Python Linear Transformation Operators (PyOp)

PyOp is a framework for creating Linear Operators in a functional form. Operators specified in this format have a few benefits.

• The memory to store the operator can be much smaller. For example, an image reconstruction done by the authors used to require storing a sparse marix that was over 150 GB. With this matrix free version, the amount of space went down to kilobytes.
• The time to perform an operation can be greatly reduced. In the same image reconstruction problem, converting only one of the operations to a matrix free linear operator, while using a gradient descent algorithm, reduced the computation time by an order of magnitude.
• For some (although by no means all) morphisms it is simpler to describe the operation in a function based form than converting the operation into a matrix.
• The matrix free operators can be converted back into matrices easily should the need for certain numerical tools be required (factorizations), although many facilities are provided that work on matrix free inputs.

This package makes it simple to create matrix free linear operators. Perhaps more importantly, it provides simple composition tools, allowing operators to be chained together with the same tools one would find in a matrix package (for example, adding operators or creating block operators).

For example, the DFT can be represented as a matrix but requires O(n^2) storage and, in the simplest formulation, O(n^2) operations to compute the FFT. However, the FFT/DFT algorithm takes only O(n log n) time and less than quadratic space (I cannot find a good reference for the space complexity of the FFT).

Package Requirements

• Python 2.7 or Python 3.3+
• numpy>=1.8
• six>=1.6

For testing you will need pytest>=2.5