  We investigate neural network image reconstruction for magnetic particle
imaging. The network performance depends strongly on the convolution effects of
the spectrum input data. The larger convolution effect appearing at a
relatively smaller nanoparticle size obstructs the network training. The
trained single-layer network reveals the weighting matrix consisted of a basis
vector in the form of Chebyshev polynomials of the second kind. The weighting
matrix corresponds to an inverse system matrix, where an incoherency of basis
vectors due to a low convolution effects as well as a nonlinear activation
function plays a crucial role in retrieving the matrix elements. Test images
are well reconstructed through trained networks having an inverse kernel
matrix. We also confirm that a multi-layer network with one hidden layer
improves the performance. The architecture of a neural network overcoming the
low incoherence of the inverse kernel through the classification property will
become a better tool for image reconstruction.
