Biomedical Engineering Reference
In-Depth Information
16.3.3 the curvelet transform
There are many other image decomposition methods, or image transforms, which operate similarly to
the Fourier transform but with a different set of basis functions. While the Fourier transform allows a
global analysis, the wavelet transform allows a local spatial analysis of image data. A further transform,
the curvelet transform [58], derives from wavelets, with the added ability to capture orientation in addi-
tion to scale.
The curvelet transform is a nonadaptive, multi-scale transform capable of representing objects in a
sparse manner while retaining information on the object's scale, location, and orientation [59]. While
curvelets have been used in mammogram-based breast cancer diagnosis [60], we present below, to our
knowledge, the first application of the curvelet transform to the problem of collagen alignment in cell
resolution image data, particularly in SHG images.
The fundamental advantage of the curvelet transform for collagen alignment analysis is the ability of
the transform to retain orientation information from the image. As shown in Figure 16.10, the curvelet
transform detects both the scale and orientation of the edges. This results in the ability to examine all
prominent edges at a particular orientation and a particular scale (varying only on location of the fixed
scale and orientation curvelet in the image). When applied to the collagen alignment analysis problem,
the curvelet transform becomes a powerful tool for detecting the presence of filamentous structures
and their location, scale, and orientation. By obtaining accurate quantitative data regarding collagen
amount, morphology, and organization/orientation, biologically relevant data can be derived. To make
use of the curvelet transform to capture information about the spatial organization of collagen, we have
developed CurveAlign analysis software.
CurveAlign is not intended to precisely follow individual fibers, but rather to determine the overall
trend in fiber alignment in an image. The measured angles can be binned according to the collagen
fiber structures in diseases such as cardiac disease [61]. As well, CurveAlign can detect trends in other
filamentous structures such as microtubules [62]. In breast cancer invasion and progression studies,
CurveAlign was used to quantitatively assess the effect of cancer cells on the remodeling of collagen
during cancer onset and progression, as well as in human pathological samples (Figure 16.11).
The integration of these concepts—the influence of stromal collagen I on the tumor microenviron-
ment, nonlinear optical imaging techniques, and digital image processing—forms the basis for a toolkit
for using quantitative SHG to understand the role of collagen. While our work to date has focused on
breast cancer, these digital image analysis techniques can be generalized, some in the spatial domain
(a)
(b)
(c)
FIgurE 16.10 Graphical representation of curvelets. (a) A curvelet whose length-wise support does not intersect
a discontinuity. The curvelet coefficient magnitude will be zero. (b) A curvelet whose length-wise support inter-
sects with a discontinuity, but not at a critical angle. The curvelet coefficient magnitude will be close to zero. (c) A
curvelet whose length-wise support intersects with a discontinuity and is tangent to that discontinuity. The curvelet
coefficient magnitude will be much larger than zero.
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