Among biomedical applications where interpolation is quite relevant, the most obvious are those where the goal is to modify the sampling rate of pixels ( picture elements) or voxels (volume elements). This operation, named rescaling, is desirable when an acquisition device — say, a scanner — has a nonhomogeneous resolution, typically a fine within-slice resolution and a coarse across-slice resolution. In this case, the purpose is to change the aspect ratio of voxels in such a way that they correspond to geometric cubes in the physical space [7,8]. Often, the across-slice resolution is modified to match the within-slice resolution, which is left unchanged. This results in a volumetric representation that is easy to handle (e.g., to visualize or to rotate) because it enjoys homogenous resolution.
A related operation is reslicing . Suppose again that some volume has a higher within-slice than across-slice resolution. In this case, it seems natural to display the volume as a set of images oriented parallel to the slices, which offers its most detailed presentation. Physicians may, however, be sometimes interested in other views of the same data; for simplicity, they often request that the volume be also displayed as set of images oriented perpendicular to the slices. With respect to the physical space, these special orientations are named axial, coronal, and sagittal, and require at most rescaling for their proper display. Meanwhile, interpolation is required to display any other orientation — in this context, this is named reslicing.
The relevance of interpolation is also obvious in more advanced visualization contexts, such as volume rendering. There, it is common to apply a texture to the facets that compose the rendered object . Although textures may be given by models (procedural textures), this is generally limited to computer graphics applications; in biomedical rendering, it is preferable to display a texture given by a map consisting of true data samples. Because of the geometric operations involved (e.g., perspective projection), it is necessary to resample this map, and this resampling involves interpolation. In addition, volumetric rendering also requires the computation of gradients, which is best done by taking the interpolation model into account .
A more banal use of interpolation arises with images (as opposed to volumes). There, a physician may want both to inspect an image at coarse scale and to study some detail at fine scale. To this end, interpolation operations such as zooming in and out are useful [12,13]. Related operations are (subpixel) translation or panning, and rotation . Less ordinary transformations may involve a change of coordinates, for example the polar-to-Cartesian scan conversion function that transforms acquired polar-coordinate data vectors from an ultrasound transducer into the Cartesian raster image needed for the display monitors. Another application of the polar-to-Cartesian transform arises in the three-dimensional reconstruction of icosahedral viruses .
In general, almost every geometric transformation requires that interpolation be performed on an image or a volume. In biomedical imaging, this is particularly true in the context of registration, where an image needs to be translated, rotated, scaled, warped, or otherwise deformed before it can match a reference image or an atlas . Obviously, the quality of the interpolation process has a large influence on the quality of the registration.
The data model associated to interpolation also affects algorithmic considerations. For example, the strategy that goes by the name of multiresolution proposes first to solve a problem at the coarse scale of an image pyramid, and then to iteratively propagate the solution at the next finer scale, until the problem has been solved at the finest scale. In this context, it is desirable to have a framework where the interpolation model is consistent with the model upon which the image pyramid is based. The assurance that only a minimal amount of work is needed at each new scale for refining the solution is only present when the interpolation model is coherent with the multiresolution model .
Tomographic reconstruction by (filtered) back-projection forms another class of algorithms that rely on interpolation to work properly. The principle is as follows: Several X-ray images of a real-world volume are acquired, with a different relative orientation for each image. After this physical acquisition stage, one is left with X-ray images (projections) of known orientation, given by data samples. The goal is to reconstruct a numeric representation of the volume from these samples (inverse Radon transform), and the mean is to surmise each voxel value from its pooled trace on the several projections. Interpolation is necessary because the trace of a given voxel does not correspond in general to the exact location of pixels in the projection images. As a variant, some authors have proposed to perform reconstruction with an iterative approach that requires the direct projection of the volume (as opposed to its back-projection) . In this second approach, the volume itself is oriented by interpolation, whereas in the first approach the volume is fixed and is not interpolated at all, but the projections are.
Interpolation is so intimately associated with its corresponding data model that, even when no resampling operation seems to be involved, it nevertheless contributes under the guise of its data model. For example, we have defined interpolation as the link between the discrete world and the continuous one. It follows that the process of data differentiation (calculating the data derivatives), which is defined in the continuous world, can only be interpreted in the discrete world if one takes the interpolation model into consideration. Since derivatives or gradients are at the heart of many an algorithm (e.g., optimizer, edge detection, contrast enhancement), the design of gradient operators that are consistent with the interpolation model [19,20] should be an essential consideration in this context.
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