Image enhancement is an important preprocessing task as the contrast is low in most of the medical images, Therefore, enhancement becomes the mandatory process before actual image processing should start. This research article proposes an enhancement of the model-based differential operator for the images in general and Echocardiographic images, the proposed operators are based on Grunwald-Letnikov (G-L), Riemann-Liouville (R-L) and Caputo (Li & Xie), which are the definitions of fractional order calculus. In this fractional-order, differentiation is well focused on the enhancement of echocardiographic images. This provoked for developing a non-linear filter mask for image enhancement. The designed filter is simple and effective in terms of improving the contrast of the input low contrast images and preserving the textural features, particularly in smooth areas. The novelty of the proposed method involves a procedure of partitioning the image into homogenous regions, details, and edges. Thereafter, a fractional differential mask is appropriately chosen adaptively for enhancing the partitioned pixels present in the image. It is also incorporated into the Hessian matrix with is a second-order derivative for every pixel and the parameters such as average gradient and entropy are used for qualitative analysis. The wide range of existing state-of-the-art techniques such as fixed order fractional differential filter for enhancement, histogram equalization, integer-order differential methods have been used. The proposed algorithm resulted in the enhancement of the input images with an increased value of average gradient as well as entropy in comparison to the previous methods. The values obtained are very close (almost equal to 99.9%) to the original values of the average gradient and entropy of the images. The results of the simulation validate the effectiveness of the proposed algorithm.

Fractional calculus is widely used in research foundation in different domains such as engineering, computer science and others [

Echocardiographic images are one of the most widely used imaging techniques for heart-related problems diagnosis. The main drawback of these images is that they have very low contrast [

The proposed filter is a fractional-order differential filter designed for image enhancement to improve the contrast of the image, consequently helpful to doctors/physicians in diagnosis.

Since echocardiographic images have intensity inhomogeneity, the algorithm is based on firstly partitioning images in terms of homogenous regions, details and edges.

The order of the filter is adaptive stating that it automatically chooses order which is a fractional value for pixels. This implies the feasibility of different order of the filter used as a fractional differential mask to above mentioned three different partitions of the image.

Also, the filter selects higher-order fractional values for a large magnitude of the gradient and vice versa, helps in fast implementation of the filter with appropriate selection of the order of the filter for the entire image, consequently speed up the entire processing of medical images.

Organization of the paper:

The following paper has been organized as follows, Section 2 discusses the related work with background, preliminary studies and literature review. Section 3 describes the materials and methods of the proposed work. Section 4 focuses on experimentation and analysis with experimental setup and parameters followed by results and discussions and Section 5 concludes.

Fractional order calculus, although proposed in history long back decades ago, the significance lies in various aspects of image processing. For the measure of a Euclidean, the frequently used common definitions are R_L, G-L and Caputo definitions. The Euclidean measure is a very frequently used and commonly applied definition for fractional calculus. The computationally complex equations are always the R–L and Caputo definitions, which are Cauchy equations. The weighted sum around the function is used to express the G_L definition. This is an obvious way to show the appropriate applications of image processing. The differential of signal s (t) according to the G_L definition [

In this equation s (t), is [a, t],

The next is to analyze the influence that fractional order differentiation has on a signal. The resultant Fourier transformation of the signal s (t) is given in

The even function in this is the amplitude characteristic and the odd function denotes the phase characteristic. The analysis of all the characteristics that belong to the fractional differential filter is for

According to the observation, the frequency response for the fractional differential filter is nonlinear when the value of

G–L, R–L, and Caputo are the very famous and popular definitions of fractional calculus. Normally R-L and G-L are the most famous definitions that are used in digital image processing (DIP) [

Methodology | Average gradient | Entropy |
---|---|---|

Fix order filters | 10.229 | 3.72 |

Histogram equalization | 6.249 | 2.77 |

Proposed method | 26.37 | 4.67 |

Original values | 5.816 | 4.61 |

This section discusses the stepwise procedure required to achieve image enhancement and a block diagram of the proposed algorithm is given in

In this paper, standard clinical database from the MIMIC (Medical Information Mart for Intensive Care) is an openly available dataset developed by the MIT Lab for Computational Physiology, comprising identified health data associated with ∼40,000 critical care patients are used for analysis. The MIMIC-III Clinical Database is available on Physio Net (doi: 10.13026/C2XW26). The analysis of the proposed method is carried out by testing and applying on the database of the images i.e., six different images are described in the next section.

An important characteristic that is used to identify the objects for any region is the texture. Based on the spatial dependencies, earlier many authors proposed computational texture features. The texture (details) of the image is characterized by major components such as the co-occurrence Matrix feature. In addition to the co-occurrence matrix, it is also required to define a few internal factors of the medical images such as the Contrast, Homogeneity (H), entropy & Local Homogeneity (LH).

If I represents the grayscale image at level m, the position of each pixel is given by

M → concurrent matrix.

Card → predefined mat lab function for the data exchange.

s → position of the pixel.

I → image data under consideration.

The Contrast, when the scale of local texture is larger the distance

Homogeneity is represented as

Entropy is the measure of randomness: close to either 0 or 1.

Local Homogeneity (LH) is given as

These various features/parameters of the medical image are very essential for the advanced image processing which will be done at the pixel level for the query and reference pixel, in this regard the first parameter contrast is given by the

It is also relevant to select every pixel from the f (x, y) image into their relevant classifications such as its edges, homogenous content and details of the image for the classification of the image data requires the operation of signal convolution with suitable mask value which is pre-defined.

With the various regions of the input data for operation, there exists further classification of pixels depending upon the intensity values of the pixels for exact differentiation. Hence providing the scope for the various embedded image features raises the requirement to be particular about the classification based on its intra and inter-pixel values of it. As an appropriate solution, this entire process demands the setting of suitable threshold values as a boundary condition for the feature extraction among the pixels. The threshold values are given by the following equations:

In which _{1} of all the pixels of the input image. λ_{1}, λ_{2 }= Eigenvalues of the image pixel. To analyze and improve the contrast of the image it is required to approach the derivative technique by defining a square matrix condition to compare inter and intra pixel levels which is defined as the Hessian matrix of the order of 2 × 2, in which the four different elements of the matrix under consideration are; the two elements correspond to the inter-pixel values and other two corresponds to the intra-pixel values which are always the second-order partial derivative function of f(x, y) which is given as a Hessian form of f(x, y) as:

In which (

with the above

Generally, two different types of derivative functions namely integer order and fractional order methods exist for the image processing applications. Due to the higher-order accuracy, fractional-order derivative is more preferable than integer-order derivative. The fractional-order derivatives are defined by three definitions namely

Grunwald–Letnikov fractional derivative (G-L).

Riemann-Liouville fractional derivative (R-L) and

Caputo (Li & Xie) et al. fractional derivative.

Out of the above-mentioned G-L fractional derivative plays a vital role in the processing of medical images without affecting their properties as represented in

The image enhancement is carried out by defining the enhancement factor ‘ν’: which acts as the deciding factor for the fuzzy logic to classify the three different regions of the pixel as a homogenous region or edge-based data or detailed data for the pixel. Which will be given by fuzzy decision logic comprising the thresholds

(*) = convolution operator.

The value of ‘ν’ defines the mask value in different directions of the pixel if the 3 × 3 mask is selected two different mask values are defined to perform masking in 8 different directions of pixel processing as shown in

As described in

For the recovery of the signal by the application of the mask and its coefficients, the interval values desired are shown in

Step by step working of the Proposed algorithm:

In this proposition, there is a proposal of a method that can improve contrast, conserve the edges and major edge features of an image, that is fractional-order image enhancement. The algorithm consists of 3 important steps.

Step-1: The enhancement method of fractional order is done based on G–L definition.

Step-2: An improved numerical method is implemented (adaptive filter order) and G–L image enhancement masks are derived.

Step-3: In the final step, the capabilities of the image enhancement process are just demonstrated using the proposed method. The contour in the low-frequency features in a smooth area is protected as shown in experimental results. In addition to that, the nonlinear method is used to maintain the texture and the high-frequency edge details in the areas, where the levels of the gray are not changing significantly.

The effectiveness of the proposed method is tested using a set of echocardiographic images which are tested both in terms of quantitative and qualitative metrics. The performance metric used for quantitative analysis is Average gradient, entropy and SSIM. Entropy is defined by an

The average gradient decides the measure of improvement in image quality. It also decides how clear the image is. Where the increased value in AG indicates an increase in enhancement and also improvement in clarity achieved in the proposed method.

AG reflects the ability to express the details of an image and can be used to measure the relative clarity of the image. AG is calculated as shown in

w.r.t ‘y’.

SSIM is a newer measurement tool that is designed based on three factors i.e., luminance, contrast, and structure as given in

If α = β = γ = 1 (the default for Exponents) then,

The proposed method uses filters for the enhancement of echocardiographic images using adaptive fractional order differentiation. The filter designed is capable of enhancing the image significantly and also preserves the structure of an input image. The proposed filter has the advantage of faster implementation of this method compared to other methods. This filter suitably adopts the fractional order of the masks concerning membership of the current pixel. The analytical test on the input echo images has proved and revealed enough the strengths and sustainability of the designed co-efficient.

If an order of the filter is integer order it is not suitable for images also when an order of the filter is fixed, in some parts of the image improvement is observed but other details in the image regions are affected. As given in

After partitioning the image into three regions accordingly order of the filter is selected for pixels in different regions and an enhanced image with improved contrast is suitable for physicians’ inaccurate diagnoses.

Parameters | MSE | Average gradient (AG) | Entropy | |||
---|---|---|---|---|---|---|

Input image name | ||||||

Lena image | 0.193 | 0.198 | 6.296 | 26.339 | 3.69 | 4.6736 |

Checkerboard image | 0.0033 | 7.7854e−05 | 5.815 | 24.139 | 4.8418 | 4.7619 |

US image | 0.00487 | 7.7855e−05 | 6.6689 | 26.26 | 3.9672 | 4.9954 |

The desired algorithm is also tested with the non-medical image database, and its comparative analysis is done with the existing and proposed algorithms as shown in

In any digital signal processing, concerning the synthetic image or medical images, it is noted that Fractional differentiation plays a vital role in the analysis of enhanced research in the domain. In this research article, improved quality of images is presented with a differentiation technique and algorithm processed with the G-L based technique for enhancement. Finally, it is demonstrated that the algorithm successfully enhanced different regions in an image resulting in increased Average gradient, entropy and SSIM with an adaptive fast selection of the order of the filter as compared to distinct image content, experimental results and analysis have proven that the algorithm is better than the existing with the various standard medical image database and other image databases. The proposed algorithm would make medical image processing speed up further and assess doctors in clinical diagnosis as a result of enhancement achieved. The adaptive nature of the filter is the novelty of the algorithm.

As a future enhancement, the applications of fractional order calculus can be used in other fields of image processing and can explore for segmentation, image restoration, image analysis etc. Also, fractional-order calculus can be a field of research for various advanced image processing techniques as discussed earlier. It has to be noted that the improvement of the proposed algorithm could have an addition of the neural network as a classifier which plays a very important role. With the developed algorithm the performance is measured using the Structural Similarity Index Matrices (SSIM), this similarity index is used as the similarity tool for the measurement of the performance and efficiency, it has to be noted that, there exists a remarkable enhancement of the similarity index among the standard database. It should also be noted that the developed algorithm can also be applied to any of the real-time Images (such as Echocardiographic images) or other untrained databases for the applications. As a future enhancement, an advanced algorithm can be designed for the video applications, satellite images and other formats of images, even as the next improvement it can also be extended to the comparative analysis among other image datasets. As every research ends with some limitations the work carried out in this paper also resulted in some limitations. Availability of the data sets that too real-time echocardiographic images used in the work are the major limitation in the work carried out.