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An augmented lagrangian method for solving a new variational model based on gradients similarity measures and high order regulariation for multimodality registration

Both authors are supported by the UK EPSRC grant EP/N014499/1.
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  • In this work we propose a variational model for multi-modal image registration. It minimizes a new functional based on using reformulated normalized gradients of the images as the fidelity term and higher-order derivatives as the regularizer. We first present a theoretical analysis of the proposed model. Then, to solve the model numerically, we use an augmented Lagrangian method (ALM) to reformulate it to a few more amenable subproblems (each giving rise to an Euler-Lagrange equation that is discretized by finite difference methods) and solve iteratively the main linear systems by the fast Fourier transform; a multilevel technique is employed to speed up the initialisation and avoid likely local minima of the underlying functional. Finally we show the convergence of the ALM solver and give numerical results of the new approach. Comparisons with some existing methods are presented to illustrate its effectiveness and advantages.

    Mathematics Subject Classification: Primary: 65M32, 35Q68, 94A08; Secondary: 65M22, 35G15.


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  • Figure 1.  Example of Reference and Template images where $ \nabla_n T\cdot \nabla_n R = 0 $ (or one of $ \nabla_n T, \ \nabla_n R $ is zero) a.e in $ \Omega $

    Figure 2.  Three examples of the triangle inequality for triangles with sides $ X $, $ Y $ and $ Z $. The left example shows a case where $ |Z| $ is much less than the sum $ |X| +|Y| $ of the other two sides, and the right example shows a case where $ |Z| $ is only slightly less than $ |X| +|Y| $

    Figure 3.  Example of a multilevel representation of images

    Figure 4.  Example 1: Comparison of three different models. Clearly only Our Model works while NGF, MI fail completely

    Figure 5.  Example 2: Comparison of different models to register T-1 and T2-MRI images. New Model performs the best

    Figure 6.  Example 3: Registration of a second pair of MRI images (T1 and T2). New Model performs the best

    Figure 7.  Comparison of $ 3 $ different models to register the MRI images fin Fig. 6. Example 3 zoomed in the red squares (see Fig. 6): From left to right; Zooms in the reference $ R $ and the registered $ T(\mathbf u) $ using New model, NGF and MI, respectively.

    Figure 8.  Example 4: High-b- and Low-b-value Diffusion-weighted MRIs (of $ 256\times 256 $) using different models. New Model performs the best

    Figure 9.  Example 5: a pair of MRI images of higher resolution $ 512\times 512 $ by $ 3 $ different models. New Model and MI perform identically, both better than NGF

    Figure 10.  Left: Log scale plot of the residual errors for $ \mathbf u $ versus ALM iteration numbers for examples 2-5. Right: Plot of the error $ S_{er} $ values versus ALM iteration numbers for examples 2-5

    Figure 11.  Left: Log scale plot of the distance Dm versus ALM iteration numbers for examples 2-5

    Figure 12.  Example 6: Registering a PET image to an MRI vimage. New model performs better than others in this example

    Table 1.  Run time comparison for all models for the pair of MRI images in Fig. 6

    $64 \times 64$ $128 \times 128$ $256 \times 256$ $512 \times 512$
    Time (s) for New Model 29.836 49.931 117.342 272.578
    Time (s) for MI Model 14.794 21.437 48.881 76.398
    Time (s) for NGF Model 22.003 42.845 100.961 264.388
     | Show Table
    DownLoad: CSV

    Table 2.  Registration results of the different models for processing Examples 1-6. The errors are computed using formula (38), (40) and (39). Here, #N is the ratio of the number of pixels where $\nabla_n T\cdot \nabla_n R \neq 0$ over the total number of pixels, whereas #G is the ratio of number pixels where GF(T, R)+TM(T, R) ≠ 0 over the total number of pixels.

    Compared Models NGF New Model
    #G #N GFer NGFer MIer GFer NGFer MIer
    Ex 1 0.2% .02% 0.540 0.964 0.446 0.032 0.932 0.993
    Ex 2 49% 24% 0.636 0.640 1.170 0.247 0.756 1.206
    Ex 3 49% 23% 0.336 0.491 1.265 0.238 0.389 1.290
    Ex 4 49% 20% 0.901 0.856 1.150 0.674 0.800 1.184
    Ex 5 43% 37% 0.741 0.656 1.163 0.454 0.623 1.178
    Ex 6 48% 23% 0.952 0.957 1.187 0.801 0.920 1.341
    Compared Models MI
    #G #N GFer NGFer MIer Note:
    Ex 1 0.2% .02% 0.370 0.97 0.381 for GFer and NGFer, the smaller the better.
    Ex 2 49% 24% 0.490 0.879 1.193
    Ex 3 49% 23% 0.463 0.579 1.265 But for MIer, The larger the better.
    Ex 4 49% 20% 0.765 0.849 1.154
    Ex 5 43% 37% 0.454 0.631 1.163
    Ex 6 48% 23% 0.836 0.970 1.254
     | Show Table
    DownLoad: CSV

    Table 3.  Registration results for $ \frac{\alpha_1}{\lambda} $-dependence tests of New Model for processing Example 3. The relative errors are computed using the normalized gradient fitting formula (38). In all cases, we set $ \alpha = 0.01\alpha_1 $

    $\frac{\alpha_1}{\lambda}$ 0.1 0.05 0.025 0.017 0.0125 0.01 0.0075 0.005
    Error 0.238 0.237 0.237 0.236 0.237 0.237 0.238 0.24
     | Show Table
    DownLoad: CSV
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