TRANSFORM METHODS in IMAGE PROCESSING: …yaro/RecentPublications/ps&pdf...TRANSFORM METHODS in...
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TRANSFORM METHODSin IMAGE PROCESSING:
L. Yaroslavsky,Dept. of Interdisciplinary Studies, Faculty of Engineering,
Tel Aviv University, Tel Aviv, Israel
Image Restoration
Target Location
Image Resampling
TRANSFORM DOMAIN ADAPTIVEFILTERS FOR IMAGE RESTORATION
• Sliding Window DCT Filters
• Wavelet shrinkage
• Hybrid SWDCT/Wavelet filters
• Comparison and Interpretation
Sliding window transform domain (SWTD) filtersR. Yu. Vitkus, L.P. Yaroslavsky, Recursive Algorithms for Local Adaptive Linear Filtration, In: Mathematical Research.Computer Analysis of Images and Patterns, ed. by L.P. Yaroslavsky, A. Rosenfeld, W. Wilhelmi, Band 40, Academie Verlag,Berlin, 1987, p. 34-39.
Filters for image deblurring and denoising (signal independent noise):
====
≠≠≠≠
−−−−
====
00
0022
r
rrrroptr
,
,AV/,thrAVmax
λλλλ
λλλλββββλλλλββββηηηη
Filters for image deblurring and denoising (signal dependent noise):
(((( ))))
====
≠≠≠≠
⋅⋅⋅⋅−−−−
====
00
002
02
r
rrrp
roptr
,
,AV/,thrAVmax
λλλλ
λλλλββββλλλλββββββββηηηη
Rejective filters:
>>>>
====otherwise,
thrAVif,/ rroptr
0
12
ββββλλλληηηη .
“Fractional” spectrum filter: 11
0
121
0
−−−−====
>>>>
====
====−−−− N,,r,
otherwise,
thrAVif, rp
rr
ββββββββηηηη
ηηηη
ADAPTIVE FILTERS WITH EMPIRICALESTIMATION OF SPECTRA
The selection of the transform for the filter implementation is governed by• A priory knowledge about image spectra in the transform domain• Accuracy of empirical spectrum estimation• Transform energy compaction capability• Computational complexity of the filtering in the transform domain
Feasible transforms: DFT, DCT, DST, Haar, Walsh, wavelets.
SELECTION OF THE TRANSFORM
2-D basis functions of (left to right) DCT, Walsh, Haar Transforms
Odds in favor of DCT:
☻Good energy compaction capability
☻Good suitability for signal/image restoration taskswith signaling and imaging system specification interms of their frequency responses
☻Suitability for multi component signal/imageprocessing
☻Low computational complexity: recursivecomputing SWDCT is possible with the complexityof O(number of coefficients required)
Sliding window DCT (odd window size).L.P. Yaroslavsky, Local adaptive image restoration and enhancement with the use of DFT and DCT in a running window,Proceedings, Wavelet Applications in Signal and Image Processing IV, 6-9 August 1996, Denver, Colorado, SPIE Proc.Series, v. 2825, pp. 1-13.
Direct transform:
(((( ))))(((( ))))
(((( ))))∑∑∑∑−−−−
====−−−−++++
++++====
1
02
2121 wN
n/Nfixnk
w
kr r
N/ncosa
Nππππαααα
Inverse transform for the window central pixel:
(((( )))) (((( ))))(((( ))))
++++++++==== ∑∑∑∑
−−−−
====
1
10
212221 wN
r
kr
kk N
r//NfixcosN
a ππππαααααααα
For odd N:
(((( ))))(((( ))))(((( ))))
−−−−++++==== ∑∑∑∑−−−−
====
21
120 12
21
/N
r
rkr
k
wk
w
Na αααααααα
Therefore, only DCT coefficients with even indices are relevant
Sliding window DCT: recursive computation
For adjacent k-th and (k+1)-th window positions, DCT spectra (((( )))){{{{ }}}}krαααα and (((( )))){{{{ }}}}1++++k
rαααα are
(((( ))))(((( ))))
(((( ))))∑∑∑∑−−−−
====−−−−++++
++++====
1
02
2121 wN
n/Nwfixnk
w
kr r
N/ncosa
Nππππαααα
and
(((( ))))(((( ))))
(((( ))))∑∑∑∑−−−−
====−−−−++++++++
++++
++++====
1
021
1 21w
w
N
n/Nfixnk
kr r
N/ncosa ππππαααα .
Introduce auxiliary spectra
(((( ))))(((( ))))
(((( ))))∑∑∑∑−−−−
====−−−−++++
++++====
1
02
21w
w
N
n/Nfixnk
kr r
N/niexpa~ ππππαααα
Spectrum (((( )))){{{{ }}}}1++++kr
~αααα can be represented through spectrum (((( )))){{{{ }}}}kr
~αααα :(((( )))) (((( )))) (((( )))) (((( ))))(((( )))) (((( ))))[[[[ ]]]] (((( ))))N/riexpaaN/riexp~~
/Nfixkr
/NfixNkwk
rk
r www21 22
1 ππππππππαααααααα −−−−−−−−−−−−++++−−−−==== −−−−−−−−++++++++ .
Therefore, axiliary spectra (((( )))){{{{ }}}}kr
~αααα can be computed recursively and local DCT spectra (((( )))){{{{ }}}}krαααα can
then be found from the relationship:(((( )))) (((( ))))(((( ))))k
rk
r~real αααααααα ==== .
SWTD DCT filtering for image restorationL.P. Yaroslavsky, Local adaptive image restoration and enhancement with the use of DFT and DCT in a running window(invited paper), in: Proceedings, Wavelet Applications in Signal and Image Processing IV, 6-9 August 1996, Denver,Colorado, SPIE Proc. Series, v. 2825, pp. 1-13
SWTD DCT filtering for image blind deblurring
SWTD DCT filtering speckle noiseLeonid P. Yaroslavsky, Ben-Zion Shaick Transform Oriented Image ProcessingTechnology for Quantitative Analysis ofFetal Movements in Ultrasound Image Sequences. In: Signal Processing IX. Theories and Applications, Proceedings ofEusipco-98, Rhodes, Greece, 8-11 Sept., 1998, ed. By S. Theodorisdis, I. Pitas, A. Stouraitis, N. Kalouptsidis, TyporamaEditions, 1998, p. 1745-1748
Sliding Window DCT 3x3
Involved transform basis functions:
====
22022000
DCTDCTDCTDCT
−−−−−−−−−−−−
−−−−
−−−−−−−−−−−−
−−−−−−−−−−−−
121242
121
111222
111121121121
111111111
;
Basis functions 20DCT , 02DCT , and 22DCT are kernels of 2-D Laplacians:horizontal, vertical and isotropic ones. One can further decompose basisfunction 22DCT into a sum of two diagonal Laplacians:
−−−−−−−−−−−−−−−−
−−−−−−−−++++
−−−−−−−−−−−−−−−−−−−−−−−−
====112121
211
211121112
22DCT
SWTD DCT3x3: Restoration of high resolutionsatellite images L. Yaroslavsky, High Resolution Satellite Image Restoration with the Use of Local Adaptive Linear Filters, Report onKeshet Program, July, University Dauphine, Ceremade, Paris,1997
Spot-image: before Spot-image: after
Potentials of SWTD DCT filtering
Initial noise levelTest image15 30 60
Rej. Fltr 6.1 8.1 10.8Emp. Wiener 6 8.1 10.8Av.Rej.Fltr 5 7.1 10.4
Aero1
Id.Wiener 3.6 5.6 9.2Rej. Fltr 5.6 6.4 10Emp. Wiener 5.6 6.4 10.4Av.Rej.Fltr 5 6 8.2
Aero2
Id.Wiener 3.3 5 8.2Rej. Fltr 10.4 15 21Emp. Wiener 9.3 14.1 19Av.Rej.Fltr 8 11.9 17.5
Lenna
Id.Wiener 5.8 8.4 13
Standard. deviation of residual noise for 3 test noisy images
(R. Oktem, L. Yaroslavsky, K. Egiazarian, Evaluation of Potentials for Local Transform Domain Filters with varyingParameters, EUSIPCO2000, Tampere, Finland, Sept. 3-8, 2000)
Wavelet Shrinkage
Low passfiltering and
downsamplingInterpolation
-
+ + Soft/hardthresholding
Interpolation
+
+ +
+
+ +
InputOutput
Interpolation
-
+
+
Interpolation
Low passfiltering and
downsampling
Low passfiltering and
downsampling
Interpolation Interpolation
Soft/hardthresholding
Soft/hardthresholding
D. L. Donoho, I. M. Johnstone, Ideal Spatial Adaptation by Wavelet Shrinkage, Biometrica, 81(3), pp. 425-455, 1994
Wavelet Shrinkage and Empirical Wiener Filtering
Empirical Wiener Denoising Filter
(((( ))))2
220
r
rroptr
,max
ββββ
ννννββββηηηη
−−−−====
Rejective filter:
>>>>====
otherwise,thrif, ropt
r0
12
ββββηηηη .
Wavelet Shrinkage Filter (soft threshold)
(((( ))))r
roptr
,thrmaxββββ
ββββηηηη
0−−−−====
Wavelet Shrinkage Filter (hard threshold)
>>>>====
otherwise,thrif, ropt
r 01 ββββηηηη .
Wavelet vs SWDCT denoising: comparisonR. Oktem, L. Yaroslavsky, K. Egiazarian, Signal and Image Denoising in Trandform domain and Wavelet Shrinkage: AComparative Study, In: In: Signal Processing IX. Theories and Applications, Proceedings of Eusipco-98, Rhodes, Greece,8-11 Sept., 1998, ed. By S. Theodorisdis, I. Pitas, A. Stouraitis, N. Kalouptsidis, Typorama Editions, 1998, p. 2269-2272
Signals ImagesFilteringmethod ECG Piece-wise
constantLena Piece-
wisecomst
antimage
Aerophot
o 1
Aerophot
o 2
MRI
RMSE MAE RMSE MAE PSNR
PSNR
PSNR
PSNR
PSNR
WL-Haar 0.06 0.042 0.042 0.024 26.72 30.0 26.9 32.3 30.9
WL-Db4 0.052 0.038 0.052 0.035 26.2 27.8 27.6 33.3 31.5
SWDCT 0.04 0.028 0.055 0.038 28 27.8 29 34.6 32.7
SWHaar 0.052 0.033 0.037 0.022 27.4 29.5 28.1 34.2 31.8
Hybrid WaveLet-SWTD DCT filtering
scale 1
scale 2
scale 3
scale 4
Effective basis functions in multi resolution DCT
Low passfiltering and
downsamplingInterpolation
-
+ + SWDCT3x3filtering
Interpolation
+
+ +
+
+ +
InputOutput
Interpolation
-
+
+
Interpolation
Low passfiltering and
downsampling
Low passfiltering and
downsampling
Interpolation Interpolation
SWDCT3x3filtering
SWDCT3x3filtering
B.Z. Shaick, L. Ridel, L. Yaroslavsky, A hybrid transform method for image denoising, EUSIPCO2000, Tampere, Finland,Sept. 5-8, 2000
SWTD-DCT, Wavelet-Shrinkage andHybrid filtering: Performance comparison
MWTD-DCT, Wavelet-Shrinkage andHybrid filtering: Performance Comparison
FilterP-W
const.image
Lennaimage MRI Air photo
Hard 8.1(3×3)
9.4(3×3)
6.7(5 ×5)
8.2(3 ×3)DCT
Soft 7.5(3×3)
8.6(5×5)
6.3(7 ×7)
7.7(3 ×3)
Hard 8.6(binom5)
10.1(binom5)
8.5(binom5)
9.3(binom5)WL-
Shrinkage. Soft 8.4
(binom5)9.0
(qmf13)7.8
(binom5)8.1
(binom5)
Hard 8.7(binom5)
9.4(binom5)
6.6(binom5)
8.2(binom5)Hyb
rid Soft 7.9(binom5)
8.6(binom5)
6.2(binom5)
7.5(binom5)
Standard. deviation of residual noise for 4 test noisy images(initial St. Dev. 13; optimized filter parameters)
SWTD “time-frequency” signal representationand subband decomposition
Signal transform {{{{ }}}}ka ( 110 −−−−==== N,,,k ) in a sliding window of width wN over set of basis functions (((( )))){{{{ }}}}nrττττ ( 110110 −−−−====−−−−==== ww N,,,r;N,,,n ),
(((( ))))(((( )))) (((( ))))∑∑∑∑
−−−−
====−−−−++++====
1
02
w
w
N
nr/Nfixnk
kr na τττταααα
Its DFT over index k: (((( )))) (((( )))) ====
====ΑΤΑΤΑΤΑΤ ∑∑∑∑
−−−−
====
1
02exp1 N
k
kr
r
Nkfi
Nfππππαααα (((( )))) (((( ))))∑∑∑∑ ∑∑∑∑
−−−−
====
−−−−
====−−−−++++
1
0
1
02/ 2exp1 w
w
N
n
N
kNfixnkr N
kfianN
ππππττττ
Then:
(((( )))) (((( ))))rf
rff
ΤΤΤΤΑΑΑΑ∝∝∝∝ΑΤΑΤΑΤΑΤ ,
where ∑∑∑∑−−−−
====
====ΑΑΑΑ
1
02exp1 N
kkf N
kfiaN
ππππ ;
(((( )))) (((( ))))∑∑∑∑−−−−
====
−−−−
====ΤΤΤΤ1
02exp1 N
nr
w
r fNnin
Nnrect
Nfππππττττ ,
SWDCT7x7 subbands
Map of standard deviations of DCT7x7subbands
0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
0 . 7
0 . 8
0 . 9
1
0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
0 . 7
0 . 8
0 . 9
1
1 . 1
0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1
0 . 2
0 . 4
0 . 6
0 . 8
1
1 . 2
1 . 4
1 . 6
0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 10
1
2
3
4
5
SWDCT7x7 subbands (1-D slice)
Frequencies within the bandwidth
SWTD DCT, Walsh and Haar: subband decompositions
Subbands (1-D) of Binom5 wavelets (left)and hybrid Binom5&SWDCT3x3 (right)
TARGET LOCATION IN CLUTTER:Optimal Adaptive Correlators
and Transform Methods
Problem formulation
S1
S2
S3
S4
S5
S6
S7Object(x0,y0)
(((( ))))××××==== ∫∫∫∫ ∑∑∑∑ ∫∫∫∫∫∫∫∫∞∞∞∞
∞∞∞∞−−−− ====
K
10000 ,)(
k Skka
k
yxwWdbbpP
(((( ))))(((( )))) 00000
,/ dydxdbyxbhb
s
∫∫∫∫∞∞∞∞
bgAVAVims
(((( )))) 1, 0000 ====∫∫∫∫∫∫∫∫ dydxyxwkS
k ; 11
====∑∑∑∑====
K
kkW
Block diagram of the localization device
Linear filter Device for locating
signal maximum
Input image
Object’s coordinates
L.P. Yaroslavsky, The Theory of Optimal Methods for Localization of Objects in Pictures, In: Progress in Optics, Ed. E.Wolf, v.XXXII, Elsevier Science Publishers, Amsterdam, 1993
Optimal Adaptive Correlators:
Additive model:(((( )))) (((( )))) (((( ))))yxayyxxayxb bg ,,, 00 ++++−−−−−−−−====
(((( )))) (((( )))) (((( ))))αααα ββββ ααααbg x y x y x yf f f f f f, , ,2 2 2
≅≅≅≅ ++++
(((( ))))====yxaopt ffH ,
(((( ))))(((( )))) (((( ))))22
,,
,
yxyx
yx
ffff
ff
ααααββββ
αααα
++++
∗∗∗∗
Implant model
≅≅≅≅),( yximplopt ffH
(((( ))))(((( )))) (((( ))))∫∫∫∫ ∫∫∫∫ −−−−−−−−
≅≅≅≅∗∗∗∗
x yF Fyxyxyyxx
yx
dpdppppfpf
ff22
,,
,
ββββ
αααα
bgW
(((( )))) (((( )))) (((( ))))yxbyyxxwyxa bgbg ,,, 00 −−−−−−−−====
(((( ))))====−−−−−−−−
objecttargetthewithin,0objecttargettheoutside,1
, 00 yyxxwbgIn the assumption of uniformdistribution of target co-ordinates:
⇓⇓⇓⇓⇓⇓⇓⇓
In the assumption of uniformdistribution of target co-ordinates:
(((( )))) (((( )))) (((( ))))∫∫∫∫ ∫∫∫∫ −−−−−−−−≅≅≅≅x yF F
yxyxyyxxyxbg dpdppppfpfff222
,,, ββββαααα bgW
Zero order approximation: imgob SS <<<<<<<<
(((( )))) (((( ))))22, ,,
00 yxyxbgyx ffffAV ββββαααα ≅≅≅≅ (((( )))) (((( ))))(((( ))))2
0
,
,,
yx
yxyxopt
pp
ffffH
ββββ
αααα∗∗∗∗
====
Optimal adaptive correlators: detection of“small” objects
Detection and enhancement of microcalcifications in a mammogram
Input mammogram
Optimal adaptive filter and matched filtercorrelators: sensitivity to object rotation
5 0 1 0 0 1 5 0 2 0 0 2 5 0
0
0 . 2
0 . 4
0 . 6
0 . 8
1
5 0 1 0 0 1 5 0 2 0 0 2 5 0
0
0 . 2
0 . 4
0 . 6
0 . 8
1
5 0 1 0 0 1 5 0 2 0 0 2 5 0
0 . 4
0 . 5
0 . 6
0 . 7
0 . 8
0 . 9
1
5 0 1 0 0 1 5 0 2 0 0 2 5 0
0 .6
0 .7
0 .8
0 .9
1
Upper row Left column
Optimal adaptive correlator
Matched filter correlator
Object tracking in video sequencies: examples
Ultrasound movie: fetus movements Video movie: infant movements
For details seehttp://www.eng.tau.ac.il/~yaro
Leonid P. Yaroslavsky, Ben-Zion Shaick Transform Oriented Image Processing Technology for Quantitative Analysis of Fetal Movements in UltrasoundImage Sequences. In: Signal Processing IX. Theories and Applications, Proceedings of Eusipco-98, Rhodes, Greece, 8-11 Sept., 1998, ed. By S.Theodorisdis, I. Pitas, A. Stouraitis, N. Kalouptsidis, Typorama Editions, 1998, p. 1745-1748
Optimal localization in spatial inhomogeneousimages:Local adaptive correlator(L. Yaroslavsky, Local Adaptive Filtering in Transform Domain for Image Restoration, Enhancement and Target Location, in: 6th
Int. Workshop on Digital Image Processing and Computer Graphics (DIP-97), Em. Wenger and L. Dimitrov, eds., 20-22 Oct.,1997, Vienna, Austria, SPIE vol. 3346
T a rg e t (h ig h lig h te d )T h e re s u lt o f lo c a liz a tio n (m a rk e d w ith a c ro s s )
5 0 1 0 0 1 5 0 2 0 0 2 5 0
-3 0
-2 0
-1 0
0
1 0
2 0
3 0
4 0
5 0
C o rre la to r´s o u tp u t c ro s s s e c tio n (c o lu m n )
2 0 4 0 6 0 8 0 1 0 0 1 2 0
-3 0
-2 0
-1 0
0
1 0
2 0
3 0
4 0
5 0
C o rre la to r´s o u tp u t c ro s s s e c tio n (ro w )
Localization of a small (8x8 pixels, highlighted) fragment in stereoscopic images bythe local adaptive filter. Running window size is 32x32 pixels; image size is 128x256pixels. The same filter applied globally rather than locally fails to locate the fragment.
Nonlinear optical correlators: 4F k-th lawcorrelator(L.P. Yaroslavsky, Optical correlators with (-k)th law nonlinearity: optimal and suboptimal solutions, Applied Optics, v. 34,No. 20 (10 July, 1995), pp. 3924-3932)
Input image Fourier lens Fourier lensSpatial filter Correlation plane
Coherentillumination
F F F F
k-th law nonlinearityF(.)=(.)k
Nonlinear optical correlators:Joint Transform Correlators(L. Yaroslavsky, E. Marom, Nonlinearity Optimization in Nonlinear Joint Transform Correlators , Applied Optics , vol. 36,No. 20, 10 July, 1997, pp. 4816-4822)
Inputimage
Referenceobject
TVcamera
Nonlinear
Amplifier
f(.)=log(.);
f(.)=(.)1/k
SLM
interface
Spatial lightmodulator
Spatial lightmodulator
Fourierlens
Fourierlens
Outputplane
Color correlators: 3-channel correlatorwith component conversion(L. Yaroslavsky, Optimal target location in color and multi component images, Asian Journal of Physics, Vol. 8,No 3 (1999) 355-369)
Discrimination capability of the colorcorrelator with component conversion
FAST SIGNALSINC-INTERPOLATION
Image resampling: basic principle
OutputimageInput
image
Sinc-interpolation: zero padding method
Interpolation functions:
(((( ))))(((( ))))∑∑∑∑−−−−
====
====1
01
1 N
k1kk Lk-kN;M;sincda
La~
1;
(((( )))) (((( ))))(((( ))))N/xsinN
N/KxsinxN;K;sincdππππ
ππππ==== ; 110 −−−−==== LN,...,,k
Shifted Discrete Fourier Transforms as discreterepresentations of the Fourier integral:
x
a(x) Sampled signal
x∆∆∆∆u
f
(((( ))))fααααSampled signal spectrum
f∆∆∆∆
v
(((( )))) (((( ))))(((( ))))xukxaxaN
kreconstr_signk ∆∆∆∆++++−−−−==== ∑∑∑∑
−−−−
====
1
0ϕϕϕϕ
(((( )))) (((( ))))(((( ))))fvrffN
kreconstr_spnr ∆∆∆∆++++−−−−==== ∑∑∑∑
−−−−
====
1
0ϕϕϕϕαααααααα
Continuos signal
Continuous signalspectrum
(((( )))) (((( )))) (((( ))))dxfxiexpxaf ∫∫∫∫∞∞∞∞
∞∞∞∞−−−−
==== ππππαααα 2 (((( ))))(((( ))))∑∑∑∑−−−−
====
++++++++====
1
021 N
kk
v,ur N
vrukiexpaN
ππππαααα
Fourier integral Shifted DFT (canonic form)Signal and spectrum sampling
(((( ))))∑∑∑∑−−−−
====
++++
====
1
0221 N
kk
v,ur N
rukiexpNkviexpa
Nππππππππαααα (((( ))))∑∑∑∑
−−−−
====
++++
−−−−====
1
0221 N
kk
v,ur N
vrkiexpNruiexpa
Nππππππππαααα
Direct and inverse Shifted DFTs (reduced form)
L.P. Yaroslavsky, Shifted Discrete Fourier Transforms, In: Digital Signal Processing, Ed. by V. Cappellini, and A. G.Constantinides, Avademic Press, London, 1980, p. 69- 74.
Block diagram of signal fast sinc-interpolationL. P. Yaroslavsky, Signal sinc-interpolation: a fast computer algorithm, Bioimaging, 4, p. 225-231, 1996
SDFT versus zero padding sinc-interpolation
Zero padding method SDFT based method
Computational complexity (generaloperations) of L-fold zooming signal of Nsamples with the use of FFT
(((( ))))NLlogNLO (((( ))))NlogNLO
Computational complexity (generaloperations) of L-fold zooming signal of Nsamples in the vicinity of an individualsample (as, for instance, in locating positionof signal maximum with subpixel accuracy )
(((( ))))NLlogNLO ,unless FFT pruned algorithms are used
(((( ))))NLO
Computational complexity (generaloperations) for signal shift by a fraction ofthe discretization interval
(((( ))))NLlogNLO ,unless FFT pruned algorithms are used;shift only by (power of 2)-th fraction ofthe discretization interval are possiblewhen the most wide spread FFTalgorithms are used.
(((( ))))NlogNO ;arbitrary shifts arepossible
Zoom factor Power of 2 for the most widely usedFFT algorithms
Arbitrary
Memory usage Requires an intermediate buffer for NLsamples
Does not need anintermediate buffer
Spectrum analysis with sub-pixel resolution
Three pass image rotation with sinc-interpolation
Initia l image Firs t pass
Second pass Third pass: rotated image
M. Unser, P. Thevenaz, L. Yaroslavsky, Convolution-based Interpolation for Fast,High-Quality Rotation of Images, IEEE Trans. on Image Processing, Oct. 1995, v. 4, No. 10, p. 1371-1382
Fast image rotation algorithm in RadonTransform and tomosynthesis
Radon transform: rotationand directional summation
Tomographicreconstruction: ramp-filtering projections, backprojecting, rotation andsummation
Image geometrical transformations by means ofsinc-interpolated image zooming (oversampling)
Sinc-interpolatedsubsampling
Initial image
Magnified image
Transformed image copies
Image geometrical transformations with sinc-interpolation
Radius
Angle
Cartesianto polar
Polar toCartesian
Signal sinc-interpolation in sliding window
1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0
- 0 . 1
0
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
0 . 7
0 . 8
0 . 9
1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0
- 0 . 1
0
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
0 . 7
0 . 8
0 . 9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
3.5
Window11 Window15
Interpolation functions for window size 11 and 15 samples Interpolation filter frequency responses
Image zoom: global versus sliding window sinc-interpolation
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