Abstract—A method for visualizing manifold-valued medical image data is proposed. The method operates on images in which each pixel is assumed to be sampled from an underlying manifold. For example, each pixel may contain a high dimensional vector, such as the time activity curve (TAC) in a dynamic positron emission tomography (dPET) or a dynamic single photon emission computed tomography (dSPECT) image, or the positive semi-definite tensor in a diffusion tensor magnetic resonance image (DTMRI). A nonlinear mapping reduces the dimensionality of the pixel data to achieve two goals: distance preservation and embedding into a perceptual color space. We use multi-dimensional scaling distance-preserving mapping to render similar pixels (e.g. DT or TAC pixels) with perceptually similar colors. The 3D CIELAB perceptual color space is adopted as the range of the distance preserving mapping, with a final similarity transform mapping colors to a maximum gamut size. Similarity between pixels is either determined analytically as geodesics on the manifold of pixels or is approximated using manifold learning techniques. In particular, dissimilarity between DTMRI pixels is evaluated via a Log-Euclidean Riemannian metric respecting the manifold of the rank 3, 2 nd order positive semi-definite DTs, whereas the dissimilarity between TACs is approximated via ISOMAP. We demonstrate our approach via artificial highdimensional, manifold-valued data, as well as case studies of normal and pathological clinical brain and heart DTMRI, dPET, and dSPECT images. Our results demonstrate the effectiveness of our approach in capturing, in a perceptually meaningful way, important features in the data. Keywords: Manifold-valued data; high dimensional data; visualization; color; nonlinear dimensionality reduction; multidimensional scaling; distance-preserving mapping; diffusion tensor magnetic resonance imaging (DTMRI); dynamic positron emission tomography (dPET); dynamic single photon emission computed tomography (dSPECT). I
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