MDL Spline Models: Gradient and Polynomial Reparameterisations

Linear and nonlinear generative probabilistic class models for shape contours

“…Illustrative Examples: Box-bump Data Fig. 1a shows a 'box-bump' data set similar to those used by Davies (2002), Hladuvka and Buhler (2005) and Ericsson and Karlsson (2006). Note that unlike most natural data sets, the distribution of the shapes is uniform given the correct correspondence (and ignoring minor effects due to alignment transformations).…”

“…RFs can be expressed as the integral of a linear combination of basis functions where both the basis functions and the coefficients are non-negative Hladuvka & Buhler, 2005). To avoid these non-negativity constraints, we use a 'monotonicity operator' (Ramsay & Silverman, 2005): take an unconstrained linear combination of arbitrary basis functions, exponentiate to ensure positivity and then integrate to ensure monotonicity.…”

“…The inclusion of an explicit low-dimensional latent space and noise model contrasts with techniques which treat major and minor components of variation separately in the objective function Ericsson & Astrom, 2003b;Thodberg & Olafsdottir, 2003;Hladuvka & Buhler, 2005). Neither the existing techniques nor PCM are designed to handle nonlinear patterns of variation and we extend PCM to NPCM to address this limitation.…”

“…Thodberg has incorporated curvature information (Thodberg & Olafsdottir, 2003) and extended the MDL technique to appearance models (Thodberg, 2003). Efficient gradient-based techniques for learning RFs have been introduced for discrete (Ericsson & Astrom, 2003b) and continuous shape representations (Hladuvka & Buhler, 2005). Ericsson and Astrom (2003a) have proposed an alternative formulation to MDL which is invariant to affine transformations.…”

“…1 and 2). Many of the previous approaches to this problem are conceptually similar (Kotcheff & Taylor, 1998;Davies et al, 2002;Ericsson & Astrom, 2003b;Thodberg & Olafsdottir, 2003;Hladuvka & Buhler, 2005): given some observed points from mul- tiple contours, fit curves (often polylines) to the data and then slide points around each curve to find the optimal correspondence with respect to an objective function. Noise in the data is usually not modeled and a 'reference shape' with fixed points/parameterization is typically required to prevent degenerate solutions whereby points cluster about a single region of the contour.…”