FM filters for modulation domain image processing

Abstract

For the first time, we demonstrate modulation domain image filters that achieve perceptually motivated image processing goals by directly manipulating the FM functions in a multi-component AM-FM image model. The action of previous modulation domain filters has been limited to modification of the AM functions based on the values of the AM and FM functions. This is because reconstruction of the modified phase from the filtered frequency modulation vectors was an unsolved problem. Here, we present two new algorithms capable of reconstructing the phase from the processed frequencies, one based on a least squares solution of the discrete Poisson equation with Neumann boundary condition and one based on cubic tensor product spline integration. New modulation domain FM filters are designed to modify both the orientations and magnitudes of the visually important emergent image frequency vectors. In our most dramatic example, we demonstrate an FM filter that autonomously changes the stripes on the pants in the well known Barbara image from vertical to horizontal.Index Terms-AM-FM image models, AM-FM image filters, modulation domain signal processing, multicomponent models

Frequency guided phase unwrapping for improved AM-FM reconstruction

“…This approach works well as long as FM filtering is not applied. However, if the FM functions have been modified then the phase initial conditions cannot be extracted directly from the the original image except in some restricted special cases (e.g., RSTimage rotation, scaling, translation) [10,11].…”

“…For a discrete image t(n), agreement with (1) may be obtained by fitting the pixel values with analytically differentiable continuous interpolating functions such as splines [8] to construct an invertible (i.e., perfect reconstruction, or PR) AM-FM transform. This has enabled the development of modulation domain filters, in which a filtered image is reconstructed from modified AM and FM functions to achieve signal processing goals such as frequency selective attenuation [9], rotation, and scaling [10,11] which would be difficult or impossible to achieve by traditional filtering methods.…”

“…Prior AM-FM image reconstruction techniques have required saving samples of the phase from the original image to be used as initial conditions for reconstructing ϕ k from the FM gradient field ∇ϕ k [7][8][9][10][11]. This can lead to serious problems when it is desired for a modulation domain filter to modify the FM functions of the image -because the phase of the filtered image is unknown after FM processing and the phase initial conditions must therefore be estimated or inferred somehow in order to reconstruct the filtered image.…”

“…Successful reconstruction after more general FM processing was demonstrated in [10], but revealed a fidelity loss that can be substantial due to the phase scaling used in the unwrapping algorithm of [8]. At the heart of this problem is the fact that phase scaling renders the phase reconstruction extremely sensitive to estimation errors in the initial phase values.…”