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How many Dof has affine transformation
12 DoF A shearing (3 DoF) à Matrix U An isotropic scaling (3 DoF) à Matrix S A rotation (3 DoF) à Matrix R A translation (3 DoF) à Vector t
What is a versor and how is it used to describe a rigid transform?
A versor is the turning factor of a quaternion. The change of one vector into another is considered in quaternions and made up of two operations; 1st, the rotation of the first vector so that it shall be parallel to the second. A versor expresses the amount and kind of the rotation. It is denoted geometrically by a line at right angles to the plane in which the rotation takes place, the length of this line being proportioned to the amount of rotation. 2d, the change of length so that the first vector shall be equal to the second. A tensor expresses the second operation. The product of the versor and tensor expresses the total operation, and is called a quaternion.
What are the four types of transformations
Rigid à simple Similarity (poly-)affine, Non-rigid à complex
What’s the difference between projective and affine transformation? Projective preserves parallelism, length, and angle Projective does not preserves parallelism, length, and angle The are actually the same
Projective does not preserves parallelism, length, and angle
What is meant by Procrustes?
Optimal fitting problem of least square type: Given two sets of N corresponding points {xi} and {yi}, find the rigid-body transformation (rotation matrix R and translation vector t) that minimizes the mean squared distance between the points Closed-form solution for rigid case. Non-rigid registration requires iterative solution
Image registration, what for?
Change detection à Look for differences in the same type of images (E.g., Mapping Pre and post contrast agent, Mapping structural changes due to function, disease/treatment progress) Image Fusion à MRI – CT, MRI-PET etc. à Correlation information across modalities Pre and post operative comparison Intra operative guided neurosurgery Atlas creation
How many DoF has anisotropic scaling?
3 DoF à Scaling in x, y, z
What are the issues of mutual information?
-       Two images not the same size but the same information on the image (circle) Joint entropy favors the left transformation, MI favors the right transformation Normally not a big issue because one machine does always the same size of image à problem for studies in which different images from different hospitals are taken. Trick à normalize the mutual information
What means mono-modal?
When we have same image modalities of two images à CT – CT, MRI – MRI
What for different dimensionality we can have?
-       2D – 2D 3D – 3D 2D – 2D
An example of multimodality is.. CT/MR Multiple CT images MR/MR CT/CT of two different patients
-       CT/ MR
Image registration is about: Comparing two images Finding a transformation that makes each image unique Finding a transformation that optimize a similarity metric Finding a transformation that minimizing a coast function
-       Finding a transformation that optimize a similarity metric
How is a joint histogram calculated?
-       Two image as intensity map Overlap the images Plot the intensity values of the first image to the intensity values of the second image Compere where both are spotted
What is the main idea of Free-Form Deformation (FFD)?
FFDs deform an underlying object by manipulating a mesh of control points control point can be displaced from their original location control points provide a parameterization of the transformation
What are the advantages and disadvantages of Thin-plate splines?
Advantages: Control points can have arbitrary (willkürlich, beliebig) spatial distribution Disadvantages: Control points have global influence since the radial basis function have infinite support
How many DoF has similarity?
9 DoF à 3 translation, 3 rotation, 3x scaling Angles are preserved Distances changed by a fixed ratio Keeps shape
How many DoF has rigid transformation?
6 DoF à 3 translation, 3 rotations Preserves distances between points Preserves angels
What is the basic method of iterative closest point algorithm (ICP)?
Surface-to-point-cloud rigid alignment
What is the aim of image registration evaluation?
-       Assess accuracy of th alignment
What are the three main motivations for registration?
-       Necessity: Minimization of difference measure can only be ill-posed. Modeling: Regularization can be used to include prior knowledge, for example about underlying tissue properties. Practical Reasons: Without regularization there are a high number of local minima in the energy function (bad for optimization).
How we have a high entropy?
If we have a uniform distribution à max entropy
How is the transformation from the grid space to the “real space” known? (three points)
Image Origin Image dimensions Voxel spacing (or voxel size)
What is a modern approach for multi-modality registration?
-       Apply sophisticated similarity measures
A similarity transformation has… 3 parameters 6 parameters 9 parameters 12 parameters
-       9 parameters
What is Normalized Cross Correlation?
Normalized Cross Correlation: Expresses the linear relationship between voxel intensities in the two volumes NCC works when there is a linear relationship between the two distributions of intensities à multimodality it will struggle à better solution mutual modality NCC is very fast to calculate and tend to be more robust than SSD E.g, if you have CT and a micro-CT of the same patient (modality with same principle à density) then NCC tents to be a good solution
What are the types of non-rigid transformation?
Parametric transformation: affine transformation polynomial transformations (cubic, bi- and linear) spline-based transformations Non-parametric transformations: optical flo type registration, elastic registration, fluid or geodesic registration
What does entropy?
-       Describes the distribution overlap
What stands FFD for?
Free-Form Deformation
What is the basic idea of interpolating spinec?
Basic idea: identify a set of points ϕi in image A identify the corresponding set of points ϕi’ in image B find a spline transformation which interpolates the displacement field at the points and produces a smoothly varying displacement field between the points points are either: anatomical or geometrical landmarks/ pseudo landmarks or control points
Name another spline transformation, which is used in computer graphics for modelling 3D deformable objects:
-       Free-Form Deformation (FFD) are a common technique in Computer Graphics for modelling 3D deformable objects (B-splines)
What forces is elastic deformation governed by?
internal forces, caused by deformation of the elastic body. internal = 0 if body is in equilibrium external forces, acts on the elastic body and causes the body to deform away from the equilibrium
What is non-rigid image registration used for?
Motion: To compensate for tissue deformation such as lung movement when breathing. Longitudinal or cross-sectional studies: - For quantification of change over time, such as brain growth. For quantification of differences btw populations. Atlas based studies: For segmentation via atlas propagation.
For what is affine registration appropriate?
Appropriate for: If not all image acquisition parameters are known àunknown: voxel size, gantry tilt If scales changes are expected à growth, inter subject registration à Limited applicability expect as initialization for non-rigid registration
Why is image registration needed?
If we want to fuse different image types or see a change between a period of time, we have to scan de patient multiple times à remove from scanner We cannot easily fix/know patient location and orientation with respect to different imaging systems Need to remove differences in patient positioning to relate information from different types of images
What is the limitation of SSD?
If there are two more or less identical images but with different illumination, we can have a very unstable SSD matric à pixel intensities are different à high penalty These differences can happen on MRI images because of a coil effect Solution: First normalization of the image and then registration
What are the two different types of non-rigid transformation? (Name the types and some examples to each type)
Parametric transformation: (#DOF << # Voxels) Affine transformation Polynomial transformation (linear, quadratic, cubic) Spline-based transformation Non-parametric transformations: (#DOF = # Voxels) Optical flow type registration (e.g., demons) Elastic registration Fluid or geodesic registration
How many DoF has translation transformation?
3 DoF à x, y, z
What is image registration? And how?
Establishing correspondence between features in sets of images, and using transformation models to infer correspondence away from those features Overlay of a two or a set of images Apply a given transformation until a given matching criteria is fulfilled à optimization process Goal is to find a transformation T, such we minimize a criteria to singularity
How can we express FFD based on B-splines?
FFDs based on B-splines can be expressed as a 3D tensor product of 1D B-splines:
Does this image show a low or high entropy?
-       Low entropy à aligned
When we have a low entropy?
If we have a non-uniform distribution
Name some image registration evaluation metrics:
Landmark based à typically uses RMSE (root-mean-square-error) metrics Hausdorff distance Uses of synthetic transformations Visualization of registration result à deformation of structured grid
Which addition we have for the mutual information?
-       Additionally, to the joint histogram, we have an histogram of the image X and the image Y
How does the landmark-based evaluation works and what are the contras of this evaluation metric?
Landmark-based typically uses RMSE (root-mean-square-error) metrics Cons: Landmark annotation
On what depends the complexity of the image registration evaluation?
Complexity of the evaluation depends on the type of registration
What is the main idea in linear interpolation?
The idea is to survey the 2 closest pixels, then draw a line between them and designate a value along that line as the output pixel value.
Image registration happens at the level of… Voxel units Real units (e.g., mm) It doesn’t matter
-       Real units (e.g., mm)
What is the main idea in bilinear interpolation?
The idea is to survey the 4 closest pixels, then create a weighted average based on the nearness and brightness of the surveyed pixels and assign that value to the pixel in the output image. Use cubic convolution if a higher degree of accuracy is needed. However, with still images, the difference between images interpolated with this method and cubic convolution methods is usually undetectable. This interpolation usually supplies a much more viable alternative than the others.
What are splines in a mathematical meaning?
Tools, used to approximate or interpolate functions from scatteres data. (1D: cruves, 2D: surfaces, 3D: volumes)
What are the different types of information?
Intensity-bases à pure intensity of pixels Non-intensity-based
What is A and B describing?
-       A: describe rotations, shear, scaling B: describe translation
What is used to compute a translation which is intensity-based?
similarity constructed based on voxel intensity information
How does the deformation of structural grid evaluation works and what are the contras of this evaluation metric?
Deformation of structured grid à check for folds, unnatural local deformations Cons: difficult to check the entire 3D space
When we need non-rigid registration in neuroimaging?
Motion: Compensation for tissue deformation à Brain deformation during neurosurgery / Fusion of functional and anatomical information (MR/PET, etc.) Atlas based studies: Segmentation via atlas propagation Longitudinal or cross-sectional studies: Quantification of change over time (brain growth/ atrophy) Quantification of differences between populations (voxel based -, deformation-based morphometry
What is the problem if we using entropy?
If we just align black background, we have a minimum entropy à max alignment but the picture is not really aligned
What does mutual information
-       It not just considers the overlap area but also the content of the image
What is used to compute a translation which is feature-based?
Uses points, landmarks, curves, etc. to compute a transformation Intrinsic (part of the anatomy) or extrinsic (fiducial point) Similarity is based on an Euclidean distance
What is the difference between a Eulerian frame and a Lagrangian frame?
Eulerian: Just object is transformed à grid not à flow of material over time Grid with object is transformed à material fix to grid
How we can optimize image registration?
Mostly based on classical optimization schemes à some form of gradient descent (E.g., Steepest gradient descent, Conjugate gradient descent) Means to find a potential minima!
What are splines in the mathematic context?
Splines are a tool used to approximate or interpolate function from scattered data 1D: Curves 2D: Surfaces 3D: Volumes
What is the difference between SSD (Sum of Squared Differences) and SAD (Sum of Absolut Differences?
SSD is better against outliers à because of the square there is a much higher penalty for the wider away points SAD à Less sensitive on large intensity differences than SSD E.g., CT scan of bones à SSD to align bone material à bones intensity are very large à if there mistake on registration then there is a very big difference between the high intensity bone and the background If we want to register two images in which one has metal artefacts à SSD is very critical because it would penalize the metal artefact extrem
For what is a joint histogram used?
joint histogram is a useful tool for visualizing the relationship between the intensities of corresponding voxels in two or more images.
What are some parameters of image registration optimization?
Maximum and minimum step size, stopping criteria, number of iterations Importance of Initialization (Manual, based on momentum, geometry of VOI) Number of samples for metric calculation (typically as a percentage of voxel number)
For what is rigid registration appropriate?
Appropriate for: Brain (constrained by skull) Bone (neck, vertebrae)
Describe the function of the ICP algorithm
ICP algorithm: Pick a set of predefined features in one data set. Choose the closest feature to each in the other data set. Solve the problem, bringing the data sets closer. Repeat the pairing selection until the distance is minimized. Converted to a local minimum Datasets must be “reasonably” close à requires a good initial guess (initialization) Local minima
What are some basic similarity metrics?
SSD/SAD (Sum of Squared Differences & Sum of Absolute Differences Cross Correlation Mutual Information Derivatives
How does the Hausdorff distance evaluation works and what are the contras of this evaluation metric?
-       doesn’t describe the entire transform
What are good similarity metric features?
Extremum for correctly aligned images Smooth, best convex Fast computation Differentiable
What is the main idea in nearest neighbor interpolation?
The idea is to use the pixel value of the data point or measurement which is closest to the current point. This is the fastest interpolation method but the resulting image may contain jagged edges.
On what is Mutual Information (MI) based?
Does not assume any type of relationship of intensities. Focuses on structure. Main idea: maximize amount of explained information between images (seen as prob. distributions)
What are the steps of ICP?
1. Compute closest points {yi} on Y 2. Register points {xi} to points {yi} 3. Apply resulting transformation to points {xi} 4. Repeat until convergence
What's the difference between projective and affine transformations?
The sole difference between these two transformations is in the last line of the transformation matrix. The projective transformation does not preserve parallelism, length, and angle. But it still preserves collinearity and incidence. Since the affine transformation is a special case of the projective transformation (the first two elements of the last line should be zeros), it has the same properties. However unlike projective transformation, it preserves parallelism. Projective transformation can be represented as transformation of an arbitrary quadrangle (i.e. system of four points) into another one. Affine transformation is a transformation of a triangle. Since the last row of a matrix is zeroed, three points are enough.
What means multi-modal?
When we have different image modalities of two images à CT – MRI, CT – PET
How can thin plate splines defined?
Thin-plate splines can be defined as a linear combination of radial basis functions θ:
Does this image show a low or high entropy?
-       High entropy à not aligned
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