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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.