tissue engineering introduction for physicists - lecture three

Tissue Engineering

Medical Imaging / Volume visualization

September 2014

Part Two

Medical Imaging / VV

“Visualization is a method of computing. It transforms the symbolic into

the geometric, enabling researchers to observe their simulations and

computations. Visualization offers a method for seeing the unseen. It

enriches the process of scientific discovery and fosters profound and

unexpected insights.”

McCormick, B. H. (1988). Visualization in scientific

computing. ACM SIGBIO Newsletter, 10(1), 15–21.

doi:10.1145/43965.43966


Sources of Error● Data acquisition

○ Sampling: are we (spatially) sampling data with enough precision to get what we need

out of it?

○ Quantization: are we converting “real” data to a representation with enough precision

to discriminate the relevant features?

● Filtering

○ Are we retaining/removing the “important/non-relevant” structures of the data?

○ Frequency/spatial domain filtering

■ Noise, clipping and cropping

● Selecting the “right” variable

○ Does this variable reflect the interesting features?

○ Does this variable allow for a “critical point” analysis?


Sources of Error● Functional model for resampling

○ What kind of information do we introduce by interpolation and approximation?

● Mapping

○ Are we choosing the graphical primitives appropriately in order to depict the kind of

information we want to get out of the data?

○ Think of some real world analogue (metapher)

● Rendering

○ Need for interactive rendering often determines the chosen abstraction level

○ Consider limitations of the underlying display technology

■ Data color quantization

○ Carefully add “realism”

■ The most realistic image is not necessarily the most informative one


Segmentation of anatomical structures of interest

Aligning multiple dataset (registration)

Three-dimensional visual representation

Medical Imaging / VV → Segmentation

Segmentation generally means dividing an image into connected regions.

Segmentation in the domain of medical imaging has some characteristic that

make the segmentation task easier and difficult at the same time.

● The imaging is narrowly focused on an anatomic region.

● The imaging context is well-defined.

● The pose variation are limited, and there is usually prior knowledge of

the number of tissues and the Region of Interest (ROI).

❖ The images produced in this field are one of the most challenging due

to the poor quality of imaging.


Depending on the level of feature extraction as an input to the

segmentation, we can methodically classify segmentation as:

● Pixel-oriented

● Edge-oriented

● Texture-oriented

● Region-oriented

Or hybrid approaches, which result from combination of single

procedures.


Pixel- Based Segmentation

Pixel-based procedures of segmentation only consider the grayscale or

color value of current pixels disregarding its surroundings.

Pixel-based approaches are NOT segmentation procedures in the strict

sense of our definition. Since each pixel is considered only isolated from its

neighborhood, it cannot be ensured that actually only connected segments

are obtained. For this reason, post-processing is required.

Most pixel-based procedures use thresholds in the histogram of an image.


Pixel- Based Segmentation → Static Thresholding

If the assignment of Pixel intensities is well known and constant for a certain

type of tissue, static thresholds are applicable.

A static threshold is independent of the individual instance in a set of similar

images.


Pixel- Based Segmentation → Static Thresholding / Example

Pixel-based segmentation in CT relies on Hounsfield Units (HU), which

allow the definition of windows for different type of tissue:

Bone [200 … 3000] Water [-200 … 200]

Fat [-500 … -200] Air [-1000 … -500]


Pixel- Based Segmentation → Adaptive Thresholding

Global adaptive thresholds result from analyzing each individual image

entirely.

The well-known method of Otsu is based on a simple object vs. background

model.



Otsu’s method is used to automatically perform clustering-based image

thresholding, or, the reduction of a gray level image to binary image.

The algorithm assumes that the image contains two classes of pixels

(foreground pixels and background pixels)

It then calculates the optimum threshold separating the two classes so that

their combined spread (intra-class variance) is minimal.



In Otsu’s method we search for the threshold that minimizes the intra-class variance (the variance

within the class), defined as a weighted sum of variances of the two classes:

Weights are the probabilities of the classes separated by a threshold t

and variances of these classes.

Otsu shows that minimizing the intra-class variance is the same as

maximizing inter-class variance



Otsu shows that minimizing the intra-class variance is the same as

maximizing inter-class variance

which is expressed in terms of class probabilities and class means



The class probability is computed from the histogram as t:

While the class mean is

where x(i) is the value at the center of ith histogram bin.


Using locally adaptive thresholds, the threshold is computed not only for each image individually,

but also for each region within an image.

This is particularly necessary if the simple object to background assumption is globally invalid

because of continuous brightness gradient.


Example: (Threshold)

Segments obtained from pixel-based analysis usually are incoherent and highly noisy. Therefore,

pot-processing is required.

Blood tracer detection with using pixel-based segmentation (Thresholding)


● Spatial information: Since intensity histograms do not preserve spatial contiguity of pixels, one

variation is to add spatial position (x,y) or (x,y,z) to form a multi-dimensional feature vector

incorporating spatial layout.

● Temporal information: If the medical images are in a time sequence, then time can be added as

an additional feature in the representation space.


Clustering

Pixel clustering is another way of pixel-based segmentation. This statistical method is particularly

suitable if more than one value is assigned to each pixel and regarded in the segmentation

process (e.g., color images).


Clustering

There is a large body of work on clustering algorithms. For our purposes, we can categorize them

into three broad classes:

● Deterministic clustering

● Probabilistic clustering (model-based clustering)

● Graph-theoretic clustering


Clustering → Deterministic algorithms

This algorithms known as the simplest methods such as

❏ K-means

❏ mean-shift

❏ agglomerative methods


Clustering → Deterministic algorithms

K-means approach, provides good results when the data is convex or blob-like.

Def. : Given a set of observations (x , x ,..., x ), where each observation is a d-dimensional real

vector, k-means clustering aims to partition the n observations into k(< n) sets S={S , S ,...,S } so

as to minimize the within-cluster sum of squares (WCSS). In other words, its objective to find:

where is the mean of points in S

1 2 n

1 2 k

i


Edge-Based Segmentation

This type of segmentation is based on the abstract level of edges and tries

to capture the objects due to their closed outline in the image.

Because of that, edge-based segmentation, only used where objects are

presented as clearly defined boundaries.


Edge-Based Segmentation→ Livewire Segmentation

In practice, edge-based segmentation is often realized semi-automatically.

By the interactive livewire segmentation, the user clicks onto or near by the edge of the Object of

Interest (OOI), and the computer determines the exact edge location based on local gradients.

Then, the computer calculates a cost function, which again is based on local gradients. For all

paths (wire) to the current position of the cursor, the path with the lowest cost is displayed in real

time (live) as the cursor is moved manually.


Edge-Based Segmentation→ Livewire Segmentation


Region-Based Segmentation

As an advantage of region-based segmentation, only connected segments are produced, and

morphological post-processing is avoided.

There are agglomerative (bottom-up) and divisive (top-bottom) approaches.

All approaches are based on a certain distance or similarity measure to guide the assignment of

neighbored pixels or regions.


Region-Based Segmentation → Agglomerative Algorithm

Region growing, in 3D also referred to as volume growing, is a well known example of an

agglomerative procedure.

Starting from seed points, which may be placed either automatically or manually, neighbored pixel

are iteratively associated to the growing areas if the distance measure is below a certain threshold.

This process is iterate until no more merges can be carried out.


Region-Based Segmentation → Divisive Algorithm

The divisive approach somehow inverts the agglomerative strategy. By splitting, the regions are

iteratively subdivided until they are considered sufficiently homogeneous in terms of the chosen

similarity measure.

Advantage:

Seed points are not require anymore, because the first split is performed throughout

the whole image.

Disadvantage:

The dividing lines are usually drawn horizontally or vertically, and this arbitrary

separation may separate the image objects.


Region-Based Segmentation → Divisive Algorithm


Model-Based Segmentation → Active Contour Model

Active contour models apply edge-based segmentation considering region-based aspects and an

object-based model of a priori knowledge.

In the medical application domain, so called snake and balloon approaches are applied for

segmentation of 2D and 3D image data and the tracing of contours in 2D image and 3D image

sequences.

The contour of the objects, which is usually closely modeled, is presented by individual nodes,

which are - in the simplest case - piecewise connected with straight lines forming a closed

polygon.



Snake approach:

In 1988, the classical snake approach was introduced by Kass et al. (Snakes: Active Contour

Models)



Snake approach:

It models an internal and an external quality criterion, both as undirected energy. The internal

energy results from a predefined elasticity and stiffness of the contour, which is high in places of

strong bends or on buckling.

The external energy is calculated from an edge-filtered image. The external energy is small, if the

contour runs along edges.

The idea behind this approach is an edge-based segmentation combined with the a priori

knowledge that biological objects rarely have sharp-bending boundaries.

With an optimal weighting of energy terms, the contour course is primarily determined by the

information of edges in the image.



Snake approach:

However, if the object’s contour is partially covered or incompletely captured, the internal energy

ensures an appropriate interpolation of the region’s shape.



Snake approach:



Balloon approach:

Balloons are based on forces rather than energies. Besides the internal and external forces, an

inner pressure or suction is modeled, which lets the contour continuously expand or shrink.


Example:

Considering brain MR imaging, as an example, we know that there are three main tissues of

interest:

● White matter

● Gray matter

● Cerebrospinal fluid

➔ We also know that in a pathological situation there may be one additional class such as

lesions or tumors.

The core operation is the division of the image into a finite set of regions, which are smooth and

homogeneous in their content and their representation.

When posed in this way, segmentation can be regarded as a problem of finding clusters in a

selected feature space.


Example: (Threshold)

Such a simple segmentation approach, is often

insufficient for medical images where the imaging

protocol can lead to variations in regional contrast

making the task of segmentation difficult.


Several variations on classical histogram

thresholding have been proposed for medical

image segmentation.

● Multi-modal or multi-sequence data

Multi-dimensional are histograms formed from the

intensity values produced by each of the imaging

protocols. It is often the case that several

acquisitions are available for the same image.

T

T1

2


References for segmentation:

● M. Sonka, J. M. Fitzpatrick, Eds., Handbook of Medical Imaging, Volume 2.

Medical Image Processing and Analysis, SPIE, 2000

● Elnakib, A., Gimel’farb, G., Suri, J. S., & El-Baz, A. (2011). Medical Image

Segmentation: A Brief Survey. In A. S. El-Baz, R. Acharya U, A. F. Laine, &

J. S. Suri (Eds.), Multi Modality State-of-the-Art Medical Image

Segmentation and Registration Methodologies. New York, NY: Springer

New York. doi:10.1007/978-1-4419-8204-9

Medical Imaging / VV → Image Registration

Image Registration

Image Registration is an important problem in image analysis with many

applications:

● Several images of the same object are taken using different imaging

modality

● Several images of the same object are taken at different time instants

● It is necessary to compare two objects

● It is desired to match an image to a model


● The problem of image registration is to determine an unknown geometric

transformation that maps one image into another (to a certain degree

accuracy)

○ In other word, after registration problem is solved, for each pixel in the

first image we know the corresponding pixel in the second image.

● This assumes that the images are similar in the sense that both images

contain the same (or similar) object, which may be rotated, translated, or

elastically deformed


● In medical applications, image registration is usually done for two-

dimensional and three-dimensional images

● In general, registration problem can be solved in any number of spatial or

temporal dimensions.


● When two images are registered it is possible to:

○ Analyze (detect) differences between the images (e.g. images taken

at two different time instants or difference between the template and a

tested product in visual inspection)

○ Combine information contained in multiple images into a single image

(image fusion) with the goal of easier interpretation by humans (e.g. in

radiology it is possible to do multimodality image registration - MR to

CT, etc. )


● Medical image registration is required for:

○ Use of different imaging modalities (e.g. MR/CT)

○ Progressive disease tracking (imaging in regular time intervals and

detection of changes, e.g. for tumor treatment evaluation)

○ In computer assisted surgery (e.g. in neurosurgery preoperative MR

images may be registered with intraoperative MR images for surgical

navigation)

○ Matching of patient images to a model


Example: Hand Registration

● X-ray image (anatomical information)

● Nuclear medicine image (functional information)

● After registration, hand image obtained by

nuclear medicine imaging is pseudocolored and

superimposed on the gray scale X-ray hand

image

● Red color corresponds to the largest isotope

concentration.


Example: Brain Registration

● MR image showing anatomy (left), PET* FDG image showing function

superimposed on MR image (right)

*Positron emission tomography (PET) is a nuclear medicine, functional imaging technique that produces a three-dimensional image of functional processes in the body.


Classification of Methods

● Registration methods can be classified with respect to several different

criteria such as:

○ Dimensionality of images that are registered (2D, 3D or 4D methods)

○ Image features being matched (extrinsic and intrinsic methods)

○ Mechanism of interaction with the user

○ Type of geometric transformation used for registration


Classification of Methods→ Dimensionality

● We can register 2D, 3D or 4D images

● 3D registration can be done for two 3D images or for a temporal sequence

of 2D images (video)

● If we want to register a sequence of 3D images this represents 4D

registration problem

● There is also a problem of 2D image (perspective projection of 3D space) to

3D image registration

○ In this case it is necessary to determine the view parameters so that the

obtained perspective transformation of 3D image matches the 2D image


Classification of Methods→ Image Features

● This classification is motivated by the type of image features used for image

registration:

○ Extrinsic methods (external objects or markers are used as reference

points for registration)

○ intrinsic methods (registration is based on pixel values - no external

objects are used)


Classification of Methods→ Image Features >> Extrinsic Methods

● Extrinsic methods use artificial external objects (markers) attached to the

object to be registered

● Markers are detected in both images and used for registration

○ Example: For brain image registration skin markers or stereotactic

frames may be used

● Disadvantage: Registration is based on external markers so accuracy

depends on the accuracy of marker detection


Classification of Methods→ Image Features >> Intrinsic Methods

● Intrinsic registration methods do not use artificial external objects

● Intrinsic methods use:

○ Anatomical landmarks (points, contours, or surfaces), or

○ Pixel values (intensity-based methods)

● Anatomical landmarks must be detected and this represents a disadvantage

(possibility of error)

● Intensity-based methods have advantage of relying only on pixel values

without the need for detection of special landmarks


Classification of Methods→ User Interaction

● With respect to user interaction, registration methods can be divided into:

○ Interactive (require user interaction to define the geometric

transformation for registration)

○ Semi-automatic (user interaction is only required for initialization,

guidance, or stopping the registration procedure)

○ Automatic (do not require any user interaction)


Classification of Methods→ Geometric Transformations

● This classification is based on the type of transformation used for

registration:

○ Rigid registration: distance between any two object points is preserved

(rotation, translation)

○ Affine transformation: A line is mapped into a line, parallelism between

lines is preserved

○ Projection transformation (e.g. perspective projection) is like affine, but

it does not preserve parallelism of lines

○ Elastic transformation: line is mapped into a curve



● Rigid transformations are a subset of affine transformations

● Affine transformations are a subset of projective transformations

● Projective transformations are a subset of elastic transformations



● In the next several slides we present an overview of the basic geometric

transformations:

○ Rigid transformations

○ Scaling transformations

○ Affine transformations

○ Projective transformations

○ Perspective transformations

○ Elastic transformations


Classification of Methods→ Geometric Transformations >> Rigid Transformations

● A rigid transformation of vector consists translation and rotation:

Where is translation vector, R is a 3x3 orthogonal rotation matrix ( ,

d and are rigid body rotation angles around z,y and x axes)


Classification of Methods→ Geometric Transformations >> Scaling Transformations

The simplest affine transformations are those that only include scaling, while the

rest of the transformation is rigid:

x’ = RSx + t

Where S= is scaling matrix in x,y and z direction, R is rotation

matrix, and t is translation vector


Classification of Methods→ Geometric Transformations >> Affine Transformations

Affine transformations preserve lines and parallel lines and are defined by

expression

x’ = Ax +t

where A is affine transformation matrix that can have any value, and t is

translation vector


Classification of Methods→ Geometric Transformations >> Affine Transformations

● For easier manipulation of matrix expressions, a representation using

homogeneous coordinates is often used

● Homogeneous coordinate vector of a 3D point is 4D


Classification of Methods→ Geometric Transformations >> Projective Transformations

● Projective transformations are similar to affine (lines are preserved), only

there is no preserved), only there is no preservation of parallel relation

● The analytical form is given by:

x’ = (Ax +t) / (p.x + )



In homogeneous coordinates we have:

where



● Perspective transformations map 3D space into a 2D image plane

● Examples: xamers imaging, X-ray imaging, microscopy

● Perspective transformations are a subset of projective transformations


Classification of Methods→ Geometric Transformations >> Elastic Transformations

● Elastic transformations do not preserve lines (i.e. a line can be mapped into

a curve)

● An elastic transformation can be defined by any non-linear mapping of

spatial coordinates

● Polynomial are often used in practice for simplicity

● For 3D case


Classification of Methods→ Geometric Transformations >> Elastic Transformations

● In practice, polynomial order is limited because of oscillations present in

high-order polynomials

● For this reason, polynomial order usually chosen so that I,J,K ≤ 2

● For the same reason it is often taken that I+J+K ≤ 5


Classification of Methods→ Image Registration Algorithms

● In the following material we present three groups of algorithms for 3D

medical image registration:

○ Algorithms using corresponding points identified in images

○ Algorithms using corresponding surfaces

○ Algorithms using voxel intensity values


Classification of Methods→ Image Registration Algorithms >> Point-based

● Point-based registration requires:

○ Identification of corresponding 3D points in images to be aligned

○ Registration of the points (to identify unknown geometric transformation

that maps one set of points into another)

○ Use determined transformation to map all other image points (i.e. to

establish point correspondences)



● In medical image registration, 3D points used for registration are often called

fiducial markers or fiducial points

● 3D points can be either external markers attached to human body or

anatomical landmarks identified in the images



● The usual approach to point-based registration problem is to find the least-

square rigid-body or affine transformation that aligns the points

● The obtained transformation can then be used to transform any point from

one image to another

● This problem is often referred to as the orthogonal Procrustes problem



Orthogonal Procrustes Problem

● Problem definition: Given two configurations of N points in D dimensions

P = {p } i Q = {q }, it is necessary to find transformation T that minimizes

error:

G(T) = | T(P) - Q|

● P i Q are NxD matrices whose rows are coordinates of points p i q , and

T(P) is the corresponding matrix of transformed points p

● The standard case is when T is a rigid-body transformation

i i

2

i i

i



Orthogonal Procrustes Problem

● If T is affine transformation, we obtain the standard least-squares problem

● In the following slides we show the solution for case when T is a rigid-body

transformation defined by rotation matrix R and translation vector t



Orthogonal Procrustes Problem >> Solution

● First replace vectors in P and Q by their demeaned versions (mean value

equal to zero):

● This reduces the problem to the orthogonal Procrustes problem in which we

need to determine orthogonal rotation matrix R




● Central to the problem is the DxD correlation matrix , which shows

how much the points in Q are predicted by points in P

● The singular value decomposition of matrix K is given by:

where U and V are orthogonal matrices containing left and right singular vectors

and D is a diagonal matrix containing singular values




Orthogonal matrix R is determined by expression:

Translation vector t may be determined by expression


Classification of Methods→ Image Registration Algorithms >> Surface-base

● The second approach to 3D registration is by using surfaces in medical

images, which are often more distinct than point landmarks

● Segmentation algorithms are used to locate surfaces:

○ For example tissue to air boundaries often have high contrast, which

makes surface detection easier

● If two corresponding surfaces can be detected in images to be matched,

then rigid-body registration can be achieved by fitting the surfaces



● Some of the best known surface registration algorithms are:

● The head and hat algorithm

● Distance transform-based algorithms

● Iterative closest point (ICP) algorithm



Head and Hat Algorithms

● Developed by Pelizzari i ostali, 1989, for 3D registration of CT, MR and PET

head images

● The first surface (head) is obtained from higher resolution modality and is

represented as a stack of image slices

● The second surface (hat) is represented as a list of unconnected 3D points

● Registration is performed by iterative transformation of the hat surface to

find the best fit onto the head surface



Head and Hat Algorithms

● Registration accuracy is measured by the square of the distance between

the point on the hat and the nearest point on the head in the direction of the

head centroid

● Iterative optimization using Powell steepest descent algorithm, which

performs a series of 1D optimizations in each of the six dimensions:

○ For 3D rigid-body registration we have six degrees of freedom (three

rotational and three translations)

● This method is useful only for spherical surfaces



Distance Transforms

● Head and hat algorithm can be improved using a distance transform to

preprocess head images

● A distance transform maps a binary image into a distance image

○ In the distance image each pixel has the value of the distance of that

pixel to the nearest surface in the binary image

● Distance transform is computed for one of the images, which makes it easy

to calculate distance from one surface to another



Iterative Closest Point Algorithm

● Iterative closest point (ICP) algorithm was developed by Besl and McKay,

1992, for 3D registration

● It was developed for general use, but is now the most widely used surface

matching algorithm for biomedical applications

● Let us assume that we have two surfaces:

○ The first surface (call it M) is the model surface

○ The second is given as a point set {p }i



Iterative Closest Point Algorithm

● Repeat:

○ For each point p identify the closest point q on the model surface M

○ Use Procrustes method to register point sets p and q

○ Apply identified geometric transformation to point set p to obtain the

new set of points, call them p’

○ Let p = p’ , for each i

● Until the change in registration mean square error falls below a defined

threshold

i i

i

i

i

i i



Intensity-based Methods

● In all previous methods registration was based on data extracted from

images (landmark points or surfaces)

○ Advantage: it is not important how images are taken (different imaging

modalities can be used) - registration is based on registration of points

or surfaces

○ Disadvantage: A necessary segmentation step was required to extract

landmark point or surfaces from images (which adds a possibility of

error)



Intensity-based Methods

● Intensity-based methods use pixel or voxel values

○ Advantage: It is not necessary to have a separate segmentation step to

extract points or surface of interest

○ A disadvantage is that images that are registered cannot be different

(e.g. recorded using different imaging modalities)

● This approach is most natural when images of the same kind (same

modality) are registered (e.g. CT to CT images)

● Intensity-based methods are various voxel similarity measures


References for image registration:

● M. Sonka, J. M. Fitzpatrick, Eds., Handbook of Medical Imaging, Volume 2.

Medical Image Processing and Analysis, SPIE, 2000

● J. V. Hajnal, D. L. G. Hill, D. J. Hawkes, Eds., Medical Image Registration,

CRC Press, 2001


● Data is discretized in space and / or time.

● Finite number of samples

○ The continuous signal is usually known only at a few points (data

points)

○ In general, data is needed in between these points

● By interpolation we obtain a representation that matches the function at the

data points

○ Evaluation at any other point possible

● Reconstruction of signal at points that are not samples

● Assumptions needed for reconstruction

○ Often smooth functions


Voronoi Diagrams and Delaunay Triangulation

● Given irregularly distributed positions without connectivity information

● Problem: obtain connectivity to find a “good” triangulation

● For a set of points there are many possible triangulations

○ A measure for the quality of a triangulation is the aspect ratio of the so-

defined triangles

○ Avoid long, thin ones



● Scattered data triangulation

○ A triangulation of data points consists of

■ Vertices (0D) = S

■ Edges (1D) connecting two vertices

■ Faces (2D) connecting three vertices

● A triangulation must satisfy the following criteria

○ faces = conv(S), i.e. the union of all faces including the boundary is the

convex hull of all vertices

○ The intersection of two triangles is either empty, or common vertex, or a

common edge, or a common face (tetrahedra)



● Triangulation with

Are NOT valid



● How to get connectivity/triangulation from scattered data?

○ Voronoi diagram

○ Delaunay triangulation



● Voronoi Diagram

○ For each sample every point within a Voronoi region is closer to it than

to every other sample

○ Given: a set of points for and a distance function

dist(x,y)

○ The Voronoi diagram Vor(x) contains for each point a cell with



● Voronoi cells

○ The half space is separated by the perpendicular bisector

between and

○ contains

○ Voronoi cell:

Voronoi cells are convex



● Delaunay graph Del(x)

○ The geometric dual (topologically equal) of

the Voronoi diagram Vor(x)

○ Points in X are nodes

○ Two nodes and are connected iff the

Voronoi cells V( ) and V( ) share same

edge

● Delaunay cells are convex

● Delaunay triangulation = triangulation of the

Delaunay graph



● Delaunay triangulation in 2D

○ Three points , , in X belong to a face from Del(X) iff no further

point lies inside the circle around

● Two points form an edge iff there is a circle around

that does not contain a third point from X

● For each triangle the circumcircle does not contain any

other sample

● Maximizes the smallest angle

● Maximizes the ratio of (radius of incircle)/(radius of

circumcircle)

● It is unique (independent of the order of sample) for all but

some very specific cases



● Local Delaunay property



● Algorithms for Delaunay triangulations

○ Edge flip algorithm

○ Plane-sweep algorithm for finding an initial triangulation

○ Bowyer-Watson algorithm

○ Other techniques:

■ Radial sweep

■ Intersecting halfspaces

■ divide and conquer (merge-based or split-based)


Univariate Interpolation

● Univariate Interpolation: Interpolation for one variable

○ Nearest neighbor (0 order)

○ Linear (first order)

○ Smooth (higher order)



● Taylor interpolation

● Basic functions: monom basis (polynomials)

● is m+1 dimensional vector space of all polynomials with

maximum degree m

● Coefficients with

● Representation problem

● Interpolation problem

f = samples

c = coefficients (to be solved)



● Properties of Taylor interpolation

○ Unique solution

○ Numerical problems / inaccuracies

○ Complete system has to be solved again if a single value is changed

Medical Imaging / VV Other basis functions result in other interpolations scheme:

● Lagrange interpolation

● Newton interpolation

● Bernstein basis: Bezier curves (approximation)

● Hermite basis

Medical Imaging / VV ● Piecewise linear interpolation

○ Simplest approach (except for nearest-neighbor sampling)

○ Fast to compute

○ Often used in visualization applications

○ continuity at segment boundaries

● Data points:

● For any point x with

described be local coordinate

that is

evaluate

Medical Imaging / VV ● First approach

○ Replace differential by “finite differences”

○ Note that approximating the derivative by

causes subtractive cancellation and large rounding errors for small h

● Second approach

○ Approximate/interpolate (locally) by differentiable function and differentiate this function

Medical Imaging / VV ● Finite differences on uniform grids with grid size h (1D case)

Medical Imaging / VV ● Finite differences on uniform grids with grid size h (1D case)

Forward differences

Backward differences

Central differences

● Error estimation

○ Forward/backward differences are first order

○ Central differences are second order

Medical Imaging / VV ● 2D or 3D uniform or rectangular grids

○ Partial derivatives

● Same as in univariate case along each coordinate axis

● Example: gradient in a 3D uniform grid

Medical Imaging / VV ● Volume rendering techniques

○ Techniques for 2-D scalar fields

○ Indirect volume rendering techniques (e.g. surface fitting)

■ Convert/reduce volume data to an intermediate representation (surface

representation), which can be rendered with traditional techniques

○ Direct volume rendering

■ Consider the data as a semi-transparent gel with physical properties and directly get a

3D representation of it

Medical Imaging / VV ● Slicing:

○ Display the volume data, mapped to colors, on a slice plane

● Isosurfacing:

○ Generate opaque/semi-opaque surfaces

● Transparency effects:

○ Volume material attenuates reflected or emitted light

Medical Imaging / VV ● Direct volume rendering techniques

○ Direct volume rendering allows for the “global” representation integrating physical

characteristics

○ But prohibits interactive display due to its numerical complexity, in general

● Indirect volume rendering techniques

○ Often result in complex representations

○ Pre-processing the surface representation might help

○ Use graphics hardware for interactive display

● Goal

○ Integrate different techniques in order to represent the data as”good” as possible

○ But, keep in mind that the most correct method in terms of physical realism must not be the

most optimal one in terms of understanding the data

Medical Imaging / VV ● Different grid structures:

○ Structured: uniform, rectilinear, curvilinear

○ Unstructured

○ Scattered data

Medical Imaging / VV ● Pixel (picture element)

● Voxel ( volume element)

○ Values are constant within a region around a grid point

● Cell

○ Values between grid points are resampled by interpolation

Medical Imaging / VV Classification

● Color table for volume visualization

● Maps raw voxel value into presentable entities:

○ color, intensity, opacity, etc.

● Transfer function

● Goals and issues:

○ Empowers user to select “structures”

○ Extract important features of the data set

○ Histogram can be a useful hint

○ Often interactive manipulation of transfer functions needed

Medical Imaging / VV Examples of different transfer functions

Medical Imaging / VV Most widely used approach for transfer functions:

● Assign to each scalar value a different color value

● Assignment via transfer function T

○ T : scalar value → color value

● Common choice for color representation : RGBA

● Alpha value is very important, describes opacity

● Code color values into a color lookup table

● On-the-fly update of color LUT

Medical Imaging / VV Shading

Simulate reflection of light

Simulate effect on color

We want to make use of the human visual system’s

ability to efficiently deal with shaded objects

Medical Imaging / VV Shading


References:

● McCormick, B. H. (1988). Visualization in scientific computing. ACM SIGBIO Newsletter, 10(1),

15–21. doi:10.1145/43965.43966

● Bartz, D., & Preim, B. (2011). Visualization and Exploration of Segmented Anatomic Structures. In

T. M. Deserno (Ed.), Biomedical Image Processing. Berlin, Heidelberg: Springer Berlin Heidelberg.

doi:10.1007/978-3-642-15816-2

● Visualization and Interactive Systems course from University of Stuttgart

tissue engineering introduction for physicists - lecture three

Engineering

domain of medical imaging

imaging context

segmentation procedures

poor quality of imaging

pixelbased approaches

segmentation task easier

real data

sampling data