This thesis presents the Adaptive Particle Representation (APR), a novel adaptive data representation that can be used for general data processing, storage, and simulations. The APR is motivated, and designed, as a replacement representation for pixel images to address computational and memory bottlenecks in processing pipelines for studying spatiotemporal processes in biology using Light-sheet Fluo- rescence Microscopy (LSFM) data. The APR is an adaptive function representation that represents a function in a spatially adaptive way using a set of Particle Cells V and function values stored at particle collocation points P∗. The Particle Cells partition space, and implicitly define a piecewise constant Implied Resolution Function R∗(y) and particle sampling locations. As an adaptive data representation, the APR can be used to provide both computational and memory benefits by aligning the number of Particle Cells and particles with the spatial scales of the function. The APR allows reconstruction of a function value at any location y using any positive weighted combination of particles within a distance of R∗(y). The Particle Cells V are selected such that the error between the reconstruction and the original function, when weighted by a function σ(y), is below a user-set relative error threshold E. We call this the Reconstruction Condition and σ(y) the Local Intensity Scale. σ(y) is motivated by local gain controls in the human visual system, and for LSFM data can be used to account for contrast variations across an image. The APR is formed by satisfying an additional condition on R∗(y); we call the Resolution Bound. The Resolution Bound relates the R∗(y) to a local maximum of the absolute value function derivatives within a distance R∗(y) or y. Given restric- tions on σ(y), satisfaction of the Resolution Bound also guarantees satisfaction of the Reconstruction Condition. In this thesis, we present algorithms and approaches that find the optimal Implied Resolution Function to general problems in the form of the Resolution Bound using Particle Cells using an algorithm we call the Pulling Scheme. Here, optimal means the largest R∗(y) at each location. The Pulling Scheme has worst-case linear complexity in the number of pixels when used to rep- resent images. The approach is general in that the same algorithm can be used for general (α,m)-Reconstruction Conditions, where α denotes the function derivative and m the minimum order of the reconstruction. Further, it can also be combined with anisotropic neighborhoods to provide adaptation in both space and time. The APR can be used with both noise-free and noisy data. For noisy data, the Reconstruction Condition can no longer be guaranteed, but numerical results show an optimal range of relative error E that provides a maximum increase in PSNR over the noisy input data. Further, if it is assumed the Implied Resolution Func- tion satisfies the Resolution Bound, then the APR converges to a biased estimate (constant factor of E), at the optimal statistical rate. The APR continues a long tradition of adaptive data representations and rep- resents a unique trade off between the level of adaptation of the representation and simplicity. Both regarding the APRs structure and its use for processing. Here, we numerically evaluate the adaptation and processing of the APR for use with LSFM data. This is done using both synthetic and LSFM exemplar data. It is concluded from these results that the APR has the correct properties to provide a replacement of pixel images and address bottlenecks in processing for LSFM data. Removal of the bottleneck would be achieved by adapting to spatial, temporal and intensity scale variations in the data. Further, we propose the simple structure of the general APR could provide benefit in areas such as the numerical solution of differential equations, adaptive regression methods, and surface representation for computer graphics.

This thesis presents the Adaptive Particle Representation (APR), a novel adaptive data representation that can be used for general data processing, storage, and simulations. The APR is motivated, and designed, as a replacement representation for pixel images to address computational and memory bottlenecks in processing pipelines for studying spatiotemporal processes in biology using Light-sheet Fluo- rescence Microscopy (LSFM) data. The APR is an adaptive function representation that represents a function in a spatially adaptive way using a set of Particle Cells V and function values stored at particle collocation points P∗. The Particle Cells partition space, and implicitly define a piecewise constant Implied Resolution Function R∗(y) and particle sampling locations. As an adaptive data representation, the APR can be used to provide both computational and memory benefits by aligning the number of Particle Cells and particles with the spatial scales of the function. The APR allows reconstruction of a function value at any location y using any positive weighted combination of particles within a distance of R∗(y). The Particle Cells V are selected such that the error between the reconstruction and the original function, when weighted by a function σ(y), is below a user-set relative error threshold E. We call this the Reconstruction Condition and σ(y) the Local Intensity Scale. σ(y) is motivated by local gain controls in the human visual system, and for LSFM data can be used to account for contrast variations across an image. The APR is formed by satisfying an additional condition on R∗(y); we call the Resolution Bound. The Resolution Bound relates the R∗(y) to a local maximum of the absolute value function derivatives within a distance R∗(y) or y. Given restric- tions on σ(y), satisfaction of the Resolution Bound also guarantees satisfaction of the Reconstruction Condition. In this thesis, we present algorithms and approaches that find the optimal Implied Resolution Function to general problems in the form of the Resolution Bound using Particle Cells using an algorithm we call the Pulling Scheme. Here, optimal means the largest R∗(y) at each location. The Pulling Scheme has worst-case linear complexity in the number of pixels when used to rep- resent images. The approach is general in that the same algorithm can be used for general (α,m)-Reconstruction Conditions, where α denotes the function derivative and m the minimum order of the reconstruction. Further, it can also be combined with anisotropic neighborhoods to provide adaptation in both space and time. The APR can be used with both noise-free and noisy data. For noisy data, the Reconstruction Condition can no longer be guaranteed, but numerical results show an optimal range of relative error E that provides a maximum increase in PSNR over the noisy input data. Further, if it is assumed the Implied Resolution Func- tion satisfies the Resolution Bound, then the APR converges to a biased estimate (constant factor of E), at the optimal statistical rate. The APR continues a long tradition of adaptive data representations and rep- resents a unique trade off between the level of adaptation of the representation and simplicity. Both regarding the APRs structure and its use for processing. Here, we numerically evaluate the adaptation and processing of the APR for use with LSFM data. This is done using both synthetic and LSFM exemplar data. It is concluded from these results that the APR has the correct properties to provide a replacement of pixel images and address bottlenecks in processing for LSFM data. Removal of the bottleneck would be achieved by adapting to spatial, temporal and intensity scale variations in the data. Further, we propose the simple structure of the general APR could provide benefit in areas such as the numerical solution of differential equations, adaptive regression methods, and surface representation for computer graphics.