**Nonlinear separation of a show-through
mixtures using a physical model trained with ICA **

(Abstract, Bi-affine separability,
Data)

**Acquired Separated**

**page****
images page images**

**Reference:**

M. S. C. Almeida and L. B. Almeida, “Nonlinear saparation of
a show-through mixing model using a physical model trained with ICA”, *Signal Processing*, Vol 92, issue 4, pp. 872-884, April,
2012. (Online)

**Abstract:**

Often, when we scan a document, the image from the
back page shows through, due to partial transparency of the paper, giving rise
to a mixture of two images. We address the problem of separating these images
through the use of a physical model of the mixture process. The model is
nonlinear but invertible, and we use the inverse model to perform the
separation. The model is trained through the MISEP technique of nonlinear ICA.
Bounded independent sources are proved to be separable through this method,
apart from offset, scale and permutation indeterminacies.

We compare our
results with those obtained with other approaches and with different separation
models that were trained with MISEP. For the latter case we test a bilinear
model and MLP-based models, using both symmetry-based regularization and the
more recently proposed minimal nonlinear distortion regularization.
Quantitative quality measures show that the approach that we propose is
superior to the other methodologies.

**Nonlinear mixing model based on the half-toning
process: 2D bi-affine symmetrical mixing model **(affine in each variable, and symmetrical)

where represent the mixtures and the
sources.

** **

**Proof of separability of a
generic nonlinear bi-affine model with nonlinear independent component analysis
(nonlinear ICA):**

Based on the geometric scheme of Fig.1, any
bi-affine mixture of independent bounded sources is proven to be separable with
the ICA criterion. The proof is based on the fact that the bi-affine
transformation of each (horizontal or vertical) line of a rectangle is affine.
The borders of any rectangle will thus transform in lines. Moreover, any point
(*X* or *X’*) of an independent limited data (that has a rectangular support),
will be transformed (mixed) as the convex combination of the corresponding
horizontal and vertical line-segments, and according to the parameters of the convex combination (see Fig. 1). As a
consequence of that, we conclude that, if the two points of two rectangles
transform by two bi-affine transformation to the same point (*T(X)=T’(X’)*),
then the points (*X * and *X’*) either are the same points of the
same rectangle or are corresponding points of two rectangles that are simply
relate by an offset and/or linear re-scale of the axis and/or rotation multiple of 90º (permutation of
sources). The mixtures are thus separable apart from offset, linear scale and
permutation indeterminacies.

The previous reasoning is extendable for
higher dimensionally data.

This proof comprises the all the sub-models of
a bi-affine mixture, namely the case bi- linear mixtures and the well know
linear model.

For more details, please see the paper. (Online)

Figure 1 – The
bi-affine transformation of two rectangles to the same quadrangle – geometric schem for proving the ICA separability
of a bi-affine mixture.

**Data:**

·
Tracing paper images:
Results
(.zip file). The mixtures can be found through the home page of Luís B.
Almeida (here).

·
Part of an old air mail letter: aligned
and flipped data (.zip file).

· Other show-through bleed through
data can be found here.

**License:** This code and data are copyright of Luís B. Almeida and Mariana S. C.
Almeida. Free permission is given for their use for nonprofit research
purposes. Any other use is prohibited, unless a license is previously obtained.
To obtain a license please contact Luís B. Almeida or Mariana S. C.
Almeida.

**See, here, another method
for separating linear/nonlinear show-through and bleed-through mixtures.**