Larrey-Ruiz et al 2018a - Revision history

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Rimni: /* 3. Results */

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‎3. Results

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Jlarrey at 12:33, 21 March 2019

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Rimni at 11:43, 21 March 2019

2019-03-21T11:43:41Z

Rimni at 11:40, 21 March 2019

2019-03-21T11:40:54Z

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 <div id='table-1'></div>
-{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"
+{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;font-size:85%;"
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-Throughout this work, the SSIM-based term uses the following parameters setting: <math display="inline">C_1=0.01</math> and <math display="inline">C_2=0.03</math>. These are the empirically obtained values suggested in the reference paper <span id='citeF-43'></span>[[#cite-43|[43]]]. Regardless, we find that in the current scenario, the performance of the proposed registration algorithm is fairly insensitive to variations of these values. Regarding the remaining parameters, which are exclusively related to our variational approach, the values of <math display="inline">\lambda=2</math> (i.e., curvature smoother, since the deformations to be corrected are notable in some cases), <math display="inline">\alpha=200</math>, <math display="inline">\beta=5</math> (only applicable to the combined SSIM- and landmark-based method) and <math display="inline">\tau=1</math> were used for all experiments —we recommend a common set which achieves a good balance between generalization and performance—. This parameter setting was obtained following the guidelines first introduced in <span id='citeF-23'></span>[[#cite-23|[23]]]. As for the number of iterations, a value of <math display="inline">\xi _{\max }=80</math> grants convergence in all cases; the cost function stabilizes after 50&#8211;70 iterations for both the CR-based and the proposed algorithm. From the results gathered in Table [[#table-1|1]], it can be concluded that the computational overhead introduced by the novel approach does not increase significantly the execution time of the CR-based algorithm. Moreover, both variational approaches outperform well-established registration methods such as ''Elastix'' in terms of efficiency. ''Elastix'' was the fastest method which provided better results among the open source methods of image registration which entered in the second phase of Empire10 Challenge <span id='citeF-30'></span>[[#cite-30|[30]]]. The overall complexity of each iteration of the compared algorithms is <math display="inline">\mathcal{O}(N)</math> —where <math display="inline">N</math> is the number of voxels of the datasets—, since doubling <math display="inline">N</math> means doubling the computational time.
+Throughout this work, the SSIM-based term uses the following parameters setting: <math display="inline">C_1=0.01</math> and <math display="inline">C_2=0.03</math>. These are the empirically obtained values suggested in the reference paper <span id='citeF-43'></span>[[#cite-43|[43]]]. Regardless, we find that in the current scenario, the performance of the proposed registration algorithm is fairly insensitive to variations of these values. Regarding the remaining parameters, which are exclusively related to our variational approach, the values of <math display="inline">\lambda=2</math> (i.e., curvature smoother, since the deformations to be corrected are notable in some cases), <math display="inline">\alpha=200</math>, <math display="inline">\beta=5</math> (only applicable to the combined SSIM- and landmark-based method) and <math display="inline">\tau=1</math> were used for all experiments —we recommend a common set which achieves a good balance between generalization and performance—. This parameter setting was obtained following the guidelines first introduced in <span id='citeF-23'></span>[[#cite-23|[23]]]. As for the number of iterations, a value of <math display="inline">\xi _{\max }=80</math> grants convergence in all cases; the cost function stabilizes after 50&#8211;70 iterations for both the CR-based and the proposed algorithm. From the results gathered in  [[#table-1|Table 1]], it can be concluded that the computational overhead introduced by the novel approach does not increase significantly the execution time of the CR-based algorithm. Moreover, both variational approaches outperform well-established registration methods such as ''Elastix'' in terms of efficiency. ''Elastix'' was the fastest method which provided better results among the open source methods of image registration which entered in the second phase of Empire10 Challenge <span id='citeF-30'></span>[[#cite-30|[30]]]. The overall complexity of each iteration of the compared algorithms is <math display="inline">\mathcal{O}(N)</math> —where <math display="inline">N</math> is the number of voxels of the datasets—, since doubling <math display="inline">N</math> means doubling the computational time.
 {|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"
 |- style="border-top: 2px solid;"
-|       Size
+| Size
 | CR
 | SSIM+LM

@@ Line 253: / Line 253: @@
 In order to assess the performance of the proposed methodology, it is tested on 15 medical experiments involving triple-phase 3D computed tomography (CT) volumes of the liver under contrast agent injection. Although the acquisition of the different phases is continuous in most cases, there is no exact correspondence between them, so they must be registered in order to show the results in a common 3D framework. First, a study without contrast agent is acquired. Once the contrast agent is injected, it reaches the arteries (arterial phase), then the portal veins and next the hepatic veins. Hepatic and portal veins are then visible in the portal venous phase. Image registration allows to locate exactly the vessels in 3D at each phase of contrast-enhanced CT data in order to measure distances and volumes and provide objective parameters of the pathology which facilitate comparisons between patients, the tracking of tumors <span id='citeF-9'></span>[[#cite-9|[9]]], to make calculations on the volume of the liver to be preserved prior to a liver resection <span id='citeF-12'></span>[[#cite-12|[12]]], or to generate vessel models for planning surgical procedures <span id='citeF-22'></span>[[#cite-22|[22]]]. The experiments carried out involve data from 5 patients. The acquisition device is a GE LightSpeed VTC (General Electric Medical Systems). The size of the volumes vary from <math display="inline">512\times{512}\times{34}</math> to <math display="inline">512\times{512}\times{51}</math> voxels, with a resolution of <math display="inline">0.9102\times{0.9102}\times{5}</math> millimeters.
-For the assessment of the proposed method, we carry out a comparison with the purely intensity-based variational method recently presented by the authors of this work in <span id='citeF-26'></span>[[#cite-26|[26]]]. This CR-based approach was reported to outperform publicly available state-of-the art methods such as ''Elastix'' and ANTs in the medical setting. As can be seen in Figure [[#img-1|1]], the proposed framework shows excellent results for the three considered registration scenarios (arterial-portal, arterial-non-contrast and portal-non-contrast), reaching average values of 1.47, 1.44 and 1.52 bits in terms of mutual information, corresponding to the arterial-portal, arterial-non-contrast and portal-non-contrast cases, respectively; this represents a mean improvement of 28.9%, 48.45% and 51.16% in relative terms of mutual information, thus outperforming the CR-based registration algorithm, which achieves a mean improvement of 26.48%, 44.22% and 43.25%, respectively. Additionally, due to the analogous behavior (i.e., comparable final values of mutual information) of the proposed method in the three scenarios, all available experiments can be grouped into one ensemble in order to assess a more comprehensive validation of the actual registration error. A ground truth was established by an expert in the form of identifiable anatomical locations (landmarks) for all experiments. The registration errors were then obtained by computing the spatial distance between the corresponding landmarks in the reference and registered template datasets. Figures [[#img-2|2]](a) and [[#img-2|2]](b) show through box plots the registration error (in millimeters) achieved by the methods under comparison, gathering the results from the three considered registration scenarios. These box plots collect the final spatial distances between corresponding landmarks, along with the median distance error and its statistical significance (notch showing the 95% confidence interval of the true median). According to Figure [[#img-2|2]](b), the proposed method significantly improves on the registration error of the CR-based approach, since it reduces the initial median error from 9.50 mm to a residual median distance between landmarks of 1.41 mm, decreasing at the same time the outliers occurrence.
+For the assessment of the proposed method, we carry out a comparison with the purely intensity-based variational method recently presented by the authors of this work in <span id='citeF-26'></span>[[#cite-26|[26]]]. This CR-based approach was reported to outperform publicly available state-of-the art methods such as ''Elastix'' and ANTs in the medical setting. As can be seen in  [[#img-1|Figure 1]], the proposed framework shows excellent results for the three considered registration scenarios (arterial-portal, arterial-non-contrast and portal-non-contrast), reaching average values of 1.47, 1.44 and 1.52 bits in terms of mutual information, corresponding to the arterial-portal, arterial-non-contrast and portal-non-contrast cases, respectively; this represents a mean improvement of 28.9%, 48.45% and 51.16% in relative terms of mutual information, thus outperforming the CR-based registration algorithm, which achieves a mean improvement of 26.48%, 44.22% and 43.25%, respectively. Additionally, due to the analogous behavior (i.e., comparable final values of mutual information) of the proposed method in the three scenarios, all available experiments can be grouped into one ensemble in order to assess a more comprehensive validation of the actual registration error. A ground truth was established by an expert in the form of identifiable anatomical locations (landmarks) for all experiments. The registration errors were then obtained by computing the spatial distance between the corresponding landmarks in the reference and registered template datasets.  [[#img-2|Figures 2]](a) and [[#img-2|2]](b) show through box plots the registration error (in millimeters) achieved by the methods under comparison, gathering the results from the three considered registration scenarios. These box plots collect the final spatial distances between corresponding landmarks, along with the median distance error and its statistical significance (notch showing the 95% confidence interval of the true median). According to  [[#img-2|Figure 2]](b), the proposed method significantly improves on the registration error of the CR-based approach, since it reduces the initial median error from 9.50 mm to a residual median distance between landmarks of 1.41 mm, decreasing at the same time the outliers occurrence.
 <div id='img-1'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 80%;max-width: 80%;"
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 70%;"
 |-
 |style="padding:10px;"|[[Image:Review_341874659107-MI.png|360px|Mean similarity between images in terms of mutual information for the three considered scenarios (arterial-portal, arterial-non-contrast and portal-non-contrast) before and after the registration.]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" style="padding:10px;"| '''Figure 1:''' Mean similarity between images in terms of mutual information for the three considered scenarios (arterial-portal, arterial-non-contrast and portal-non-contrast) before and after the registration.
+| colspan="1" style="padding:10px;"| '''Figure 1'''. Mean similarity between images in terms of mutual information for the three considered scenarios (arterial-portal, arterial-non-contrast and portal-non-contrast) before and after the registration
 |}
@@ Line 274: / Line 274: @@
 | (b) Close-up view of (a)
 |- style="text-align: center; font-size: 75%;"
-| colspan="2" style="padding:10px;"| '''Figure 2:''' Spatial distances (in millimeters) between corresponding landmarks before and after the registration process.
+| colspan="2" style="padding:10px;"| '''Figure 2'''. Spatial distances (in millimeters) between corresponding landmarks before and after the registration process
 |}
-In addition to the previous measurements, the visual outcomes of two of the experiments are shown in Figures [[#img-3|3]] and [[#img-4|4]], whose purpose is to highlight the most illustrative differences (from a medical point of view) between the results provided by the compared methods. In Figure [[#img-3|3]], we observe a normal size of the liver, with discretely irregular contours and homogeneous signal intensity. In hepatic segment II, there is a lesion of 40 mm of maximum axis, encapsulated and with well-defined contours and heterogeneous enhancement in arterial phase (after administration of intravenous contrast), suggestive of hepatocellular carcinoma (HCC). In this slice of the CT scan, we can also observe the aorta that shines in the arterial phase, the lower area of the stomach and the upper area of the spleen. In Figure [[#img-4|4]], the liver has a normal size with discretely irregular contours in relation to changes due to chronic liver disease. In hepatic segment IV, a 36 mm diameter focal lesion is identified, which has arterial phase enhancement with a small area of necrosis of 13 mm; it corresponds to a HCC previously chemoembolized with partial necrosis. In this slice of CT, we can also observe the aorta, the gastric chamber and the spleen. When comparing the two methods under study, it can be seen how in Figure [[#img-3|3]] the resulting registered datasets are very similar. However, looking closely, it can be noticed that in the right part of the image (left side of the patient) the shape and width of the structures corresponding to the stomach and the spleen in Figure [[#img-3|3]](d) match better those in the reference dataset. Likewise, the part of the rib at the upper right of the image is more similar to the same region in the reference dataset by using the proposed method. Regarding the experiment shown in Figure [[#img-4|4]], it can be easily appreciated how the geometrical matching (with respect to the reference dataset, Figure [[#img-4|4]](a)) of the structures in the right side of the image (specially the gastric chamber) is visually more satisfactory in Figure [[#img-4|4]](d). Moreover, the area of tumor necrosis which results from the proposed method is also slightly better aligned.
+In addition to the previous measurements, the visual outcomes of two of the experiments are shown in [[#img-3|Figures 3]] and [[#img-4|4]], whose purpose is to highlight the most illustrative differences (from a medical point of view) between the results provided by the compared methods. In  [[#img-3|Figure 3]], we observe a normal size of the liver, with discretely irregular contours and homogeneous signal intensity. In hepatic segment II, there is a lesion of 40 mm of maximum axis, encapsulated and with well-defined contours and heterogeneous enhancement in arterial phase (after administration of intravenous contrast), suggestive of hepatocellular carcinoma (HCC). In this slice of the CT scan, we can also observe the aorta that shines in the arterial phase, the lower area of the stomach and the upper area of the spleen. In  [[#img-4|Figure 4]], the liver has a normal size with discretely irregular contours in relation to changes due to chronic liver disease. In hepatic segment IV, a 36 mm diameter focal lesion is identified, which has arterial phase enhancement with a small area of necrosis of 13 mm; it corresponds to a HCC previously chemoembolized with partial necrosis. In this slice of CT, we can also observe the aorta, the gastric chamber and the spleen. When comparing the two methods under study, it can be seen how in  [[#img-3|Figure 3]] the resulting registered datasets are very similar. However, looking closely, it can be noticed that in the right part of the image (left side of the patient) the shape and width of the structures corresponding to the stomach and the spleen in  [[#img-3|Figure 3]](d) match better those in the reference dataset. Likewise, the part of the rib at the upper right of the image is more similar to the same region in the reference dataset by using the proposed method. Regarding the experiment shown in  [[#img-4|Figure 4]], it can be easily appreciated how the geometrical matching (with respect to the reference dataset,  [[#img-4|Figure 4]](a)) of the structures in the right side of the image (specially the gastric chamber) is visually more satisfactory in  [[#img-4|Figure 4]](d). Moreover, the area of tumor necrosis which results from the proposed method is also slightly better aligned.
@@ Line 300: / Line 300: @@
 | (d) Registered template, <math display="inline">T_\mathbf{u}</math> (proposed method)
 |- style="text-align: center; font-size: 75%;"
-| colspan="2" style="padding:10px;"| '''Figure 3:''' Visual outcomes of experiment 1 (slice 9): registration of arterial and non-contrast phases of patient 1.
+| colspan="2" style="padding:10px;"| '''Figure 3'''. Visual outcomes of experiment 1 (slice 9): registration of arterial and non-contrast phases of patient 1
 |}
@@ Line 322: / Line 322: @@
 | (d) Registered template, <math display="inline">T_\mathbf{u}</math> (proposed method)
 |- style="text-align: center; font-size: 75%;"
-| colspan="2" style="padding:10px;"| '''Figure 4:''' Visual outcomes of experiment 5 (slice 12): registration of arterial and portal venous phases of patient 2.
+| colspan="2" style="padding:10px;"| '''Figure 4'''. Visual outcomes of experiment 5 (slice 12): registration of arterial and portal venous phases of patient 2
 |}

@@ Line 22: / Line 22: @@
 ==1. Introduction==
-Image registration is the process of finding the spatial correspondence between two different datasets (images or volumes), which typically represent different views of the same object or similar ones; in other words, register is equivalent to ''align'' in this context. Particularly, in medical imaging there are several applications that require a registration step (e.g. image fusion, atlas matching, pathological diagnosis). Classical reviews of image registration can be found, for example, in the references <span id='citeF-6'></span>[[#cite-6|[6]]], <span id='citeF-28'></span>[[#cite-28|[28]]], <span id='citeF-47'></span>[[#cite-47|[47]]] or <span id='citeF-29'></span>[[#cite-29|[29]]]. More recent comprehensive overviews about image registration methods can be found in <span id='citeF-37'></span>[[#cite-37|[37]]] and <span id='citeF-36'></span>[[#cite-36|[36]]], where the state-of-the-art advances in this field are described in a systematic fashion, and the techniques applied to medical datasets are emphasized as well. When classified according to the image characteristics used to estimate the transformation which geometrically relates the involved datasets, registration algorithms are usually divided into three groups: geometric feature-based, which rely on the segmentation, done generally before the registration process itself, of part or all of the images, therefore obtaining objects that are registered by minimizing some geometrical distance between them; intensity-based, which maximize a intensity similarity measure computed between points lying at the same spatial position; and iconic feature-based, which can be understood as intermediate between the two previous categories, since they use explicitly some type of geometrical distance in addition to the intensity similarity measure, see <span id='citeF-8'></span>[[#cite-8|[8]]]. The performance of intensity-based image registration methods is highly influenced by the choice of the similarity measure. This measure can be defined directly on image intensities as, e.g., sum of squared differences (SSD), correlation coefficient (CC), correlation ratio (CR), or mutual information (MI). A complete analysis of all these metrics can be found in  <span id='citeF-27'></span>[[#cite-27|[27]]]. These measures often rely on the assumption of independence and stationarity of the intensities from pixel to pixel without considering their spatial dependencies. Alternative metrics have been proposed in order to consider intensity nonstationarities and complex spatially-varying intensity distortions. For instance, in <span id='citeF-31'></span>[[#cite-31|[31]]] a similarity measure based on the residual complexity is described. Another example is the structural similarity index (SSIM), introduced by <span id='citeF-43'></span>[[#cite-43|[43]]], which achieves a good trade-off between speed and performance and is able to cope with complex relationships between image intensity values. Due to the mathematical properties of SSIM <span id='citeF-7'></span>[[#cite-7|[7]]] and its potentials in both theoretical development and practical applications, it is being incorporated into optimization frameworks in order to improve perceived image/video quality in a number of image processing problems, as e.g. <span id='citeF-17'></span>[[#cite-17|[17]]] or <span id='citeF-34'></span>[[#cite-34|[34]]]. This similarity measure has been previously used in image registration in <span id='citeF-4'></span>[[#cite-4|[4]]], <span id='citeF-1'></span>[[#cite-1|[1]]], <span id='citeF-41'></span>[[#cite-41|[41]]], <span id='citeF-42'></span>[[#cite-42|[42]]], <span id='citeF-45'></span>[[#cite-45|[45]]], or <span id='citeF-19'></span>[[#cite-19|[19]]], but its use is still not widespread —for example, the evaluation addressed in <span id='citeF-33'></span>[[#cite-33|[33]]] does not include SSIM and the review in <span id='citeF-36'></span>[[#cite-36|[36]]] does not deal with similarity measures based on the SSIM—.
+Image registration is the process of finding the spatial correspondence between two different datasets (images or volumes), which typically represent different views of the same object or similar ones; in other words, register is equivalent to ''align'' in this context. Particularly, in medical imaging there are several applications that require a registration step (e.g. image fusion, atlas matching, pathological diagnosis). Classical reviews of image registration can be found, for example, in the references <span id='citeF-6'></span>[[#cite-6|[6]]], <span id='citeF-28'></span>[[#cite-28|[28]]], <span id='citeF-47'></span>[[#cite-47|[47]]] or <span id='citeF-29'></span>[[#cite-29|[29]]]. More recent comprehensive overviews about image registration methods can be found in <span id='citeF-37'></span>[[#cite-37|[37]]] and <span id='citeF-36'></span>[[#cite-36|[36]]], where the state-of-the-art advances in this field are described in a systematic fashion, and the techniques applied to medical datasets are emphasized as well. When classified according to the image characteristics used to estimate the transformation which geometrically relates the involved datasets, registration algorithms are usually divided into three groups: geometric feature-based, which rely on the segmentation, done generally before the registration process itself, of part or all of the images, therefore obtaining objects that are registered by minimizing some geometrical distance between them; intensity-based, which maximize a intensity similarity measure computed between points lying at the same spatial position; and iconic feature-based, which can be understood as intermediate between the two previous categories, since they use explicitly some type of geometrical distance in addition to the intensity similarity measure, see <span id='citeF-8'></span>[[#cite-8|[8]]]. The performance of intensity-based image registration methods is highly influenced by the choice of the similarity measure. This measure can be defined directly on image intensities as, e.g., sum of squared differences (SSD), correlation coefficient (CC), correlation ratio (CR), or mutual information (MI). A complete analysis of all these metrics can be found in  <span id='citeF-27'></span>[[#cite-27|[27]]]. These measures often rely on the assumption of independence and stationarity of the intensities from pixel to pixel without considering their spatial dependencies. Alternative metrics have been proposed in order to consider intensity nonstationarities and complex spatially-varying intensity distortions. For instance, in <span id='citeF-31'></span>[[#cite-31|[31]]] a similarity measure based on the residual complexity is described. Another example is the structural similarity index (SSIM), introduced by <span id='citeF-43'></span>[[#cite-43|[43]]], which achieves a good trade-off between speed and performance and is able to cope with complex relationships between image intensity values. Due to the mathematical properties of SSIM <span id='citeF-7'></span>[[#cite-7|[7]]] and its potentials in both theoretical development and practical applications, it is being incorporated into optimization frameworks in order to improve perceived image/video quality in a number of image processing problems, as e.g. <span id='citeF-17'></span>[[#cite-17|[17]]] or <span id='citeF-34'></span>[[#cite-34|[34]]]. This similarity measure has been previously used in image registration in <span id='citeF-1'></span>[[#cite-1|[1]]], <span id='citeF-4'></span>[[#cite-4|[4]]],  <span id='citeF-41'></span>[[#cite-41|[41]]], <span id='citeF-42'></span>[[#cite-42|[42]]], <span id='citeF-45'></span>[[#cite-45|[45]]], or <span id='citeF-19'></span>[[#cite-19|[19]]], but its use is still not widespread —for example, the evaluation addressed in <span id='citeF-33'></span>[[#cite-33|[33]]] does not include SSIM and the review in <span id='citeF-36'></span>[[#cite-36|[36]]] does not deal with similarity measures based on the SSIM—.
 Intensity-based approaches are in general fully automatic and usually yield good registration results. However, they may perform poorly for specific, important locations such as anatomical landmarks. On the other hand, geometric feature-based registration techniques are designed to accurately match user specified landmarks, see e.g. <span id='citeF-35'></span>[[#cite-35|[35]]]. Recent examples of landmark-based registration methods in medical imaging scenarios can be found in <span id='citeF-44'></span>[[#cite-44|[44]]] or <span id='citeF-21'></span>[[#cite-21|[21]]]. A common drawback of most landmark-driven registration methods is the fact that the intensities of the datasets are completely neglected. Consequently, the registration outcome away from the landmarks may be very poor. In the literature, the registration results are typically obtained by interpolating the transformation with a ''thin-plate spline'' (TPS) model <span id='citeF-5'></span>[[#cite-5|[5]]] or by combining iterative closest point (ICP) registration and parametric relaxation <span id='citeF-38'></span>[[#cite-38|[38]]], among other techniques. Although these approaches produce a smooth transformation from one dataset to another, they do not define a consistent correspondence between the two datasets except at the landmark points. Some previous attempts to combine the sparse correspondences with a intensity-based registration method can be found in <span id='citeF-18'></span>[[#cite-18|[18]]] or <span id='citeF-39'></span>[[#cite-39|[39]]]. However, to the authors knowledge, there has been no attempt to formulate a registration method which combines in the distance term both a SSIM-based term and a landmark-based term. In recent works, such as <span id='citeF-32'></span>[[#cite-32|[32]]] or <span id='citeF-26'></span>[[#cite-26|[26]]], the authors of this paper dealt with intensity-based registration approaches within the variational framework first presented in <span id='citeF-25'></span>[[#cite-25|[25]]], whose formulation in the frequency domain allowed for implementations of high efficiency <span id='citeF-40'></span>[[#cite-40|[40]]]. Please refer to <span id='citeF-2'></span>[[#cite-2|[2]]] or <span id='citeF-46'></span>[[#cite-46|[46]]] for further details regarding variational image registration. The aforementioned approaches involved similarity terms derived from SSD- or CR-based measures, so none of them included a SSIM-based nor a landmark-based distance term. Regardless, the method presented in <span id='citeF-26'></span>[[#cite-26|[26]]] achieved better results than publicly available state-of-the art approaches such as ''Elastix'' <span id='citeF-20'></span>[[#cite-20|[20]]] and ANTs (<math display="inline">http://stnava.github.io/ANTs/</math>).

@@ Line 541: / Line 541: @@
 <div id="cite-1"></div>
-[[#citeF-1|[1]]]  Amintoosi, M., Fathy, M.,   Mozayani, N. (2009). Precise image registration with structural similarity   error measurement applied to superresolution. EURASIP Journal on Advances in Signal Processing   2009, 305479. <div id="cite-2"></div>
+[[#citeF-1|[1]]]  Amintoosi, M., Fathy, M.,   Mozayani, N. Precise image registration with structural similarity error measurement applied to superresolution. EURASIP Journal on Advances in Signal Processing, 2009:1&#8211;7, 2009. <div id="cite-2"></div>
-[[#citeF-2|[2]]]  Amit, Y. (1994). A nonlinear variational problem for image matching. SIAM Journal of Scientific Computing   15, 207&#8211;224. <div id="cite-3"></div>
+[[#citeF-2|[2]]]  Amit, Y. A nonlinear variational problem for image matching. SIAM Journal of Scientific Computing, 15:207&#8211;224, 1994. <div id="cite-3"></div>
-[[#citeF-3|[3]]]  Bai, J., Feng, X.C. (2007). Fractional-order anisotropic diffusion for image   denoising. IEEE Transactions on Image Processing   16, 2492&#8211;2502. <div id="cite-4"></div>
+[[#citeF-3|[3]]]  Bai, J., Feng, X.C. Fractional-order anisotropic diffusion for image   denoising. IEEE Transactions on Image Processing, 16:2492&#8211;2502, 2007. <div id="cite-4"></div>
-[[#citeF-4|[4]]]  Ben&nbsp;Sassi, O., Delleji, T.,   Taleb-Ahmed, A., Feki, I.,   Ben&nbsp;Hamida, A. (2008). Mr image monomodal registration using structure   similarity index, in: Image Processing Theory, Tools and   Applications, 2008. IPTA 2008. First Workshops on, pp.   1&#8211;5. <div id="cite-5"></div>
+[[#citeF-4|[4]]]  Ben&nbsp;Sassi, O., Delleji, T.,   Taleb-Ahmed, A., Feki, I.,   Ben&nbsp;Hamida, A. Mr image monomodal registration using structure   similarity index. Image Processing Theory, Tools and Applications, 2008:1&#8211;5, 2008. <div id="cite-5"></div>
-[[#citeF-5|[5]]]  Bookstein, F.L. (1989). Principal warps: thin-plate splines and the   decomposition of deformations. IEEE Transactions on Pattern Analysis and Machine   Intelligence 11, 567&#8211;585. <div id="cite-6"></div>
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@@ Line 172: / Line 172: @@
 with <math display="inline">i_r\,,i_t\in \mathcal{I}\subset \mathbb{R}</math> being the intensities of <math display="inline">R</math> and <math display="inline">T_\mathbf{u}</math>, respectively.
-In [[#5 Variational formulation of SSIM|5]], the first addend in equation ([[#eq-5|5]]) is deduced. The proof of the internal forces field of the Euler-Lagrange equation (i.e., the last addend in eq.([[#eq-5|5]])) can be found in <span id='citeF-24'></span>[[#cite-24|[24]]]. Finally, the addend which corresponds to the landmark-based term can be easily obtained by applying the definition of the ''Gateaux'' derivative to the last addend in eq.([[#eq-2|2]]):
+In [[#5 Variational formulation of SSIM|5]], the first addend in equation ([[#eq-5|5]]) is deduced. The proof of the internal forces field of the Euler-Lagrange equation (i.e., the last addend in Eq.([[#eq-5|5]])) can be found in <span id='citeF-24'></span>[[#cite-24|[24]]]. Finally, the addend which corresponds to the landmark-based term can be easily obtained by applying the definition of the ''Gateaux'' derivative to the last addend in Eq.([[#eq-2|2]]):
 <span id="eq-11"></span>
@@ Line 189: / Line 189: @@
 where <math display="inline">\langle \cdot ,\cdot \rangle _{\mathbb{R}^d}</math> denotes the inner (or ''dot'') product in <math display="inline">\mathbb{R}^d</math>.
-In order to solve the Euler-Lagrange equation ([[#eq-5|5]]), an iterative time-marching scheme can be used. For doing so, an artificial time <math display="inline">t</math> has to be added to the E-L equation and then the steady-state solution has to be computed, i.e., <math display="inline">\frac{\partial }{\partial t}\mathbf{u}+\mathbf{f}+\alpha \,(-\Delta )^\lambda \mathbf{u}=\mathbf{0}</math> (where <math display="inline">\mathbf{f}</math> gathers the first two addends in eq.([[#eq-5|5]])). In the steady-state, the displacement field <math display="inline">\mathbf{u}</math> converges and hence <math display="inline">\frac{\partial }{\partial t}\mathbf{u}=\mathbf{0}</math>. The time <math display="inline">t</math> is discretized, <math display="inline">t=\xi \,\tau </math> (with <math display="inline">\xi \in \mathbb{N}</math> being the iteration index and where <math display="inline">\tau{>0}</math> is the time-step), and finally the temporal derivative is replaced with its discrete approximation (first backward difference). Due to the fact that digital datasets are usually encoded by uniformly distributed spatial elements in each dimension (e.g. pixels if <math display="inline">d=2</math> or voxels if <math display="inline">d=3</math>), the discretization of the spatial variable becomes a natural approach too. Therefore in the following the notation <math display="inline">\mathbf{u}^{(\xi )}[\mathbf{n}]=\mathbf{u}[\mathbf{n},\xi \,\tau ]</math> is used, where <math display="inline">\mathbf{n}=(n_1,\ldots ,n_d)</math> is the index of the discrete spatial position, with <math display="inline">n_i=0,\ldots \,N_i-1</math>, and <math display="inline">N_i</math> being the number of discrete spatial elements in the <math display="inline">i</math>-th dimension. Using this notation, the resulting semi-implicit iteration for the <math display="inline">l</math>-th component of the displacement field is the following:
+In order to solve the Euler-Lagrange equation ([[#eq-5|5]]), an iterative time-marching scheme can be used. For doing so, an artificial time <math display="inline">t</math> has to be added to the E-L equation and then the steady-state solution has to be computed, i.e., <math display="inline">\frac{\partial }{\partial t}\mathbf{u}+\mathbf{f}+\alpha \,(-\Delta )^\lambda \mathbf{u}=\mathbf{0}</math> (where <math display="inline">\mathbf{f}</math> gathers the first two addends in Eq.([[#eq-5|5]])). In the steady-state, the displacement field <math display="inline">\mathbf{u}</math> converges and hence <math display="inline">\frac{\partial }{\partial t}\mathbf{u}=\mathbf{0}</math>. The time <math display="inline">t</math> is discretized, <math display="inline">t=\xi \,\tau </math> (with <math display="inline">\xi \in \mathbb{N}</math> being the iteration index and where <math display="inline">\tau{>0}</math> is the time-step), and finally the temporal derivative is replaced with its discrete approximation (first backward difference). Due to the fact that digital datasets are usually encoded by uniformly distributed spatial elements in each dimension (e.g. pixels if <math display="inline">d=2</math> or voxels if <math display="inline">d=3</math>), the discretization of the spatial variable becomes a natural approach too. Therefore in the following the notation <math display="inline">\mathbf{u}^{(\xi )}[\mathbf{n}]=\mathbf{u}[\mathbf{n},\xi \,\tau ]</math> is used, where <math display="inline">\mathbf{n}=(n_1,\ldots ,n_d)</math> is the index of the discrete spatial position, with <math display="inline">n_i=0,\ldots \,N_i-1</math>, and <math display="inline">N_i</math> being the number of discrete spatial elements in the <math display="inline">i</math>-th dimension. Using this notation, the resulting semi-implicit iteration for the <math display="inline">l</math>-th component of the displacement field is the following:
 <span id="eq-12"></span>

@@ Line 367: / Line 367: @@
 ==Appendix A. Variational formulation of SSIM==
-In this work, we propose the opposite of the structural similarity index as the intensity-based part of the disparity term of the joint energy functional (first addend in equation ([[#eq-2|A.2]])). Thus, a maximization problem is transformed into the minimization of the following cost function:
+In this work, we propose the opposite of the structural similarity index as the intensity-based part of the disparity term of the joint energy functional (first addend in equation ([[#eq-2|2]])). Thus, a maximization problem is transformed into the minimization of the following cost function:
 <span id="eq-1"></span>
@@ Line 434: / Line 434: @@
 where <math display="inline">p_R</math> and <math display="inline">p_{T_\mathbf{u}}</math> denote the marginal intensity distributions estimated from <math display="inline">R</math> and <math display="inline">T_\mathbf{u}</math>, respectively, and <math display="inline">p_{R,T_\mathbf{u}}</math> stands for an estimate of the joint intensity distribution. The intensities of the datasets, <math display="inline">i_r\,,i_t\in \mathcal{I}\subset \mathbb{R}</math>, are considered as random variables whose probability densities are respectively given by <math display="inline">p_R</math> and <math display="inline">p_{T_\mathbf{u}}</math>.
-In order to obtain the external forces field related to the SSIM-based energy term (i.e., the first addend in the Euler-Lagrange equation ([[#eq-5|A.5]])), the computation of the corresponding ''Gateaux'' derivative <math display="inline">\mathrm{d}\mathcal{D}^{\mbox{SSIM}}</math> is required. Therefore, we carry out an explicit computation from the definition of the ''Gateaux'' derivative (see Eq.([[#eq-4|A.4]])):
+In order to obtain the external forces field related to the SSIM-based energy term (i.e., the first addend in the Euler-Lagrange equation ([[#eq-5|5]])), the computation of the corresponding ''Gateaux'' derivative <math display="inline">\mathrm{d}\mathcal{D}^{\mbox{SSIM}}</math> is required. Therefore, we carry out an explicit computation from the definition of the ''Gateaux'' derivative (see Eq.([[#eq-4|4]])):
 <span id="eq-7"></span>
@@ Line 534: / Line 534: @@
 |}
-and where the definitions of <math display="inline">\gamma _0</math>, <math display="inline">\gamma _1</math>, <math display="inline">\gamma _2</math> and <math display="inline">\gamma _3</math> can be found in Eqs.([[#eq-7|A.7]])-([[#eq-10|A.10]]).
+and where the definitions of <math display="inline">\gamma _0</math>, <math display="inline">\gamma _1</math>, <math display="inline">\gamma _2</math> and <math display="inline">\gamma _3</math> can be found in Eqs.([[#eq-7|7]])-([[#eq-10|10]]).
 ==References==