞
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2D systems – description in the frequency domain
g ( x, y ) = ∫
∞
∞
∫ h(x − s, y − t ) f (s, t )ds dt = h * f (x, y )
−∞ −∞
G (u, v ) = FT{h * f ( x, y )}
=∫
∞
∫ ⎡⎢⎣∫ ∫ h(x − s, y − t ) f (s, t ) ds dt ⎤⎥⎦ e
∞
−∞ −∞
G (u, v ) = ∫
∞
∞
∞
∫ f (s, t )⎡⎢⎣∫ ∫ h(x − s, y − t )e
∞
−∞ −∞
G (u , v ) = ∫
∞
−∞ −∞
∫
∞
−∞ −∞
∞
∞
− ∞ −∞
− j (ux + vy )
− j (ux + vy )
dx dy
dx dy ⎤ ds dt
⎥⎦
f (s, t ) H (u , v ) e − j (us + vt ) ds dt
H(u,v)=FT{h(x,y)} – frequency response
G (u , v ) = H (u , v ) F (u , v )
(or frequency transfer)
of the (space-invariant) system
Alternative possibility of calculating the output of the system:
F (u , v ) = FT2D { f ( x, y )}
-1
{H (u, v ) F (u, v )}
g ( x, y ) = FT2D
examples of image processing by linear space-invariant systems (2D filters)
original image
filter responses
processed
images
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(cont. of the examples)
original spectrum
frequency response
frequency response
(cont. of the examples)
frequency response
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Stochastic images as realisations of a stochastic field
Basic concepts:
(generalisation of 1D case
of stochastic process)
– family of functions
– associated experiment
– local stochastic variables
– local characteristics
– stochastic field
(2D stochastic process)
– correlation between two
and more points
– correlation functions
– homogeneous (stationary) fields
– ergodic fields
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