Interpretable and Frugal Learning Systems Employing Multiresolution Pyramids and Volterra Kernels

Deep learning models are widely used to process multidimensional signals such as time series, images, and volumetric medical images, but their learned representations often lack explicit signal structure and are difficult to inspect. This thesis develops model-based, signal-theoretic learning systems guided by data and task objectives. It combines multiresolution analysis, wavelets and filter banks, multirate representations, nonlinear Volterra systems, and neural computation graphs. Scale, directional geometry, memory, and nonlinear input-output interactions are represented as differentiable operator modules trainable by backpropagation. The design keeps intermediate variables tied to kernels, subbands, recursions, and transform-domain coefficients rather than only to opaque feature channels. The thesis formulates fast GPU-compatible D-dimensional convolution layers, multirate sampling layers, Volterra-kernel layers in natural and wavelet coefficient domains, rational polynomial cascade heads, stability-constrained multidimensional IIR filters, wavelet banks, and digital shearlet layers with learnable gains. These modules are composed into task-specific architectures for inverse modeling, classification, and segmentation across atmospheric, audio, texture, and medical-imaging problems. In microwave radiometric inversion, InVeRt retrieves vertical temperature and humidity profiles from microwave brightness temperature observations using learnable Volterra kernels in wavelet bases. Multiresolution filter-bank encoders with Volterra heads are used for efficient classification. WaveletViT and ShearViT serve as subband transformer blocks for WaveNETR and ShearNETR, direction-sensitive segmenters for image and MRI segmentation. MRILong deploys trained 3D T1-weighted brain MRI segmenter checkpoints for automatic segmentation and longitudinal analysis of ischemic stroke MRI volumes.