Image reconstruction technique based on Ftransform uses clearly defined basic functions. These functions have strong impact on the quality of reconstruction. We can use some predefined shape and radius, but also we can create a new one from the scratch. The aim of this paper is to analyze the creating process and based on that find best basic function for input set of damaged testing images.
Image reconstruction aims at recovering damaged parts. In real situation this damage can be caused by various sources on various surfaces. Input information has to be digitalized as a first step and after that divided on damaged and nondamaged parts. Typical situation lays on photography. We can distinguish among many types of damage like stains, scratches, text, or noise. We can say that every kind of damage covers typical way of damaging process. Unwanted time or date stamp is covered by
(a) Inpaint damage; (b) text damage; (c) noise damage.
Target of reconstruction is removing damaged parts from input image and replacing them by the parts with recomputed values. These values are computed from the neighborhood ones. The technique mentioned in this paper is based on Ftransform which brings particular way of valuation neighborhood pixels and their usage in the computation [
In this paper, we will focus on the valuation part of the computation. Quality of reconstruction will be measured by RMSE value (RMSE stands for the root mean square error). In Figure
(a) Original image; (b) damaged image; (c) reconstructed image.
In image reconstruction, a discrete version of the Ftransform is used. Details can be seen in [
Let
for all
The condition (
The shape of the basic functions is not predetermined, and therefore, it can be chosen according to additional requirements (e.g., smoothness). Let us give examples of various fuzzy partitions with the Ruspini condition. In Figure
(a) Ruspini partitions with triangular basic function; (b) Ruspini partitions with cosine basic function.
We say that a Ruspini partition of
An
As an example, we notice that the function
We say that the
The elements
The
We use discrete basic functions
Template for basic function definition.
You can see white squares in Figure
(a) Result after selection of fourth pixel in
Because of using Ruspini condition, we can use copies of the basic function for covering the whole range. Height of the template determines values
Template for basic function definition.
The overlapping part of basic functions have functional values equal 1 in each column. For better visualization, black vertical lines are also plotted.
Shapes of the basic functions can be
choose radius
choose active column as
choose current row as
based on those values create a basic function by mirroring and with respect to Ruspini condition formula (
use the basic function for image reconstruction,
compare reconstructed image with original undamaged one by RMSE,
if
change current row as
continue by step 4.
choose current radius as
choose current column as
choose current row as
based on those values create a basic function,
use the basic function for image reconstruction,
compare reconstructed image with the original undamaged by RMSE,
if
if
choose current row as
continue by step 4.
First way
RMSE values of the








Lena  Picture  Nature  
1  62.99  5.16  46.72  4.01  71.25  7.55 
2  49.99  5.04  36.63  3.89  56.39  7.39 
3  49.98  4.97  36.62  3.82  56.37  7.30 
4  49.97  4.93  36.62  3.79  56.36  7.27 
5  49.97  4.92  36.61  3.80  56.35  7.27 
6  49.96  4.95  36.61  3.84  56.35  7.32 
Second way of basic function creation
RMSE values of the

1  2  3  4  5  6 

Lena  
1  6.12  5.71  5.58  5.51  5.50  5.52 
2  5.16  4.97  4.85  4.79  4.78  4.81 
3  5.12  4.96  4.86  4.81  4.80  4.83 
4  5.11  4.97  4.88  4.83  4.83  4.86 
5  5.13  5.00  4.92  4.87  4.87  4.90 
6  5.16  5.04  5.97  4.93  4.92  4.95 
 
Picture  
1  4.93  4.32  4.21  4.16  4.15  4.19 
2  3.93  3.76  3.65  3.61  3.61  3.65 
3  3.91  3.76  3.67  3.63  3.64  3.68 
4  3.92  3.79  3.71  3.67  3.68  3.72 
5  3.96  3.83  3.76  3.73  3.73  3.77 
6  4.01  3.89  3.82  3.80  3.80  3.84 
 
Nature  
1  8.50  7.68  7.45  7.33  7.30  7.33 
2  7.63  7.33  7.16  7.07  7.06  7.10 
3  7.53  7.30  7.15  7.09  7.09  7.13 
4  7.50  7.30  7.18  7.13  7.14  7.19 
5  7.51  7.34  7.23  7.19  7.20  7.25 
6  7.55  7.39  7.30  7.27  7.27  7.32 
(a) Basic function for radius
(a) Basic function for radius
As a result we can say that step by step process converges to oscillating or linear shape.
This work was supported by SGS14/PRF/2013 (advanced techniques of applications of soft computing methods in image processing).