Algorithm:GATech:DWMRI Musings - Revision history

Melonakos at 15:37, 20 September 2007

2007-09-20T15:37:03Z

Melonakos at 17:10, 27 July 2007

2007-07-27T17:10:14Z

Melonakos at 17:08, 27 July 2007

2007-07-27T17:08:47Z

Melonakos at 16:59, 27 July 2007

2007-07-27T16:59:19Z

Melonakos at 16:58, 27 July 2007

2007-07-27T16:58:55Z

Melonakos at 16:57, 27 July 2007

2007-07-27T16:57:33Z

Melonakos at 16:57, 27 July 2007

2007-07-27T16:57:15Z

Tannenba at 15:27, 27 July 2007

2007-07-27T15:27:53Z

Melonakos at 19:37, 26 July 2007

2007-07-26T19:37:30Z

Melonakos at 19:03, 26 July 2007

2007-07-26T19:03:21Z

← Older revision		Revision as of 15:37, 20 September 2007
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	'''Algorithm Categorization'''		'''Algorithm Categorization'''
		+
		+	[[media:DWMRIAlgorithms.pdf\|A Nice PDF View of DWMRI Categorization]]

	Algorithms designed to answer the primary DW-MRI questions listed above come in a variety of flavors. There are many possible paths to go from raw DW-MRI data to clinical answers. In fact, most algorithms do not even attempt to go all the way from the DW-MRI data to a clinical comparison of different populations. Instead, algorithms focus on particular subsets of the various possible pipelines to get from DW-MRI data to clinical answers. In attempting to find method in the madness, it is useful to consider the key distinguishing factors by which algorithms may be categorized. The following is an attempt at listing a few of categories of distinguishing algorithm characteristics (please refer to the ''Synchronizing Algorithm Terminology'' section for explanations of the terms):		Algorithms designed to answer the primary DW-MRI questions listed above come in a variety of flavors. There are many possible paths to go from raw DW-MRI data to clinical answers. In fact, most algorithms do not even attempt to go all the way from the DW-MRI data to a clinical comparison of different populations. Instead, algorithms focus on particular subsets of the various possible pipelines to get from DW-MRI data to clinical answers. In attempting to find method in the madness, it is useful to consider the key distinguishing factors by which algorithms may be categorized. The following is an attempt at listing a few of categories of distinguishing algorithm characteristics (please refer to the ''Synchronizing Algorithm Terminology'' section for explanations of the terms):

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 '''DW-MRI vs DT-MRI'''
-A Preliminary Note: I should mention that I have taken the position that for the purposes of an algorithms person, it is the DW-MRI data that is handed to us by the acquisition folks, so it is with the DW-MRI data that image processing begins.  A more adventurous algorithms person may attempt to look back at reconstruction algorithms which are tied to the physics of MR in the hope of parlaying even more information from the scanner to the end result.  But for our purposes, we can probably assume that our hands are already full attempting to analyze the DW-MRI data.
+A Preliminary Note: I should mention that I have taken the position that for the purposes of an algorithms person, it is the DW-MRI data that is handed to us by the acquisition folks, so it is with the DW-MRI data that image processing begins.  A more adventurous algorithms person may attempt to look back at reconstruction algorithms which are tied to the physics of MR in the hope of parlaying even more information from the scanner to the end result... (I'm not that adventurous!).
 In answering the questions posed above, it is common for people to assume from the onset that we are using terminology related to tensors (i.e. DT-MRI, DTI).  The raw data coming out of the scanner is referred to as DW-MRI or DWI.  The difference between the "T" and the "W" is that the "T" has undergone a preprocessing step in which tensors are computed at each voxel that are a function of the original "W" data.  Hence, the data resulting from this preprocessing step is referred to at DT-MRI.  There are advantages and disadvantages to using this preprocessing tensor construction step.  The point here is that when we think about trying to find ways to go from the scanner data to clinical answers, we are talking about going from DW-MRI data to clinical answers.  In other words, it is ''okay'' to think outside the box and to consider ways in which ellipsoidal constraints arising from the tensor model may be meaningfully relaxed.

← Older revision		Revision as of 17:08, 27 July 2007
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	'''DW-MRI vs DT-MRI'''		'''DW-MRI vs DT-MRI'''
		+
		+	A Preliminary Note: I should mention that I have taken the position that for the purposes of an algorithms person, it is the DW-MRI data that is handed to us by the acquisition folks, so it is with the DW-MRI data that image processing begins. A more adventurous algorithms person may attempt to look back at reconstruction algorithms which are tied to the physics of MR in the hope of parlaying even more information from the scanner to the end result. But for our purposes, we can probably assume that our hands are already full attempting to analyze the DW-MRI data.

	In answering the questions posed above, it is common for people to assume from the onset that we are using terminology related to tensors (i.e. DT-MRI, DTI). The raw data coming out of the scanner is referred to as DW-MRI or DWI. The difference between the "T" and the "W" is that the "T" has undergone a preprocessing step in which tensors are computed at each voxel that are a function of the original "W" data. Hence, the data resulting from this preprocessing step is referred to at DT-MRI. There are advantages and disadvantages to using this preprocessing tensor construction step. The point here is that when we think about trying to find ways to go from the scanner data to clinical answers, we are talking about going from DW-MRI data to clinical answers. In other words, it is ''okay'' to think outside the box and to consider ways in which ellipsoidal constraints arising from the tensor model may be meaningfully relaxed.		In answering the questions posed above, it is common for people to assume from the onset that we are using terminology related to tensors (i.e. DT-MRI, DTI). The raw data coming out of the scanner is referred to as DW-MRI or DWI. The difference between the "T" and the "W" is that the "T" has undergone a preprocessing step in which tensors are computed at each voxel that are a function of the original "W" data. Hence, the data resulting from this preprocessing step is referred to at DT-MRI. There are advantages and disadvantages to using this preprocessing tensor construction step. The point here is that when we think about trying to find ways to go from the scanner data to clinical answers, we are talking about going from DW-MRI data to clinical answers. In other words, it is ''okay'' to think outside the box and to consider ways in which ellipsoidal constraints arising from the tensor model may be meaningfully relaxed.

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 '''Hamilton-Jacobi vs Hamilton-Jacobi-Bellman vs Eikonal Equations'''
-These are the partial-differential equations (PDEs) which are typically solved when attempting to find optimal paths in a domain, such as optimal tract in the brain.
+These are the partial-differential equations (PDEs) which are typically solved when attempting to find optimal paths in a domain, such as optimal tracts in the brain.
 Let U(p): R^n->R be a scalar field.  Now let the DW-MRI data coming out of the scanner be the gradient of U(p), denoted grad(U(p)), which is equivalent to the directional information.  General Hamilton-Jacobi equations are a given as a function of U(p), grad(U(p)), and p as follows:  H(U(p),grad(U(p)),p)=0.  However, for functionals built upon DW-MRI data, we are not given the scalar field U(p).  Hence, we end up solving a subset of Hamilton-Jacobi equations, which does not depend upon U(p).  This subset is termed Hamilton-Jacobi-Bellman equations, given by:  H(grad(U(p)),p)=0.  So any energy functional attempting to find optimal paths in DW-MRI data is effectually solving a PDE of the Hamilton-Jacobi-Bellman class.

← Older revision		Revision as of 16:58, 27 July 2007
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	'''Hamilton-Jacobi vs Hamilton-Jacobi-Bellman vs Eikonal Equations'''		'''Hamilton-Jacobi vs Hamilton-Jacobi-Bellman vs Eikonal Equations'''
		+
		+	These are the partial-differential equations (PDEs) which are typically solved when attempting to find optimal paths in a domain, such as optimal tract in the brain.

	Let U(p): R^n->R be a scalar field. Now let the DW-MRI data coming out of the scanner be the gradient of U(p), denoted grad(U(p)), which is equivalent to the directional information. General Hamilton-Jacobi equations are a given as a function of U(p), grad(U(p)), and p as follows: H(U(p),grad(U(p)),p)=0. However, for functionals built upon DW-MRI data, we are not given the scalar field U(p). Hence, we end up solving a subset of Hamilton-Jacobi equations, which does not depend upon U(p). This subset is termed Hamilton-Jacobi-Bellman equations, given by: H(grad(U(p)),p)=0. So any energy functional attempting to find optimal paths in DW-MRI data is effectually solving a PDE of the Hamilton-Jacobi-Bellman class.		Let U(p): R^n->R be a scalar field. Now let the DW-MRI data coming out of the scanner be the gradient of U(p), denoted grad(U(p)), which is equivalent to the directional information. General Hamilton-Jacobi equations are a given as a function of U(p), grad(U(p)), and p as follows: H(U(p),grad(U(p)),p)=0. However, for functionals built upon DW-MRI data, we are not given the scalar field U(p). Hence, we end up solving a subset of Hamilton-Jacobi equations, which does not depend upon U(p). This subset is termed Hamilton-Jacobi-Bellman equations, given by: H(grad(U(p)),p)=0. So any energy functional attempting to find optimal paths in DW-MRI data is effectually solving a PDE of the Hamilton-Jacobi-Bellman class.

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 Note that in the absence of directional information, you are left with the Eikonal equation, which is a further subset of Hamilton-Jacobi equations.  A common temptation of algorithm designers is to attempt to solve the Eikonal equation for DW-MRI problems, because the Eikonal equation is so popular in other directionally isotropic cases.  However, when considering solutions to energy functionals based upon directionally dependent DW-MRI data, it is inappropriate to attempt to solve the directionally isotropic Eikonal equation.
-'''Numerical Solutions to Hamilton-Jacobi Equations'''
+'''Numeric Solutions to Hamilton-Jacobi Equations'''
 First of all, note that Hamilton-Jacobi-Bellman and Eikonal Equations are also Hamilton-Jacobi (HJ) equations.  The following contains generic ideas which may be applied to numeric solutions of any class of HJ equations.

← Older revision		Revision as of 16:57, 27 July 2007
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	Note that in the absence of directional information, you are left with the Eikonal equation, which is a further subset of Hamilton-Jacobi equations. A common temptation of algorithm designers is to attempt to solve the Eikonal equation for DW-MRI problems, because the Eikonal equation is so popular in other directionally isotropic cases. However, when considering solutions to energy functionals based upon directionally dependent DW-MRI data, it is inappropriate to attempt to solve the directionally isotropic Eikonal equation.		Note that in the absence of directional information, you are left with the Eikonal equation, which is a further subset of Hamilton-Jacobi equations. A common temptation of algorithm designers is to attempt to solve the Eikonal equation for DW-MRI problems, because the Eikonal equation is so popular in other directionally isotropic cases. However, when considering solutions to energy functionals based upon directionally dependent DW-MRI data, it is inappropriate to attempt to solve the directionally isotropic Eikonal equation.
		+
		+	'''Numerical Solutions to Hamilton-Jacobi Equations'''
		+
		+	First of all, note that Hamilton-Jacobi-Bellman and Eikonal Equations are also Hamilton-Jacobi (HJ) equations. The following contains generic ideas which may be applied to numeric solutions of any class of HJ equations.
		+
		+	Assume that one wishes to find the numerical solution to an HJ equation that yields an optimal open curve connecting two ROIs in a domain. Numerically, there are two standard ways in which these solutions are computed:
		+
		+	*Gradient descent along the solution of a PDE, taking either one of the ROIs as the seed and the other as the target region (see for instance the Live Wire algorithm by Mortensen, et al.).
		+	*Summation of the solutions of two PDEs, where the only difference between the two PDEs is the seed domain -- one PDE for each ROI (see for instance the work by Yezzi and Prince on computing Tissue Thickness in IEEE TMI, Oct 2003).
		+
		+	These two approaches give equivalent results (assuming certain minimal requirements such as symmetry in the underlying metric, etc). Therefore, the choice of approach typically depends upon computational complexity of the added PDE solver versus the gradient descent step.

@@ Line 11: / Line 11: @@
 '''DW-MRI vs DT-MRI'''
-In answering the questions posed above, it is common for people to assume from the onset that we are using terminology related to tensors (i.e. DT-MRI, DTI).  The raw data coming out of the scanner is referred to as DW-MRI or DWI.  The difference between the "T" and the "W" is that the "T" has undergone a preprocessing step which computes tensors at each voxel which are a function of the original "W" data.  Hence, the data resulting from this preprocessing step is referred to at DT-MRI.  There are advantages and disadvantages to using this preprocessing tensor construction step.  The point here is that when we think about trying to find ways to go from the scanner data to clinical answers, we are talking about going from DW-MRI data to clinical answers.  In other words, it is ''okay'' to think outside the box and to consider ways in which ellipsoidal constraints arising from the tensor model may be meaningfully relaxed.
+In answering the questions posed above, it is common for people to assume from the onset that we are using terminology related to tensors (i.e. DT-MRI, DTI).  The raw data coming out of the scanner is referred to as DW-MRI or DWI.  The difference between the "T" and the "W" is that the "T" has undergone a preprocessing step in which tensors are computed at each voxel that are a function of the original "W" data.  Hence, the data resulting from this preprocessing step is referred to at DT-MRI.  There are advantages and disadvantages to using this preprocessing tensor construction step.  The point here is that when we think about trying to find ways to go from the scanner data to clinical answers, we are talking about going from DW-MRI data to clinical answers.  In other words, it is ''okay'' to think outside the box and to consider ways in which ellipsoidal constraints arising from the tensor model may be meaningfully relaxed.
 '''A Spaghetti Fancy'''
-In this line of research, the most common figures one can expect to see in the key Neuroimage, Science, Nature, etc. publications are pretty pictures of a bunch of open curves (or fibers) sprawling through the brain.  These images are so common that some people have come to presuppose that this is the kind of output one should expect from algorithms which process DW-MRI or DT-MRI data.  Of course, these images are the result of streamline techniques, which is one important branch of DW-MRI research.  But if you consider recent 2006-2007 publications on DW-MRI and DT-MRI algorithms, you quickly realize that the bent of research has recently experienced a shift toward other optimal path / volumetric segmentation type approaches, which do not produce these kinds of sprawling fiber figures.  So, in current and future publications, one can expect to see a greater variety of visualizations which reflect the other branches of research emerging in DW-MRI processing.
+In this line of research, the most common figures one can expect to see in the key Neuroimage, Science, Nature, etc. publications are pretty pictures of a bunch of open curves (or fibers) sprawling through the brain.  These images are so common that some people have come to presuppose that this is the kind of output one should expect from algorithms which process DW-MRI or DT-MRI data.  Of course, these images are the result of streamline techniques, which is one important branch of DW-MRI research.  But if you consider recent 2006-2007 publications on DW-MRI and DT-MRI algorithms, you quickly realize that this research trend has recently experienced a shift toward other optimal path / volumetric segmentation type approaches, which do not produce these kinds of sprawling fiber figures.  So, in current and future publications, one can expect to see a greater variety of visualizations which reflect the other branches of research emerging in DW-MRI processing.
 '''Synchronizing Algorithm Terminology'''
@@ Line 67: / Line 67: @@
 '''Beware of the Segmentation-Driven Statistical Comparison Paradox'''
-A common temptation among DW-MRI researchers is to use the statistical measures (such as FA) which they wish to compare between two populations as an input in their tractography or fiber bundle segmentation algorithms.  The paradox here is the following:  If you use a statistic to give you a segmentation that you like, then when you are comparing two populations based on this statistic you are unable to determine whether any statistical significance arising from the study is due to the fact that the populations are truly statistically distinct or due to the fact that the segmentations you achieved in the first place were biased towards favoring that statistic.  Therefore, it is highly important that the statistics used in the population comparisons be independent of the statistics used to achieve the tractography or segmentation results.
+A common temptation among DW-MRI researchers is to use the statistical measures (such as FA) with which they wish to compare between two populations as an input in their tractography or fiber bundle segmentation algorithms.  The paradox here is the following:  If you use a statistic to give you a segmentation that you like, then when you are comparing two populations based on this statistic you are unable to determine whether any statistical significance arising from the study is due to the fact that the populations are truly statistically distinct or due to the fact that the segmentations you achieved in the first place were biased towards favoring that statistic.  Therefore, it is highly important that the statistics used in the population comparisons be independent of the statistics used to achieve the tractography or segmentation results.
 '''Beware of Mr. Legendre'''

@@ Line 45: / Line 45: @@
 '''Algorithm Categorization'''
-Algorithms designed to answer the primary DW-MRI questions listed above come in a variety of flavors.  There are many possible paths to go from raw DW-MRI data to clinical answers.  In fact, most algorithms do not even attempt to go all the way from the DW-MRI data to a clinical comparison of different populations.  Instead, algorithms focus on particular subsets of the various possible pipelines to get from DW-MRI data to clinical answers.  In attempt to find method in the madness, it is useful to consider the key distinguishing factors by which algorithms may be categorized.  The following is an attempt at listing a few of categories of distinguishing algorithm characteristics (please refer to the ''Synchronizing Algorithm Terminology'' section for explanations of the terms):
+Algorithms designed to answer the primary DW-MRI questions listed above come in a variety of flavors.  There are many possible paths to go from raw DW-MRI data to clinical answers.  In fact, most algorithms do not even attempt to go all the way from the DW-MRI data to a clinical comparison of different populations.  Instead, algorithms focus on particular subsets of the various possible pipelines to get from DW-MRI data to clinical answers.  In attempting to find method in the madness, it is useful to consider the key distinguishing factors by which algorithms may be categorized.  The following is an attempt at listing a few of categories of distinguishing algorithm characteristics (please refer to the ''Synchronizing Algorithm Terminology'' section for explanations of the terms):
 *Is an explicit segmentation provided?
@@ Line 57: / Line 57: @@
 **If so, does the connectivity map explain fiber connectivity or fiber bundle connectivity (i.e. does it tell how well a volumetric fiber bundle connects two ROIs or does it tell the ''best'' connectivity available among all possible infinitesimally small paths connecting the two ROIs).
 **If so, which metrics are optimized in the computation of the connectivity map and is the algorithm flexible to determine connectivity under a variety of metrics?
 *What statistics are provided for comparison of clinical populations?