Generating Suitable Basic Functions Used in Image Reconstruction by F-Transform

Abstract

Image reconstruction technique based on F-transform 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.

1. Introduction

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 text. Scratches, cracks, and folds are covered by inpaint. These are shown in Figure 1 which includes also noise.

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Figure 1 

(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 F-transform which brings particular way of valuation neighborhood pixels and their usage in the computation [1, 2]. Other methods [3, 4] differ by the choice at the technique.

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 2 you can see damaged input images and reconstructed ones with usage of the ideal basic function described later. We will analyse influence of changing parameter on target quality.

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Figure 2 

(a) Original image; (b) damaged image; (c) reconstructed image.

2. F-Transform

In image reconstruction, a discrete version of the F-transform is used. Details can be seen in [5, 6]. In this section two-dimensional (2D) variant and also conditions for proper functioning will be briefly introduced.

2.1. Fuzzy Partition with Ruspini Condition

Let 𝑥₁<⋯<𝑥_𝑛 be fixed nodes within [𝑎,𝑏] such that 𝑥₁=𝑎, 𝑥_𝑛=𝑏 and 𝑛≥2. We say that the fuzzy sets 𝐴₁,…,𝐴_𝑛, identified with their membership functions defined on [𝑎,𝑏], establish a fuzzy partition with Ruspini condition of [𝑎,𝑏] if they fulfill the following conditions for 𝑘=1,…,𝑛:(1)𝐴_𝑘∶[𝑎,𝑏]→[0,1],  𝐴_𝑘(𝑥_𝑘)=1;(2)𝐴_𝑘(𝑥)=0 if 𝑥∉(𝑥_𝑘−1,𝑥_𝑘+1), where for uniformity of notation, we set 𝑥₀=𝑎 and 𝑥_𝑛+1=𝑏;(3)𝐴_𝑘(𝑥) is continuous;(4)𝐴_𝑘(𝑥), for 𝑘=2,…,𝑛, increases on [𝑥_𝑘−1,𝑥_𝑘], and 𝐴_𝑘(𝑥), for 𝑘=1,…,𝑛−1, strictly decreases on [𝑥_𝑘,𝑥_𝑘+1];(5)for all 𝑥∈[𝑎,𝑏], ^𝑛_𝑘=1𝐴_𝑘(𝑥)=1. (1)

The condition (1) is known as the Ruspini condition. The membership functions 𝐴₁,…,𝐴_𝑛 are called basic functions. A point 𝑥∈[𝑎,𝑏] is covered by basic function 𝐴_𝑘 if 𝐴_𝑘(𝑥)>0.

2.2. Shape of the Basic Function

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 3, two such partitions with triangular and cosine basic functions are shown. The formulas given below represent generic fuzzy partitions with the Ruspini condition and triangular functions: 𝐴₁(𝑥)=1−𝑥−𝑥₁ℎ₁,𝑥∈𝑥₁,𝑥₂,𝐴0,otherwise,_𝑘(𝑥)=𝑥−𝑥_𝑘−1ℎ_𝑘−1,𝑥∈𝑥_𝑘−1,𝑥_𝑘,1−𝑥−𝑥_𝑘ℎ_𝑘,𝑥∈𝑥_𝑘,𝑥_𝑘+1,𝐴0,otherwise,_𝑛(𝑥)=𝑥−𝑥_𝑛−1ℎ_𝑛−1,𝑥∈𝑥_𝑛−1,𝑥_𝑛,0,otherwise, (2) where 𝑘=2,…,𝑛−1 and ℎ_𝑘=𝑥_𝑘+1−𝑥_𝑘.

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Figure 3 

(a) Ruspini partitions with triangular basic function; (b) Ruspini partitions with cosine basic function.

We say that a Ruspini partition of [𝑎,𝑏] isℎ-uniform if its nodes 𝑥₁,…,𝑥_𝑛, where 𝑛≥3, are ℎ-equidistant; that is, 𝑥_𝑘=𝑎+ℎ(𝑘−1), for 𝑘=1,…,𝑛, where ℎ=(𝑏−𝑎)/(𝑛−1), and two additional properties are met:(6)𝐴_𝑘(𝑥_𝑘−𝑥)=𝐴_𝑘(𝑥_𝑘+𝑥), for all 𝑥∈[0,ℎ], 𝑘=2,…,𝑛−1;(7)𝐴_𝑘(𝑥)=𝐴_𝑘−1(𝑥−ℎ), for all 𝑘=2,…,𝑛−1 and 𝑥∈[𝑥_𝑘,𝑥_𝑘+1], and 𝐴_𝑘+1(𝑥)=𝐴_𝑘(𝑥−ℎ), for all 𝑘=2,…,𝑛−1 and 𝑥∈[𝑥_𝑘,𝑥_𝑘+1].

An ℎ-uniform fuzzy partition of [𝑎,𝑏] can be determined by the so called generating function 𝐴₀∶[−1,1]→[0,1], which is assumed to be even (The function 𝐴₀∶[−1,1]→ℝ is even if for all 𝑥∈[0,1],  𝐴₀(−𝑥)=𝐴₀(𝑥)), continuous, have a bell shape, and fulfill 𝐴₀(0)=1. Basic functions 𝐴_𝑘 of an ℎ-uniform fuzzy partition with generating function 𝐴₀ are shifted copies of 𝐴₀ in the sense that 𝐴₁(𝑥)=𝐴₀𝑥−𝑥₁ℎ,𝑥∈𝑥₁,𝑥₂,0,otherwise, (3) and for 𝑘=2,…,𝑛−1, 𝐴_𝑘(𝑥)=𝐴₀𝑥−𝑥_𝑘ℎ,𝑥∈𝑥_𝑘−1,𝑥_𝑘+1,𝐴0,otherwise,_𝑛(𝑥)=𝐴₀𝑥−𝑥_𝑛ℎ,𝑥∈𝑥_𝑛−1,𝑥_𝑛,0,otherwise. (4)

As an example, we notice that the function 𝐴₀(𝑥)=1−|𝑥| is a generating function for any ℎ-uniform triangular partition. In the sequel, we will be using ℎ-uniform fuzzy partitions only and refer to ℎ as to a radius of partition [7].

2.3. Discrete 2D F-Transform

We say that the 𝑛×𝑚-matrix of real numbers [𝑈_𝑘𝑙] is called the (discrete) F-transform of𝑢with respect to {𝐴₁,…,𝐴_𝑛} and {𝐵₁,…,𝐵_𝑚} if for all 𝑘=1,…,𝑛,𝑙=1,…,𝑚, 𝑈_𝑘𝑙=∑^𝑀_𝑗=1∑^𝑁_𝑖=1𝑢𝑝_𝑖,𝑞_𝑗𝐴_𝑘𝑝_𝑖𝐵_𝑙𝑞_𝑗∑^𝑀_𝑗=1∑^𝑁_𝑖=1𝐴_𝑘𝑝_𝑖𝐵_𝑙𝑞_𝑗. (5)

The elements 𝑈_𝑘𝑙 are called components of the F-transform.

The inverse F-transform is defined as follows: 𝑢𝑖,𝑗=^𝑛𝑚_𝑘=1𝑙=1𝑈_𝑘𝑙𝐴_𝑘(𝑖)𝐵_𝑙𝑗. (6)

We use discrete basic functions 𝐴_𝑘 and 𝐵_𝑙. Our results show that sufficient image reconstruction is enough to use maximal radius 4. Discrete basic function with respect to Ruspini condition is building on top of template in Figure 4.

Figure 4 

Template for basic function definition.

You can see white squares in Figure 4. These squares can be marked as parts of the basic function. One marked per column. After that is automatically computed mirrored part of the basic function as can be seen in Figure 5. We choose 𝑦=4 for the column 𝑎. Value in column 10 is the same, and because of Ruspini condition, values in columns 5 and 7 are easily computable as 𝑓𝑦₅=1−𝑓𝑦_𝑎,𝑓𝑦₇=1−𝑓𝑦_𝑎=𝑓𝑦₅. (7)

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Figure 5 

(a) Result after selection of fourth pixel in 𝑎 column; (b) result after selection of fifth pixel in 𝑏 column.

Because of using Ruspini condition, we can use copies of the basic function for covering the whole range. Height of the template determines values 𝑓(𝑦) on the 𝑦 axis. For better usability we choose value 0 for first row, 1 for last one, and twice 0.5 on the middle. Two basic functions with marked values can be seen in Figure 6.

Figure 6 

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.

3. Influence of Basic Functions

Shapes of the basic functions can be nondecreasing, oscillating, or nonincreasing. Differences and influences on computation are described in [8]. We choose basic functions with radius 2 in the first step of reconstruction and radius 4 in the second step because of fully sufficient usage on the input set of testing images. In this paper, we focus on building shape of basic function step by step based on results provided by RMSE. We identified two ways of this process which will be demonstrated on template in Figure 4 where max=6.

Column by Column. Find the best value for current column and continue with next column:(1)choose radius 𝑟=2, (2)choose active column as 𝑐=“𝑎,”(3)choose current row as 𝑦=1, (4)based on those values create a basic function by mirroring and with respect to Ruspini condition formula (7), (5)use the basic function for image reconstruction, (6)compare reconstructed image with original undamaged one by RMSE, (7)if 𝑦=max, then 𝑟=4, 𝑐=“𝑏” and continue by step 3, (8)change current row as 𝑦=𝑦+1, (9)continue by step 4.

Radius by Radius. Find the best value for current radius and continue with next radius: (1)choose current radius as 𝑟=2, (2)choose current column as 𝑐=“𝑎,”(3)choose current row as 𝑦=1, (4)based on those values create a basic function, (5)use the basic function for image reconstruction, (6)compare reconstructed image with the original undamaged by RMSE, (7)if 𝑦=max and 𝑟=4, then 𝑐=“𝑏” and continue by step 3, (8)if 𝑦=max, then 𝑟=4 and continue by step 3, (9)choose current row as 𝑦=𝑦+1, (10)continue by step 4.

First way column by column is computed in Table 1. You can see that there is for column 𝑎 best RMSE value for 𝑦=6. It means that we choose sixth row for column 𝑎. Next column 𝑏 is the best RMSE value in row 𝑦=5 for the Lena and Nature images and 𝑦=4 for the Picture image. We choose basic function with values 𝑦=6 for column 𝑎 and 𝑦=5 for column 𝑏. This function is in Figure 8.

Table 1 

RMSE values of the column by column basic function creation.

Second way of basic function creation radius by radius is computed in Table 2. From the table comes that the best value is 𝑦=2 for 𝑎 and 𝑦=5 for column 𝑏. This basic function is shown in Figure 7.

Table 2 

RMSE values of the radius by radius basic function creation.

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Figure 8 

(a) Basic function for radius 2; (b) basic function for radius 4.

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Figure 7 

(a) Basic function for radius2; (b) basic function for radius 4.

4. Conclusion

As a result we can say that step by step process converges to oscillating or linear shape. Column by column way is the best solution oscillating basic function. Value 𝑦=6 for column 𝑎 and 𝑦=5 for column 𝑏 provides RMSE values 4.92 for Lena, 3.80 for Picture, and 7.27 for Nature. Because of very close results for Picture image between 𝑏=4 and 𝑏=5, we choose value 𝑏=5 for all of them. Radius by radius is the best linear function with 𝑦=2 for column 𝑎 and 𝑦=5 for column 𝑏. For Lena RMSE is equal to 4.78, for Picture is equal to 3.61, and for Nature is equal to 7.06.

Acknowledgment

This work was supported by SGS14/PRF/2013 (advanced techniques of applications of soft computing methods in image processing).