CDT Signal Space P Transform Space ~ P Q P = {p|p = h ′(p0 o h), ∀h ∈ C} Q = {q|q = h ′(q0 o h), ∀h ∈ C} C = {h|h(x) = x + t, t ∈R} ~ Q CDT ~ ~ p(x) = p0(x) + t √I0(x) q(x) = q0(x + t) CDT ~ ~ q(x) = q0(x) + t √I0(x) p(x) = p0(x + t) ~ P~ Q P Q Projection of the Data Onto a 3-D Discriminant Subspace Projection of the Transformed Data Onto a 3-D Discriminant Subspace (a) (b) Random CDT Image Space Transform Space Class 1 Class 1 Class 2 Class 2 Class 1 Class 2 Class 1 Class 2 Projection of the Data Onto a 2-D Discriminant Subspace Projection of the Transformed Data Onto a 2-D Discriminant Subspace (c) (d) FIGURE 4. Examples for the linear separability characteristic of the CDT and the Radon CDT. The discriminant subspace for each case is calculated using the penalized-linear discriminant analysis. It can be seen that the nonlinear structure of the data is well captured in the transform spaces. (a) and (b) The linear separation property of the CDT. (c) A facial expression data set with two classes and its corresponding representations in the LDA discriminant subspace and (d) the Radon CDT of the data set and the corresponding representation of the transformed data in the LDA discriminant subspace. 3-D: three-dimensional. (Face portraits courtesy of the public CMU Pose, Illumination, and Expression database.) 50 IEEE SIGNAL PROCESSING MAGAZINE | July 2017 |