Filtering Objective –improve SNR and CNR Challenges – blurs object boundaries and smears out...
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Transcript of Filtering Objective –improve SNR and CNR Challenges – blurs object boundaries and smears out...
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Filtering• Objective
– improve SNR and CNR • Challenges
– blurs object boundaries and smears out important structures
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Averaging using rotating mask• Consider each image pixel (i,j). Calculate dispersion in the mask
for all possible mask rotations about (i,j).• Choose the mask with minimum dispersion.• Assign to the pixel (i,j) in the output image the average
brightness in the chosen mask.
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Anisotropic diffusive filtering• An iterative process in which intensity diffusion V
takes place between adjacent spels in a nonlinear fashion with gradients F as follows:
V = GF, where G is diffusion conductance
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Diffusion
• A diffusion process can be defined using the divergence operator “div” on a vector field. A mathematical formulation of the diffusion process on a vector field V at a point c
0div lim
s
f dt
V V s
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• D(c,d): the unit vector for c toward d
2 2
0
( ) ( )( , ) ( , )( )
l ll n
i i ii
f c f dc d c dc d
F D
• Intensity gradient at l-th iteration:
Diffusion flow: GV F: Nonlinear diffusion conductanceG
Mathematical formulation
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• Diffusion conductance function:2
2( , )2( , )
l c d
lG c d e
F
• Diffusion flow
( , ) ( , ) ( , )l l lc d c d G c dV F
Continued…
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• Iterative diffusion process
1 1 1( )
( ), if l 0,( ) ( ) ( , ), otherwise.l l d l
d N c
f cf c f c K c d
V
• Diffusion constant1min( )d c C
KN c
• Diffusion constant used here1517
, in 2D,, in 3D.dK
Continued …
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Examples
Original MR image VOI from original image Anisotropic diffusion
MS 2D
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Examples
Original MR image VOI from the original image Anisotropic diffusion
MS 3D
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Use of structure scale in diffusive filtering
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Examples
Original MR image VOI from original image Anisotropic diffusion
MS 2D
Scale-based anisotropic
diffusion
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Examples
Original MR image VOI from the original image Anisotropic diffusion
MS 3D Scale-based
anisotropic diffusion
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Results
original
s-scale t-scale
aniso. diff.
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Zoomed Display
original
s-scale t-scale
aniso. diff.
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Canny’s edge detection• The detection criterion expresses the fact that
important edges should not be missed and that there should be no spurious responses
• The localization criterion minimizes the distance between the actual and the located edge position
• The response criterion minimizes multiple response to a single edge
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Scale in edge detection• Scale is a resolution or a range of resolution
needed to provide a sufficient yet compact representation of the object or a target information
• In a Gaussian smoothing or edge detection kernel the parameter σ resembles with scale
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Canny’s edge localization• It seeks out zero-crossings of
• In one-dimension a closed form solution may be found using calculus of variation.
• In two or higher dimension, the best solution is obtained by a numerical optimization, called non-maximal suppression, that essentially seeks for the best solution for
2 2( * ) / / (( * ) ) /G f G f nn n G n n
2 2( * ) / 0G f n
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Non-maximal suppression1. Quantize edge directions eight ways according to 8-
connectivity2. For each pixel with non zero edge magnitude, inspect the
two adjacent pixels indicated by the direction of its edge3. If the magnitude of either of these two exceeds that of the
pixel under inspection, mark it for deletion4. When all pixels have been inspected, re-scan the image
and erase to zero all edge data marked for deletion
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Examples
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Hysteresis to filter output of an edge operator
1. Mark all edges with magnitude greater than tH as correct
2. Scan all pixels with edge magnitude in the range [tL,tH]
3. If such a pixel is adjacent to another already marked as an edge, then mark it too.
4. Repeat from step 2 until stability
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Examples
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Canny’s edge detector1. Convolve an image f with a Gaussian of scale σ2. Localize edge points using the Non-Maximal
Suppression algorithm3. Compute edge magnitude of the edge at each locations4. Apply the Hysteresis algorithm to filter edge locations
eliminating spurious responses5. Repeat steps (1) through (5) for ascending values of
scales σ of a range [σmin, σmax]
6. Aggregate the edge information at different scale using feature synthesis
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Examples
LoG DoG
Roberts Sobel Prewitt
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Examples
σ=1 σ=2 σ=3
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Examples
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Parametric edge detection
• Facet model: a piecewise continuous function representing the intensity in the neighborhood of a pixel, e.g., a bi-cubic faced model
21 2 3 4 5
2 3 2 2 36 7 8 9 10
( , )
f i j c c x c y c x c xy
c y c x c x y c xy c y
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Parametric edge detection• Model parameters may be computed using a least-
squares method with singular value decomposition• Facet model is computationally expensive but give
more accurate localization of edges with sub-pixel accuracy
• Haralick and Shairo shown that for a 5x5 facet model, the parameters may be directly computed using ten 5x5 kernels