## Noise-Adaptive Nonlinear Diffusion Filtering of MR Images With Spatially Varying Noise Levels (2004)

Venue: | Magnetic Resonance in Medicine |

Citations: | 8 - 0 self |

### BibTeX

@INPROCEEDINGS{Samsonov04noise-adaptivenonlinear,

author = {Alexei A. Samsonov and Chris R. Johnson},

title = {Noise-Adaptive Nonlinear Diffusion Filtering of MR Images With Spatially Varying Noise Levels},

booktitle = {Magnetic Resonance in Medicine},

year = {2004},

pages = {52--798}

}

### OpenURL

### Abstract

Anisotropic diffusion filtering is widely used for MR image enhancement. However, the anisotropic filter is nonoptimal for MR images with spatially varying noise levels, such as images reconstructed from sensitivity-encoded data and intensity inhomogeneity-corrected images. In this work, a new method for filtering MR images with spatially varying noise levels is presented. In the new method, a priori information regarding the image noise level spatial distribution is utilized for the local adjustment of the anisotropic diffusion filter. Our new method was validated and compared with the standard filter on simulated and real MRI data. The noise-adaptive method was demonstrated to outperform the standard anisotropic diffusion filter in both image error reduction and image signal-to-noise ratio (SNR) improvement. The method was also applied to inhomogeneity-corrected and sensitivity encoding (SENSE) images.

### Citations

1274 | Scale-space and edge detection using anisotropic diffusion
- Perona, Malik
- 1990
(Show Context)
Citation Context ...n anisotropic diffusion filter (19). The filtering problem is formulated as an estimation of a piecewiseconstant image function from noisy data, using a robust error norm �: min I �� ����I�, � e� d�, =-=[2]-=- where � e is a parameter bounding the values of the possible outliers. In MR images, these outliers correspond to image gradients generated by tissue boundaries. The formulation is equivalent to the ... |

201 | Anisotropic diffusion in image processing
- Weickert, Teubner
- 2012
(Show Context)
Citation Context ...le to preserve the boundaries of anatomical structures of small contrast in MR images. Too small a value of k, however, would result in the preservation and enhancement of highnoise-related gradients =-=(20)-=-. Figure 1 demonstrates the effect of optimal and suboptimal choices of k on the result of the anisotropic diffusion filtering. The optimal anisotropic diffusion filter should diminish the enhancement... |

78 |
SENSE: sensitivity encoding for fast MRI. Magn Reson Med
- KP, Weiger, et al.
- 1999
(Show Context)
Citation Context ...e outline methods for obtaining noise maps for SENSE and inhomogeneitycorrected images. FIG. 3. Four-point discretization scheme for the diffusion equation approximation. �t� �t� �Imp � g��Imp, kmp�, =-=[9]-=- where �(m) is the discretization neighborhood of pixel m, (t) �Imp is an finite difference approximation of the derivative of the image function, and �t establishes the diffusion rate. The range of a... |

62 |
Manning WJ. Simultaneous acquisition of spatial harmonics (SMASH): fast imaging with radiofrequency coil arrays. Magn Reson Med
- DK
- 1997
(Show Context)
Citation Context ...ws: �t� kmp� SENSE Image Noise The properties of noise in SENSE images reconstructed from Cartesian data (Cartesian SENSE) are described by an image noise matrix given by (9): X � 1 (S nk H��1S) �1 , =-=[10]-=-sNoise-Adaptive Nonlinear Diffusion Filtering 801 FIG. 4. Simulation study data. a: Digital brain phantom b: Noise map. where S is the sensitivity matrix, n k is the number of sampling positions in k-... |

42 |
Pernus F. Retrospective correction of MR intensity inhomogeneity by information minimization
- Likar, MA
(Show Context)
Citation Context ...x, k)�x � g( x, k) is zero: ���x, c�e�� � 0. [6] �x x��e For example, for the exponential diffusivity function (2) c is �2. This yields g���I�, k� � exp�����I�/k� 2 �, [7] k mn � �2 � �� m 2 � �n 2 . =-=[8]-=- This result agrees with the choice of the conductance parameter in Ref. 5 for filtering the MR images with spatially uniform noise levels. In that work, k for the exponential diffusion function was e... |

41 |
Generalized autocalibrating partially parallel acquisitions (GRAPPA). Magn Reson Med 2002;47:1202–1210
- MA, PM, et al.
(Show Context)
Citation Context ...plicative image intensity correction. The procedure creates images with spatially varying noise levels. The spatial distribution of noise variances in the corrected images is given by: N R � G 2 �r�. =-=[14]-=- Again, the noise distribution should be scaled (Eq. [13]) to reflect the overall noise level. Data For validation, we used a realistic digital phantom of the human brain that is available on the Inte... |

31 |
The Rician distribution of noisy MRI data. Magnetic resonance in medicine 1995;34(6):910. of captions
- Gudbjartsson, Patz
(Show Context)
Citation Context ... boundaries. The formulation is equivalent to the Perona-Malik filter under a gradient descent minimization of Eq. [2], if the diffusivity function is chosen as g�x, k� � 1 x ���x, �e� , �x k � c� e, =-=[3]-=- where c is a scaling coefficient. Gerig et al. (5) demonstrated that the filter is highly efficient for magnitude-reconstructed MR images. They pointed out that k should be kept as small as possible ... |

31 |
Advances in sensitivity encoding with arbitrary k-space trajectories. Magnetic Resonance in Medicine
- Pruessmann, Weiger, et al.
- 2001
(Show Context)
Citation Context .... The application of the proposed method to SENSE images reconstructed from arbitrary trajectories is challenging due to the nonavailability of the image noise matrix in feasible iterative approaches =-=(28)-=-. One solution is a simulation-based noise map determination. Another application of the new filter might be to produce the image estimate needed for feedback reconstruction techniques (16,17). The de... |

21 | The NMR phased array. Magn Reson Med - PB, WA, et al. - 1990 |

17 |
Kikinis R, Jolesz FA: Nonlinear anisotropic filtering of MRI data
- Gerig, Kubler
- 1992
(Show Context)
Citation Context ...of such outliers in the robust error norm as the population standard deviation (SD) �e��mn. Correspondingly, the local conductance parameter is found from Eqs. [3] and [4]: k mn � c � �� m 2 � �n 2 . =-=[5]-=- FIG. 2. Illustration of the robust statistical analysis of the anisotropic diffusion filter.s800 Samsonov and Johnson Parameter k is now dependent on the local noise properties. Image gradients that ... |

13 |
CF, Bao SM, Mulkern RV, Jolesz FA. Sensitivity profiles from an array of coils for encoding and reconstruction
- WE, LP, et al.
(Show Context)
Citation Context ...ing to �, find the noise map N R,� in the area, and normalize I � as I � norm � I�/�N R,�. [12] norm 3) Find the SD �n of �I� �, and use it to obtain the normalization constant: � � � n 2 /�2 � �/2�. =-=[13]-=- Here the denominator accounts for a change in the variance of pixels in the air background due to the magnitude reconstruction (22). Noise in Inhomogeneity-Corrected Images Intensity inhomogeneity co... |

13 |
Nonlinear diffusion for image filtering and monotonicity enhancement [PhD thesis
- Mrazek
- 2001
(Show Context)
Citation Context ...ecting a sufficient number of signal-free samples (with radiofrequency pulses turned off) along with data acquisition. This new method could possibly be extended to the filtering of image derivatives =-=(29)-=-, which could be beneficial for images that strongly deviate from the piecewise constant (slowly varying) model (i.e., images obtained from surface coil images by a sum-of-squares approach). The princ... |

11 |
Fully automatic segmentation of the brain
- Atkins, Mackiewich
- 1998
(Show Context)
Citation Context ...iltering (Eq. [2]). If pixels m and n belong to the same tissue type, the distribution of the difference � mn is zeromean Gaussian (Fig. 2): e Difference �mn � mn � N�0, � mn�, � mn � �� m 2 � �n 2 . =-=[4]-=- across an intensity discontinuity, such as a tissue boundary, may be considered an outlier in the dise tribution (�mn�N(0, �mn)) because it is formed by values from various populations (Fig. 2). We c... |

11 |
A generalized approach to parallel magnetic resonance imaging Med Phys 2001;28:1629–1643
- DK, CA
(Show Context)
Citation Context ...tion times and amplified noise errors lead to increased and spatially varying image noise levels. The problem becomes worse at high speedup factors. Special techniques, such as numerical conditioning =-=(15)-=-, second-pass reconstruction (16), and phase-constrained image refinement (17), have been proposed in an effort to overcome noise amplification. These methods may significantly decrease noise amplific... |

6 |
Parallel imaging with localized sensitivities (PILS). Magn Reson Med 2000;44:243–251
- MA, PM, et al.
(Show Context)
Citation Context ...rix is not singular (different linear trends for each coil may be used). 2) Reconstruct the pixels I � belonging to �, find the noise map N R,� in the area, and normalize I � as I � norm � I�/�N R,�. =-=[12]-=- norm 3) Find the SD �n of �I� �, and use it to obtain the normalization constant: � � � n 2 /�2 � �/2�. [13] Here the denominator accounts for a change in the variance of pixels in the air background... |

6 |
AC, Pike GB. MRI simulation-based evaluation of imageprocessing and classification methods
- RK, Evans
- 1999
(Show Context)
Citation Context ...as segmented and preprocessed to create the anatomical brain model. The available datasets were created by calculating the NMR signal from a simulation of pulse sequences based on the Bloch equations =-=(23,24)-=-. Zero-mean Gaussian noise (SD � 0.1 of white matter intensity) was modulated by a Gaussian function (max � 1 in the image center; Fig. 4b), and then added to real and imaginary image channels. Filter... |

5 |
Feedback regularization for SENSE reconstruction
- Tsao, KP, et al.
- 2002
(Show Context)
Citation Context ...rors lead to increased and spatially varying image noise levels. The problem becomes worse at high speedup factors. Special techniques, such as numerical conditioning (15), second-pass reconstruction =-=(16)-=-, and phase-constrained image refinement (17), have been proposed in an effort to overcome noise amplification. These methods may significantly decrease noise amplification; however, they are not capa... |

4 |
A signal-to-noise calibration procedure for NMR imaging systems. Med Phys 1984;11:180
- WA, PA, et al.
(Show Context)
Citation Context ... use it to obtain the normalization constant: � � � n 2 /�2 � �/2�. [13] Here the denominator accounts for a change in the variance of pixels in the air background due to the magnitude reconstruction =-=(22)-=-. Noise in Inhomogeneity-Corrected Images Intensity inhomogeneity correction is often applied to MR images before other postprocessing techniques, such as thresholding-based image segmentation, are us... |

4 |
Zijdenbos A, Kollokian V, Sled JG, Kabani NJ, Holmes CJ, Evans AC. Design and construction of a realistic digital brain phantom
- DL
- 1998
(Show Context)
Citation Context ...as segmented and preprocessed to create the anatomical brain model. The available datasets were created by calculating the NMR signal from a simulation of pulse sequences based on the Bloch equations =-=(23,24)-=-. Zero-mean Gaussian noise (SD � 0.1 of white matter intensity) was modulated by a Gaussian function (max � 1 in the image center; Fig. 4b), and then added to real and imaginary image channels. Filter... |

4 |
A three-coil comparison for MR angiography. J Magn Reson Imaging
- JR, BE, et al.
(Show Context)
Citation Context ...ion’s policies regarding human subjects. Data were acquired on a 1.5T GE SIGNA MR scanner (GE Medical Systems, Milwaukee, WI) using a custom-built, four-element (N c � 4), bilateral phased-array coil =-=(25)-=-. Phantom data were obtained with a standard fast spin-echo (FSE) sequence (ETL � 16, BW � 32 kHz, TE � 20 ms, TR � 1000 ms, image matrix � 256 � 256). Brain data were obtained with a dual-contrast FS... |

3 |
Sapiro G, Marimont DH, Heeger D. Robust anisotropic diffusion
- MJ
- 1998
(Show Context)
Citation Context ...emoval in homogeneous image regions, while preserving tissue boundaries and enhancing edge sharpness. For analysis purposes, we use a robust statistical formulation of an anisotropic diffusion filter =-=(19)-=-. The filtering problem is formulated as an estimation of a piecewiseconstant image function from noisy data, using a robust error norm �: min I �� ����I�, � e� d�, [2] where � e is a parameter boundi... |

3 |
PM, Edelman RR, Manning WJ. Signal-to-noise ratio and signal-to-noise efficiency
- DK, MA, et al.
- 1999
(Show Context)
Citation Context ... application of PPI techniques. We used SENSE images to test our noise-adaptive filter. A similar approach can be taken for other PPI techniques that provide information on spatial noise distribution =-=(27)-=-. Noise adaptation for SENSE data does not incur significant computational overhead, since the calculation of the partial image noise matrix (Eq. [10]) is a part of the reconstruction procedure (9). T... |

2 |
Gullberg GT. Signal-to-noise efficiency in magnetic resonance imaging. Med Phys 1990;17:250–257
- DL
- 1990
(Show Context)
Citation Context ...vercome noise amplification. These methods may significantly decrease noise amplification; however, they are not capable of improving image SNR behind the fundamental limits established by MRI theory =-=(18)-=-. Retrospective denoising with a nonlinear technique, such as anisotropic diffusion filtering, is an appealing option for improving the SNR of PPI images. This work addresses the problem of anisotropi... |

2 |
Scientific Computing: An Introduction with Parallel Computing
- GH, JM
- 1993
(Show Context)
Citation Context ...ly uniform, then � n �� m �� noise, and k mn � 2� noise. Algorithm: Noise-Adaptive Nonlinear Diffusion MATERIALS AND METHODS Implementation Details We chose to use an explicit final difference method =-=(21)-=- to discretize Eq. [1]: I m �t�1� � Im �t� � �t � � p���m� for each pixel m 2 2 Find kmp � �2 � ��m � �p, p � �l,n,i,j� �Fig. 3� end I�0� � Noisy Image for t � 0...�Number of Iterations-1� for each pi... |

1 |
Brechbuhler C, Szckely G, Gerig G. Parametric estimate of intensity inhomogeneities applied to MRI
- Styner
(Show Context)
Citation Context ...contribution of image gradients to the functional in Eq. [2] decreases at � e (19). This occurs when the derivative of the influence, or flow, function �( x, k)�x � g( x, k) is zero: ���x, c�e�� � 0. =-=[6]-=- �x x��e For example, for the exponential diffusivity function (2) c is �2. This yields g���I�, k� � exp�����I�/k� 2 �, [7] k mn � �2 � �� m 2 � �n 2 . [8] This result agrees with the choice of the co... |

1 |
Robust myelin water quantification: averaging vs. spatial filtering. Magn Reson Med 2003
- CK, KP, et al.
(Show Context)
Citation Context ...influence, or flow, function �( x, k)�x � g( x, k) is zero: ���x, c�e�� � 0. [6] �x x��e For example, for the exponential diffusivity function (2) c is �2. This yields g���I�, k� � exp�����I�/k� 2 �, =-=[7]-=- k mn � �2 � �� m 2 � �n 2 . [8] This result agrees with the choice of the conductance parameter in Ref. 5 for filtering the MR images with spatially uniform noise levels. In that work, k for the expo... |

1 |
Kholmovski EG. Method for quality improvement of images reconstructed from sensitivity-encoded data
- AA
- 2002
(Show Context)
Citation Context ...image noise levels. The problem becomes worse at high speedup factors. Special techniques, such as numerical conditioning (15), second-pass reconstruction (16), and phase-constrained image refinement =-=(17)-=-, have been proposed in an effort to overcome noise amplification. These methods may significantly decrease noise amplification; however, they are not capable of improving image SNR behind the fundame... |

1 |
Schnabelt JA. Enhancement of anisotropic diffusive filtering of MR images using approximate entropy
- GJM
- 1999
(Show Context)
Citation Context ....e., images obtained from surface coil images by a sum-of-squares approach). The principles outlined in this work could be easily extended to other techniques based on the nonlinear diffusion process =-=(30,31)-=-. Generalization of the method to the 3D case and the filtering of multicontrast data is straightforward. CONCLUSIONS FIG. 7. Application of filters to inhomogeneity-corrected phantom image (25 iterat... |

1 |
Total variation based convex filters for medical imaging
- SL
(Show Context)
Citation Context ....e., images obtained from surface coil images by a sum-of-squares approach). The principles outlined in this work could be easily extended to other techniques based on the nonlinear diffusion process =-=(30,31)-=-. Generalization of the method to the 3D case and the filtering of multicontrast data is straightforward. CONCLUSIONS FIG. 7. Application of filters to inhomogeneity-corrected phantom image (25 iterat... |