Trigonometric spline for medical image interpolation

* Corresponding author (samreen5830@gmail.com) Medical imaging deals with visualising internal structures of the human body to diagnose and treat diseases. Several image processing schemes and algorithms are introduced to overcome the challenges regarding medical image analysis. In this paper, a cubic trigonometric spline scheme with multiple quality control parameters is proposed for medical image processing. In comparison, the proposed scheme is found to be better than both the conventional bilinear and bicubic image processing techniques in terms of peak signal-to-noise ratio, root mean square error and structural similarity index measure.


INTRODUCTION
In the present era of health care, medical image processing plays a vital role in the entire process from the diagnosis of diseases and treatment planning, to surgical procedures with follow up case studies.Some of the problems in the production of medical images are poor quality, blurriness, unwanted noise, and low contrast.Thus, image enhancement or resolution is necessitated when a surgeon needs to interpret a medical image, which gets deteriorated in quality during the process of acquiring, as it helps to improve image quality or sharpen its features such as edges, boundaries, or contrast to make a graphic display more useful for investigation.
A number of mathematical schemes or algorithms based on spatial domain methods and frequency domain methods are used for digital image processing to improve the visual and objective quality of a medical image (Lau et al et al., 2013).
Most of these algorithms are based on interpolation and manipulation techniques such as histogram manipulation, empirical mode decomposition, and spline and wavelet interpolation techniques.
Splines are considered to be the most suitable kernels in order to attain better quality in medical images, which are widely used in computer aided designing and computer graphics for different tasks.The most popular splines used for medical imaging are convolution based splines.A number of reasons are presented in literature splines with convolution based splines for medical imaging is also provided, which concludes that splines with a higher degree produce better results as compared to other methods including sinc, linear and convolution based splines (Meijiering, 2000).
In this paper, a cubic trigonometric spline function is proposed for medical image interpolation.Three standard medical images are selected here as test images to evaluate the performance of the proposed scheme in comparison with other conventional image processing techniques, including bilinear and bicubic schemes.Experimental results are conducted both in subjective (visual) and objective measures.

continuity (
) with quality control parameters is established.The basis functions and the cubic trigonometric function with their properties are presented.The detailed implementation strategy of the proposed scheme is described and the experiments and simulation results are presented.

March 2017
Journal of the National Science Foundation of Sri Lanka 45(1)

METHODOLOGY
Trigonometric splines play an important role in electronics and medicine (Qi et al., 2009), shape designing, and data visualisation (Ibraheem et al., 2012;Hussain et al., 2014).A piecewise cubic trigonometric spline of continuity with control parameters has been proposed for medical imaging.The basis functions and the properties are also discussed.
Cubic trigonometric basis functions for given knots ; as follows: ...  Figures 2 and 3, respectively show the graphical view of convex hull property and the variation diminishing property for the proposed cubic trigonometric spline function.

Implementation strategy
This section describes the implementation strategy of the proposed cubic trigonometric spline function to still digital images.Since digital images are formed from by the number of sample points and sample spacing.
March 2017 Journal of the National Science Foundation of Sri Lanka 45 (1) The proposed trigonometric spline function was applied to the selected image data in two dimensional spatial domain using tensor products.
Let us consider a set of image sample data, chosen from an arbitrary rectangular domain in the xy-plane where x and y are non-negative real valued coordinates and presents the image intensity or pixel value at the spatial location .Over each rectangular sub-division , the proposed cubic trigonometric spline function ... (5) where and , whereas , for .

RESULTS AND DISCUSSION
Three different medical test images, named as CT scan-× 512 with 8 bits per pixel were selected to compare the performance of the proposed scheme with conventional techniques including bilinear and bicubic schemes for medical image interpolation.The proposed scheme was applied on the selected test images to get the resulting interpolated image.First the images were down-sampled 512 × 512.Since quality is a major concern with medical images, the quality of the resulting image in contrast to the original test image was evaluated in terms of peak signal-to-noise ratio (PSNR) (Fevralev et al., 2011), root mean square error (RMSE) (Lalitha & Latte, 2011) and structural similarity index measure (SSIM) (Wang et al., 2004).
PSNR is the most commonly used objective metric and is very simple to calculate.It is usually expressed in units of decibel (dB).On comparison of the resulting image with the test image, normally a higher PSNR value indicates a high quality resulting image and a low PSNR value indicates a low quality resulting image.In some cases a resulting image may appear to be closer to the test image although it has a low PSNR value.RMSE is a measure of the residuals (differences) of the test and the resulting images.It is used to combine the residuals into a single measure of analytical power.
an objective image quality metric, is based on structure similarity information of both the test and resulting images.It is the most powerful quality metric among other conventional quality metrics.SSIM is a decimal fraction between 0 and 1, where value 1 is only reachable in the case of two identical sets of image data.Mathematical descriptions of each of these metrics are presented as follows: ... (7) ... (8) where N is the number of pixels in the image and M is the number of bits, which is used to quantise intensity values of the image.X presents the test image whereas Y is the resulting image.and are the mean intensities of X and Y, whereas and are the standard deviations used in the estimation of the contrast.corresponds to the covariance and 1 , C 2 are constants to avoid instability. of PSNR, RMSE and SSIM versus bite per pixel for the test images.From Figure 4, one can see that the bilinear scheme produces the lowest PSNR.Likewise, the proposed scheme has better PSNR values than both bilinear scheme depicts the highest RMSE and lowest SSIM values, respectively, whereas the results attained by the proposed scheme are better than both bilinear and bicubic schemes.Table 1   the proposed scheme along with bilinear and bicubic schemes for the test images in term of PSNR (dB), RMSE and SSIM.From the table it can be seen that the proposed scheme produces better results than the other conventional schemes.Figures 7 to 9 show the details for the sequence of the resulting medical images and their SSIM maps produced using bilinear, bicubic and the proposed image interpolation scheme, respectively.One can easily observe that the edges in the medical images interpolated by the bilinear and bicubic interpolation schemes are comparatively blurred, whereas the results shown in the medical images interpolated by the proposed Subjective To investigate the behaviour of the proposed scheme with different values of quality control parameters in terms of PSNR and SSIM, an arbitrary collection of 248 × 248 pixels of the test image is selected.Table 2 shows the values of the corresponding PSNR and SSIM for different values of quality control parameters using the proposed scheme.

March 2017
Journal of the National Science Foundation of Sri Lanka 45(1)

CONCLUSION
In this paper, a piecewise cubic trigonometric spline scheme was proposed for medical image interpolation and compared with other conventional techniques.The proposed scheme uses multiple parameters to achieve the desired quality of medical images (Table 2).Here three medical test images were selected for experimental work and the results were collected in terms of three different objective quality metrics PSNR, RMSE and SSIM, along with their graphical representation for the selected test images.The experimental results show that the proposed scheme yields comparatively better results than the other conventional techniques both in objective and subjective quality metrics.
x [x i ,x i+1 ]; i=1,2,...trigonometric function (2) takes the form: ...(3) Here, the values of may be given or approximated by using any approximation scheme such as arithmetic mean, geometric mean and harmonic mean approximations et al., 1997).In this work the arithmetic mean approximation is utilised for this purpose.As the values of in cubic trigonometric spline function (3) is viewed values of these s can be obtained by applying iThe cubic trigonometric spline function (4) has the following properties: a. Terminal properties: for each subinterval where presents the derivative of the cubic trigonometric spline function with respect to variable .b. Convex hull property: the entire segment of piecewise cubic trigonometric spline function must lie inside the control polygon spanned by , , and .

Figure 2
Figure 2 Convex Hull Property Graphical view of convex hull property for the proposed cubic trigonometric spline function Graphical representation of objective quality compression, PSNR (dB) vs bite per pixel for the images (a) CT scan-spine; (b) MRI-brain 1; (c) MRI-brain 2 representation of objective quality compression, RMSE vs. bite per pixel for the images (a) CT scan-spine; (b) MRI-brain 1; (c) MRI-brain 2 Graphical representation of objective quality compression, SSIM vs bite per pixel for the images (a) CT scan-spine; (b) MRI-brain 1; (c) MRI-brain 2 (b) (c) (a)

Figure 7 Figure 8
March 2017 Journal of the National Science Foundation of Sri Lanka 45(1) Subjective quality comparison for selected CT Scan-Spine image (a) bilinear; (b) bicubic; Subjective quality comparison for a selected CT scan-spine image (a) bilinear; (b) bicubic; (c) proposed, SSIM maps where brightness indicates the magnitude of the SSIM index; (d) bilinear; (e) bicubic; (f) proposed Subjective quality comparison for selected MRI-Brain 1 image (a) bilinear; (b) bicubic; (c) Subjective quality comparison for a selected MRI-brain 1 image (a) bilinear; (b) bicubic; (c) proposed, SSIM maps where brightness indicates the magnitude of the SSIM index; (d) bilinear; (e) bicubic; (f) proposed Journal of the National Science Foundation of Sri Lanka 45(1) March 2017 PSNR (dB) and SSIM measurements for the test images with different values of quality control parameters

Figure 9
scheme indicates smooth edges with better subjective quality than the others.
gives the results obtained from (1)rnal of the National Science Foundation of Sri Lanka 45(1)March 2017