Validation on Simulated and Real MR Images

Figure 5.2 presents the results of denoising a particular slice from volumetric T1-weighted simulated BrainWeb data. The proposed MRI-denoising algorithm acts conservatively, reducing the RMS error by about $40 \%$. Figure 5.2(d) shows the difference between the corrupted and the uncorrupted images. The shift in the intensity PDF introduced by Rician noise is evident in the lighter background region (higher intensity on the average) corresponding to low signal intensities. The intensities in this difference image also possess a very low degree of spatial correlation. Figure 5.2(e) shows the difference between the denoised and the uncorrupted images. We see that algorithm reduces the Rician-noise-introduced shift in intensities in the low-intensity background region--fewer bright spots. Empirical analysis shows that denoised image effectively corrects the for the shift in the corrupted-intensity PDF caused by Rician noise--as measured by the average value of the background intensities in the uncorrupted, corrupted, and denoised images. For the case of T1-weighted BrainWeb data with $5$% noise and $40$% bias in Figure 5.2 the average background values are: (a) $0.1$ for the uncorrupted image, (b) $3.1$ for the corrupted image, and (c) $0.03$ for the denoised image. The difference images in Figure 5.2(e) show low magnitudes for errors in the background region. The difference image also possesses low correlation indicating that the proposed algorithm retained the significant image features more-or-less intact. The power spectrum of the difference image in Figure 5.2(f) shows the whiteness [81] of the residual.

Figure 5.2: Results with T1-weighted simulated BrainWeb data (intensity range $0:100$) with the Rician noise level $\sigma _R = 5$ and a $40 \%$ bias field. (a) Uncorrupted image. (b) Rician-noise corrupted image: RMSE = $5.53$. (c) Denoised image: RMSE = $3.3$. (d) Difference between the corrupted and uncorrupted images. (e) Difference between the denoised and uncorrupted images. (f) Power spectrum of the image in (e): close to white.

Figure 5.3: MRI-denoising results. (a)-(c) Three different brain slices from the BrainWeb dataset (only T1 modality shown; intensity range $0:100$). (d)-(f) Graphs indicating RMS errors for denoised and noisy images, with $0$% and $40$% bias fields, for T1, T2, and PD modalities on the three slices above.

Figure 5.3 gives the performance of the proposed algorithm on three different slices of the BrainWeb MR data for varying noise and bias levels. We observe that the performance on biased and unbiased data is equivalent. This stems from the ability of adaptive-MRF model to effectively infer the appropriate Markov statistics for each case and denoise based on the inferred model. We also observe that for very low Rician noise, i.e., $\sigma_R \approx 1$, the algorithm does not effectively reduce the RMS error. This may be because of a similar level of variability inherent in the data, and in the estimated uncorrupted-signal Markov PDFs, which makes the algorithm not clearly identify the noise. As the amount of noise increases, the proposed method can clearly differentiate the structure underlying the data from the noise. Figure 5.4 shows the performance of proposed algorithm on real data that depicts a significant inhomogeneity/bias.

Figure 5.4: MRI-denoising results. (a),(b) Noisy slices from a real MR volume. (c),(d) Denoised images.

Figure 5.5 compares, qualitatively and quantitatively, the performance of the proposed algorithm with several other recent and popular filtering algorithms. We have manually tuned all the free parameters in these other algorithms in order to give the best possible results. The proposed algorithm does better qualitatively, with an RMS error of $3.3$ (RMS error for noisy image is $5.53$) as compared to the RMS errors produced by other algorithms of around $4.0$ or more. Qualitatively too, the proposed algorithm gives a residual (difference between denoised and uncorrupted image) that is significantly less correlated. The state-of-the-art wavelet-based denoising algorithm [129] also seems to introduce artifacts in the denoised image.

Figure 5.6 show the qualitative and quantitative comparison of the proposed method with a state-of-the-art wavelet-based MRI-denoising algorithm [129]. We see that the proposed method produces lower RMS errors at all noise levels except with one image at the 9$\%$ noise level. Although the RMS error for the proposed method is a little more for this high-noise case, Figure 5.6(c) and Figure 5.6(d) show that the residual for the wavelet-based method is significantly more correlated. This residual also indicates the presence of artifacts in the wavelet-denoised image.

Figure 5.5: Results with T1-weighted simulated BrainWeb data (intensity range $0:100$) with the Rician noise level $\sigma _R = 5$ and a $40 \%$ bias field. The noisy image in Figure 5.2(b) (RMSE = $5.53$) denoised using (a) anisotropic diffusion [127]: RMS error $4.03$, (b) curvature flow [153]: RMS error $3.93$, (c) UINTA [9]: RMS error $4.0$, and (d) the state-of-the-art wavelet-based MRI denoiser [129]: RMS error $5.64$, (e)-(h) show the differences between the denoised images in (a)-(d) and the uncorrupted image in Figure 5.2(a).

Figure 5.6: Comparison of proposed MRI-denoising method with the state of the art. (a) Quantitative comparison of the proposed method with a state-of-the-art wavelet-based MRI-denoiser [129] for the three different slices of T1 BrainWeb data, shown in Figure 5.3 (intensity range $0:100$), with varying noise levels and a 40$\%$ bias field. (b) Corrupted T1 data with 9$\%$ noise and 40$\%$ bias field. (c) and (d) show the difference between the denoised and uncorrupted images for the proposed and wavelet-based [129] methods, respectively, when these methods are applied to the corrupted data in (b).