1 fusion of brushlet and wavelet denoising methods for nuclear images elsa angelini 1, yinpeng jin...
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FUSION OF BRUSHLET AND WAVELET DENOISING METHODS FOR
NUCLEAR IMAGES
Elsa Angelini1, Yinpeng Jin1, Peter Esser2, R. Van Heertum2, Andrew Laine1
1 Department of Biomedical Engineering 2 Department of Radiology
Columbia University, New York, NY, USA
ISBI 2004 Washington, DC
April 17, 2004
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Sample PET and SPECT Images
SPECT Liver PET Brain
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Previous Work on Multi-Scale Processing of Previous Work on Multi-Scale Processing of PET and SPECTPET and SPECT
• Local reconstruction to improve spatial resolution within a region of interestLocal reconstruction to improve spatial resolution within a region of interest– T. Olson and J. De Stefano, "Wavelet Localization of the Radon Transform." IEEE Trans. Image
Processing, vol. 42, pp. 2055-2067, 1994.– F. Rashid-Farrokhi, K. Liu, C. Berenstein, and D. Walnut, "Wavelet-based Multiresolution Local
Tomography." IEEE Trans. Image Processing, vol. 22, pp. 1412-1430, 1997.– S. Zhao, G. Wang, and J. Hsieh, "Wavelet Sampling and Localization Schemes for the Radon
Transform in Two Dimensions." SIAM Journal on Applied Mathematics, vol. 57, pp. 1749-1762, 1997.– M. Bottema, B. Morean, and S. Suorova, "An Application of Wavelets in Tomography." Digital Signal
Processing, vol. 8, pp. 244-254, 1998.– W. Maldych, "Tomography, Approximate Reconstructions, and Continuous Wavelet Transforms."
Journal of Applied Computation and Harmonic Analysis, vol. 7, pp. 54-100, 1999.
• Accelerating implementation of the traditional FBP algorithmAccelerating implementation of the traditional FBP algorithm– A. Delaney and Y. Bresler, "Multi-resolution Tomographic Reconstruction Using Wavelets." IEEE Trans.
Image Processing, vol. 4, pp. 799-813, 1995.– L. Blanc-Feraud, P. Charbonnier, P. Lobel, and M. Barlaud, "A Fast Tomographic Reconstruction
Algorithm in the 2-D Wavelet Transform Domain." IEEE International Conference on Acoustics, Speech and Signal Processing, pp. 305-308, 1994.
• Post-filtering or regularization/constraints in tomographic reconstructionPost-filtering or regularization/constraints in tomographic reconstruction– E. Kolaczyk, "A Wavelet Shrinkage Approach to Tomographic Image Reconstruction." Journal of
American Statistics Association, vol. 91, pp. 1079-1090, 1996.– N. Lee and B. Lucier, "Wavelet Methods for Inverting the Radon Transform with Noisy Data." IEEE
Trans. Image Processing, vol. 10 (1), pp. 79-94, 2001.– J. Lin, A. Laine, and S. Bergmann, "Improving PET-based Methods Using the Wavelet Transform for
Positron Emission Tomography." IEEE Trans. Biomedical Engineering, vol. 48, pp. 202-212, 2001.– J. Kalifa, A. Laine, and P. Esser, "Regularization in Tomographic Reconstruction Using Thresholding
Estimators." IEEE Trans. Medical Imaging, vol. 22 (3), pp. 351-359, 2003.
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Image De-Noising in PET and SPECT
• Directional and Textural Noise
Edge-Based
De-noising (3D Wavelet Modulus Analysis)
Image Image FusionFusion
• Over-Smooth
Structure Edges
Texture-Based De-noising
(Brushlet Analysis)
SPECT PET
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Image De-Noising in PET and SPECT
• Directional and Textural Noise
Edge-Based
De-noising (3D Wavelet Modulus Analysis)
Image Fusion
• Over-Smoothed
Structure Edges
Texture-Based De-noising
(Brushlet Analysis)
SPECT PET
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Fourier Tiling to construct expansion basis
Brushlets Basis Functions
Expansion
Brushlet and Textural Analysis
Reconstruction
2D Analysis Function
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- Compact representation of textured signals.
- Adaptive tiling of frequency plane.
- Adaptive directional selectivity.
- Fast implementation with folding operators and FFT.
- Orthogonal basis.
Advantages of Brushlets Analysis
Brushlet and Textural Analysis
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De-noising with Brushlet Basis Functions (ISBI 02)
Isolate oriented textures Isolate oriented textures via thresholdingvia thresholding
frequency
Brushlet and Textural Analysis
- Minimax threshold level based on noise variance, estimated in the background.
- Spatial adaptivity of thresholding for 3 types of regions: texture, smooth, edges [Vetterli].
- De-noising via hard thresholding of low frequency coefficients.
Data
Mean
Regions
Map Variance
Noise
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Examples of Brushlet De-Noising: SPECT Brain Data
Brushlet and Textural AnalysisO
rigin
alD
enoi
sed
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Image De-Noising in PET and SPECT
• Directional and Textural Noise
Edge-Based
De-noising. (3D Wavelet Modulus Analysis)
Image Fusion
• Over-Smoothed
Structure Edges
Texture-Based De-noising
(Brushlet Analysis)
SPECT PET
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Edge-Based De-noising
• 3D Dyadic Wavelet Thresholding.– Feature selection based on spatial orientation of
contours in three dimensions.
• Cross-Scale Regularization (MICCAI’ 03)
– Explore correlations of signal features across spatial-frequency scales.
– Effective signal recovering from noise-dominated multi-scale expansions.
Wavelet and Edge De-noising
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3D Dyadic Wavelets and Wavelet Modulus
3D:3D: 1 2 3( , , )f n n n 1 2 31 2 3 1 2 3 1 2 3 1 2 3 [1, ]( , , ),{ ( , , ). ( , , ), ( , , )}M m m m m MS f n n n W f n n n W f n n n W f n n n
Wavelet Modulus in 3D:2 2 21 2 3
m m m mM f W f W f W f
Input Data
Wavelet Edge De-noising
DC
Wavelet Coefficients
m,1 m,2 m,3
N,1 N,2 N,3
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Traditional* Dyadic Wavelet Thresholding (3D)
DC
Threshold
Input Image
Enhanced(Denoised)
Image
Wavelet Decomposition Wavelet Reconstruction
Threshold
Threshold
Threshold
Threshold
Threshold
Wavelet Edge De-noising
* [Mallat 92]
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Dyadic Wavelet Modulus Thresholding (3D)
DC
Enhanced(Denoised)
Image
Wavelet Decomposition Wavelet Reconstruction
Modulus Thresholding
Modulus Thresholding
Wavelet Voxel De-noising
Input Image
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Cross-scale Regularization (CSR)
Input Image
-1 -0.5 0 0.5 1-5
-4
-3
-2
-1
0
1
2
3
4
5
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
+x
N
- “Pre-processing” of higher level sub-bands:
– De-correlation of noise in spatial-frequency expansion
- “Windowed” Normalization- Avoid attenuation of weak edges
- 50% Max rule (brain, liver data)
- 70% Max rule (bone data)
Wavelet Modulus atExpansion Levels 1 and 2
Wavelet Edge De-noising
Level 2Level 1
“Regularization Map”
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Comparison: CSR vs. Soft Threshold
Input Data CSR De-noising Soft Thresholding
Wavelet Edge De-noising
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Image De-Noising in PET and SPECTImage De-Noising in PET and SPECT
• Directional and Textural Noise.
Edge-Based
De-noising. (3D Wavelet Modulus Analysis)
Image Fusion
• Over-Smoothed Structure Edges.
Texture-Based De-noising
(Brushlet Analysis)
SPECT PET
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Multi-Scale Image Fusion
• Fusion: To combine different or incomplete To combine different or incomplete representations into a unified form with integrated representations into a unified form with integrated information.information.
• Motivation of fusion in the context of denoising:– Brushlet analysis provides better enhancement of “harmonic
textures”, representing physiological activities inside target organs.– Wavelet modulus thresholding provides better enhancement of
“anatomical edges”, or delineation of anatomical structures of clinical interest.
– Both types of information are important for accurate diagnostic decisions and image interpretation.
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Fusion Process
F1 F2 F3
A1 A2 A3
Wavelet Expansion
A B
B1 B3B2
Wavelet Expansion
FWavelet Reconstruction
Fusion Rule:FFii(x,y,z) = Max(A(x,y,z) = Max(Aii(x,y,z), B(x,y,z), Bii(x,y,z))(x,y,z))
Multi-Scale Image Fusion
Both [A] and [B] expanded and reconstructed with 3D Dyadic Transform
De-noised Data Sets
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Example: Fusion of coefficient features at the most detailed expansion level.
Wavelet Modulus De-Noising
Brushlet De-Noising Fused Image Features
Multi-Scale Image Fusion
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Input Data
Brushlet De-Noising
Wavelet Modulus De-Noising
Image Fusion Result
Example Cases: Fusion of denoised images
Multi-Scale Image Fusion
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Brushlet De-Noising
Wavelet Modulus De-Noising
For comparison:Reconstructed Using OSEM
Multi-Scale Image FusionExample: Fusion of denoised images
Input Data:Clinical PET Brain
Reconstructed Using FBP Image Fusion Result
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• Brushlet de-noising:Brushlet de-noising: – Beneficial for enhancing “harmonic activity”, e.g. anatomical or
physiological variations within the target organs.
• Wavelet modulus analysis with cross-scale regularization:Wavelet modulus analysis with cross-scale regularization:– Beneficial for enhancing “anatomical edges”, with a better definition
and delineation of the organ contours.
• Fused images:Fused images: – Effectively combined important features from both processed images,
without introducing artifacts. – When compared to OSEM reconstructions, provided significantly
improved image quality in terms of both lower noise level and improved contrast for key anatomical and physiological features.
Preliminary Clinical Evaluation
Multi-Scale Image Fusion
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Conclusion
• Multi-scale fusion of two expansions– Selected predominant wavelet coefficient modulus
from distinct de-noising expansions.– Effective integration of de-noising methods for
enhancement of anatomical and physiological features.
• Potential improvements of the method– Preservation of the linearity of the nuclear measures.– Refinement of fusion rule.
• Further evaluation studies– Clinical phantom data.– Clinical data with pathological ground truth.
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References• Y. Jin, E. Angelini, P. Esser, and A. Laine, "De-noising SPECT/PET images using cross-
scale regularization," MICCAI, pp. 32-40, Montreal, Canada, 2003.• F. Meyer and R. R. Coifman, "Brushlets: A tool for directional image analysis and image
compression," Applied and Computational Harmonic Analysis, vol. 4, No. 1, pp. 147-187, 1997.
• E. D. Angelini, J. Kalifa, and A. F. Laine, "Harmonic multiresolution estimators for denoising and regularization of SPECT-PET data," International Symposium on Biomedical Imaging, pp. 697-700, Washington, D.C., USA, 2002.
• S. Mallat and S. Zhong, "Signal characterization from multiscale edges," 10th International Conference on Pattern Recognition, pp. 891-896, Atlantic City, NJ, USA, 1990.
• E. Angelini, A. Laine, S. Takuma, J. Holmes, and S. Homma, "LV volume quantification via spatio-temporal analysis of real-time 3D echocardiography," IEEE Transactions on Medical Imaging, vol. 20, No. 6, pp. 457-469, 2001.
• S. G. Chang, B. Yu, and M. Vetterli, "Spatially adaptive wavelet thresholding with context modeling for image denoising," IEEE International Conference on Image Processing, pp. 535 -539, Chicago, IL, USA, 1998.
• S. G. Nikolov, D. R. Bull, C. N. Canagarajah, M. Halliwell, and P. N. T. Wells, "Fusion of 2-D images using their multiscale edges," IEEE International Conference on Pattern Recognition, pp. 41-44, Barcelona, Spain, 2000.
• I. Koren, A. Laine, and F. Taylor, "Image fusion using steerable dyadic wavelet transform," IEEE International Conference on Image Processing, pp. 232-235, Washington, D.C., USA, 1995.
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Acknowledgements
This study was supported in part by Siemens Medical Solutions, Inc.