  Deep image prior (DIP), which utilizes a deep convolutional network (ConvNet)
structure itself as an image prior, has attracted attentions in computer vision
and machine learning communities. It empirically shows the effectiveness of
ConvNet structure for various image restoration applications. However, why the
DIP works so well is still unknown, and why convolution operation is useful for
image reconstruction or enhancement is not very clear. In this study, we tackle
these questions. The proposed approach is dividing the convolution into
``delay-embedding'' and ``transformation (\ie encoder-decoder)'', and proposing
a simple, but essential, image/tensor modeling method which is closely related
to dynamical systems and self-similarity. The proposed method named as manifold
modeling in embedded space (MMES) is implemented by using a novel
denoising-auto-encoder in combination with multi-way delay-embedding transform.
In spite of its simplicity, the image/tensor completion, super-resolution,
deconvolution, and denoising results of MMES are quite similar even competitive
to DIP in our extensive experiments, and these results would help us for
reinterpreting/characterizing the DIP from a perspective of ``low-dimensional
patch-manifold prior''.
