3D and 4D Adaptive Filtering for Image Denoising and Patient Safety

The aim of this medical image science project is to increase patient safety in terms of improved image quality and reduced exposure to ionizing radiation in CT. The means to achieve these goals is to develop and evaluate an efficient adaptive filtering (denoising/image enhancement) method that fully explores true 3D and 4D image acquisition modes. Four-dimensional (4D) medical image data are captured as a time sequence of image volumes. During 4D image acquisition, a 3D image of the patient is recorded at regular time intervals. The resulting data will consequently have three spatial dimensions and one temporal dimension. Increasing the dimensionality of the data impose a major increase the computational demands. The initial linear filtering which is the cornerstone in all adaptive image enhancement algorithms increase exponentially with the dimensionality. On the other hand the potential gain in Signal to Noise Ratio (SNR) also increase exponentially with the dimensionality. This means that the same gain in noise reduction that can be attained by performing the adaptive filtering in 3D as opposed to 2D can be expected to occur once more by moving from 3D to 4D.

• Staff:

Hans Knutsson , Prof		Project Manager, Supervisor		Medical Informatics
Mats Andersson , PhD		Operative project leader, Co-supervisor		Medical Informatics
Michael Sandborg , Ass Prof		Radiation Physicist		Radiation Physics
Gunnar Farnebäck , PhD		Senior Scientist		ContextVision AB
Örjan Smedby , Prof, MD		Radiologist		Radiology

• Former Staff:

• Project Description:

  Preliminary results

















a) A 2D slice of the synthetic 4D test image. b) Test image after degradation by 4D noise. c) Result after 2D adaptive filtering. d) Result after 3D adaptive filtering. e) Result after 4D adaptive filtering, see text for details



A synthetic 4D image is used to illustrate the power of adaptive filtering with respect to dimensionality.  To the left in the above figure a 2D image out of the synthetic 4D volume is shown. The size of the 4D volume is 127 x 127 x 9 x 9 and the 4D test image has no signal variation in the last two dimensions. There are three events that can be observed in this image: a large step close to the diagonal, a thin white line a little bit to the right of the step and finally a shading from the upper left corner to the lower right corner. All these events share the same orientation and the adaptive filtering parameters can be set globally. Finally the test image is degraded with a massive amount of 4D additive noise, (second figure).

Next a 3D volume of size 127 x 127 x 9 and a 2D image of size 127 x 127 is cropped out from the original 4D volume. For these 4D, 3D and 2D images an adaptive filtering is performed using the same filter parameters except for the dimensionality.  The third figure shows the result after 2D adaptive filtering. A massive amount of noise remains but some faint traces of the edge and the shading can be discerned. The last two figures show the result for 3D and 4D adaptive filtering. In 3D the noise level is significantly reduced but the white line is not discernible. In the 4D case the thin white line which has a magnitude of a 1/4 compared to the edge is clearly visible. The events in this images varies in just one dimension (as for most images independent of dimensionality). The power of 3D and 4D image enhancement is to explore the difference in dimensionality between the a local image event and the dimensionality of the image.






Filter Networks




Additionally, more advanced signal processing operations such as adaptive filtering involve more than one filter either used within the algoritmh or used as a pre-processing to increase performance. As an example a single linear filter is traditionally applied to a small signal neighborhood, say of size Nⁿ when applied to an n-dimensional signal. Thus the number of multiplications required is Nn per data sample, i.e. for a typical image size of 512 x 512 pixels, a typical filter of size 11x11 yields approximatley 3.2·10⁷ multiplications.


Unfortunately, both signal and filter size grows exponentially with signal dimension n, i.e. carrying out the corresponding operation in 3-D requires 1.8·10¹¹ multiplications. Assuming that the computational time is linear with the number of multiplications required, this operation takes 5.6·10³ longer. On a modern computer such an operation takes approximately an hour. Evidently, the practical use of multi-dimensional filtering is severely limited by the computational burden.

 As the name implies filter networks for fast convolution is a structure of filters, designed for efficient computation of a set of multi-dimensional filters. The efficiency is due to decomposition of multi-dimensional filter sets into a structure of smaller sparse filters called sub-filters. The structure used constrained to a directed acyclic graph allows the sub-filter to contribute to several filters in the set. The use of filter networks involves non-trivial design, i.e. choosing the network structure and optimizing each sub-filter. The design problem associated with filter networks is described and solutions found has been implemented for extracting features like signal orientation, local frequency and local phase from volumetric data.


Filter networks has many potential applications and the primary target in this project is local structure analysis, involving features like local frequency, phase, bandwidth and anisotropy. The implemented filter networks shows a computational gain of a factor exceeding 50 for estimation of local 3-D structure compared to standard convolution. As a proof of concept showing use in medical applications, filter networks for enhancement of medical 3-D data have been developed.

GPU-acceleration


Today a large variety of methods and hardware solutions are developed that have the potential to increase the performance of multidimensional image enhancement. Some of the the most promising are multi-core processors, image processing on Graphics Processing Units (GPU) and Linux based PC clusters. Note however that the filter network approach is highly compatible with all of these methods and the computational gain of the filter network structure can be attained together with all of the above implementations.




The project is supported by VR - The Swedish Research Council (Vetenskapsrådet, diarienr 2008-3813  ) 




Publications

A Multidimensional Filtering Framework with Applications to Local Structure Analysis and Image Enhancement by B. Svensson, PhD Thesis, Linköping University, Sweden 2008. [Abstract]



Fast Multi-dimensional Filter Networks, Design, Optimization and Implementation by B. Svensson, Licentiate Thesis, Linköping University, Sweden, April 2006. [Abstract]

Efficient 3-D Adaptive Filtering for Medical Image Enhancement by B. Svensson, M. Andersson, Ö. Smedby and H. Knutsson, Proceedings of the {ISBI} IEEE International Symposium on Biomedical Imaging, Arlington, USA, April 2006. [Abstract]

Radiation Dose Reduction by Efficient 3D Image Restoration by B. Svensson, M. Andersson, Ö. Smedby and H. Knutsson, Proceedings of the {ECR} European Congress on Radiology, Vienna, Austria, March 2006. [Abstract]

Sparse Approximation for FIR Filter Design by B. Svensson, M. Andersson and H. Knutsson, Proceedings of the {SSBA} Symposium on Image Analysis, Umeå, Sweden, March 2006. [Abstract]

Performance Analysis of Filternets by H. Einarsson, Master Thesis, Linköping University, Sweden, February 2006. [Abstract]

Filter Networks for Efficient Estimation of Local 3D Structure by B. Svensson, M. Andersson and H. Knutsson, Proceedings of the Proceedings of the IEEE-ICIP, Genoa, Italy, September 2005. [Abstract]

Implications of Invariance and Uncertainty for Local Structure Analysis Filter Sets by H. Knutsson and M. Andersson, Signal Processing: Image Communications, July 2005. [PDF]

A Graph Representation of Filter Networks by B. Svensson, M. Andersson and H. Knutsson, Proceedings of the 14th Scandinavian conference on image analysis (SCIA'05), Joensuu, Finland, June 2005. [Abstract]

Design of Fast Multidimensional Filters Using Genetic Algorithms by M. Langer and B. Svensson and A. Brun and M. Andersson and H. Knutsson, Proceedings of EvoIASP, 7th European Workshop on Evolutionary Computing in Image Analysis and Signal Processing, Lausanne, Schweiz, March 2005. [PDF]

Design of Fast Multidimensional Filters by Genetic Algorithms by M. Langer, Master Thesis, Linköping University, Sweden, November 2004. [PDF]

Issues on Filter Networks for Efficient Convolution by B. Svensson, M. Andersson, J. Wiklund and H. Knutsson, Proceedings of the {SSBA} Symposium on Image Analysis, Uppsala, Sweden, March 2004. [PDF]


 

Related Articles

What's So Good About Quadrature Filters? by H. Knutsson and M. Andersson, Proceedings of the {ICIP} IEEE International Conference on Image Processing, Barcelona, Spain, September 2003. [PDF]
 
Loglets: Generalized Quadrature and Phase using Spherical Harmonics by H. Knutsson and M. Andersson, Proceedings of the {SCIA} Scandinavian Conference on Image Analysis, Göteborg, Sweden, June 2003.  [PDF]