## Images in Scalismo 0.90

#### Marcel Lüthi

Lecturer, Department of Mathematics and Computer Science, University of Basel### Introduction

The recently released version of Scalismo - version 0.90 - comes with a number of important changes in its core classes. In this blog post, we will look at images.

In older versions of Scalismo images had a special status in the library. While conceptually they were thought to be just discrete fields, they were implemented using a number of special classes, representing the differnet types of images. This led to inconsistencies in the API and complicated the type hierarchy. Even worse, it enforced the wrong notion that image are conceptually different from other representations of intensities used in Scalismo. In version 0.90 we cleaned up the hierarchy and removed all the special classes. Discrete images are now simply a special instantiation of a discrete field, whose domain is a regular grid. In the following we explain the underlying concepts, show how we can create images and how we can obtain a continuous from a discrete representations and vice versa.

### Continuous and discrete Images

Similar to other types of representations, images come in two types: Discrete images and continuous images.
Continuous images are modeled as a `Field`

; I.e. they are functions that have a domain $D \subset \mathbb{R}^d$ and
map each point of the domain to some values. The mapped values can be scalars, vectors or even more complicated objects.

Discrete images in turn are a special case of a discrete field. Discrete fields are defined as a finite set of points, where for each point we have an associated value. A discrete image is a discrete field, whose domain is constrained to be a regular grid; I.e. whose domain points are equally spaced. That the points are equally spaces makes it possible to represent the domain points implicitly by a mathematical formula rather than having to explicitly store them. Furthermore, accessing the image values and looking up closest points becomes a constant time operation.

### Structured Points

The basic object to represent such a set of structured points is the class `StructuredPoints`

.
We can create a set of points on a grid as follows:

This creates a grid of points $32 \times 64 \times 92$ points, where the bottom left point is at the `origin`

,
and the points are the in the $x, y, z$ direction $0.1mm$ apart.

Note that the grid of points is aligned to the coordinate axis. In case you would like to have a different alignment, it is possible to specify a rotation of the points. The rotation is specified by 1 or 3 Euler angles, depending on whether there is a 2 or 3 dimensional image.

### Image domain

The image domain represents a domain, whose points are aligned in a rectangular grid.
Naturally, it uses `StructuredPoints`

as a representation of the underlying points of the
domain. We can create an image domain from structured points as follows:

For convenience, Scalismo also offers the possibility to specify the origin, spacing and size directly, as we did for the structured points.

Note however, that this still creates a structured points object internally.

As for structured points, we can also define a rotation, by specifying the corresponding Euler angles.

Finally, we can specify the points by specifying its bounding box together with the information about the spacing or size:

or

This last creation method is particularly useful for changing the resolution of an image, as we will see later.

### Creating images

To create an image, we need to specify a value for each point in the domain. In this example, we create an image, which assigns a zero value to each point.

Alternatively, we could have specified the values using an array, as follows:

Note that an image is just another name for a discrete field with a image domain. We could have equally well constructed the image as:

### Interpolation and discretization.

It is often more convenient to work with a continuous representation of the image.
To obtain a continuous image, we use the `interpolate`

method and specify a suitable
interpolator:

The resulting object is defined on all the points within the bounding box of the image domain.
To go back to a discrete representation, we can specify a new image domain and use the
`discretize`

method. As the new domain could be bigger than the domain of the continuous image,
we need to specify a value that is assigned to the points, which fall outside this domain.

Of course, we could also resample the continuous image using a different type of domain. Assume for example that we have a CT image of the upper leg, but we are only interested in representing the intensities for the femur bone. We could then discretize the (interpolated) image using a tetrahederal mesh, and thus obtain a representation of the field which is restricted to the femur bone only.

### Summary

We have discussed the new design of images in Scalismo. Discrete images are modelled as discrete fields, and thus have a domain and
associated values attached to it. The points of the domain are represented using the class `StructuredPoints`

, which
represent points that lie on a regular grid. Exploiting this special structure, we can efficiently access values
associated to the grid points in the image, or use dedicated interpolation methods to swich from a discrete to a
continuous representation. Once we have the continuous representation, we can discretize using a different domain, which
allows us for example to resample the image in a different resolution, restrict the image to a part of the domain or even change the
type of the domain.