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Due to the development of CT (Computed Tomography), MRI (Magnetic Resonance Imaging), PET (Positron Emission Tomography), EBCT (Electron Beam Computed Tomography), SMRI (Stereotactic Magnetic Resonance Imaging), etc. has enhanced the distinguishing rate and scanning rate of the imaging equipments. The diagnosis and the process of getting useful information from the image are got by processing the medical images using the wavelet technique. Wavelet transform has increased the compression rate. Increasing the compression performance by minimizing the amount of image data in the medical images is a critical task. Crucial medical information like diagnosing diseases and their treatments is obtained by modern radiology techniques. Medical Imaging (MI) process is used to acquire that information. For lossy and lossless image compression, several techniques were developed. Image edges have limitations in capturing them if we make use of the extension of 1-D wavelet transform. This is because wavelet transform cannot effectively transform straight line discontinuities, as well geographic lines in natural images cannot be reconstructed in a proper manner if 1-D transform is used. Differently oriented image textures are coded well using Curvelet Transform. The Curvelet Transform is suitable for compressing medical images, which has more curvy portions. This paper describes a method for compression of various medical images using Fast Discrete Curvelet Transform based on wrapping technique. After transformation, the coefficients are quantized using vector quantization and coded using arithmetic encoding technique. The proposed method is tested on various medical images and the result demonstrates significant improvement in performance parameters like Peak Signal to Noise Ratio (PSNR) and Compression Ratio (CR).

The foremost important purpose of image compression is to ease spectral and spatial redundancy to save or to communicate information. Another purpose is to maintain the quality of the image even at lower bit rate to represent an image and to reconstruct it without any degrade in its visual quality. In lossless compression scheme, the reconstructed image will be similar to the original image. In lossy compression scheme, the reconstructed image may not be similar to the original image. But, lossy compression scheme gives higher compression ratio than lossless compression scheme. The proposed scheme is a kind of lossless compression. In the proposed scheme, the coefficients are obtained using second generation of Curvelet Transform. In the quantization stage, the coefficients carrying least information are rounded off. The transform based compression spatially distributes the energy into less number of data samples so that no information is lost. The compressed image is reconstructed into spatial domain by the inverse transformation process [

Mohammed Hussien Miry [

Over the past years, several researchers accepted wavelet transform for researches on image compression. Here artifacts are avoided at high compression ratios, but on images with different contents, wavelet decomposition produces poor compression ratio. A new compression technique called CALIC was developed for the purpose of context formation, quantization, and modeling. It did not gain popularity because of its poor performance and complex operation on binary and continuous modes.

The above-mentioned methods are inefficient because they use many number of co-efficient to reconstruct edges along curves and to characterize edge discontinuities along a curve. So in order to overcome these drawbacks a new multi resolution transform called the Curvelet Transform was introduced [

1) It is optimally sparse representation of objects with edges.

2) It is optimal image reconstruction in severely ill posed issues.

3) it is optimal sparse representation of wave propagators.

In this paper, wrapping based Curvelet Transform is used. The obtained coefficients after transformation are subjected to vector quantization where quantization process happens in a group. Finally, arithmetic encoding is performed to encode stream of characters. The parameters like Compression Ratio and Peak Signal to Noise Ratio (PSNR) are measured.

There are several techniques for image compression and observation infers the wavelet may not be the best choice to compress medical image. This is because wavelets have the tendency to ignore edge smoothness. The drawback of wavelet is overcome by the introduction of a technique developed by Candes and Donoho in 1999. This was designed originally to represent edges and curves well than wavelet. The representation of an edge using wavelet is shown in

Two generations of Curvelet Transform has been developed. First generations Curvelet Transform (Continuous Curvelet Transform) were obtained from sub band filter theory and Ridgelet theory. The curvelet decomposition takes place in four stages [

Ridgelet analysis of radon transform is used in the first generation of Curvelet Transform. Ridgelet transform performance is very slow. Thus, the use of Ridgelet transform was eliminated and a new method to curvelet as tight frame is taken. Using tight frame, an individual curvelet has frequency support in a parabolic wedge area of the frequency domain [

In an experimental argument, all curvelet fall into one of these three categories [

1) The magnitude of curvelet coefficient will be zero. Whose discontinuities will not intersect with the length wise support (

2) A curvelet whose discontinuity intersects with length-wise support, but not at its critical angle. At this point magnitude of the curvelet coefficient is approximately equal to zero (

3) A curvelet whose length-wise support intersects with a discontinuity, at there the curvelet coefficient magnitude will be greater than zero. (

Second generation of Curvelet Transform (Fast Discrete Curvelet Transform) have the advantage of FFT (Fast Fourier transform). Using FFT, the image is represented in Fourier domain [

All the coefficients of the orientation and scale are in increasing order. The frequency response of the curvelet is a trapezoidal wedge. The digital curvelet tiling of space and frequency is shown in

In the Curvelet Transform theory, there are two methods to obtain the curvelet coefficients:

1) USFFT (Unequal Space Fast Fourier Transform) method,

2) Wrapping method.

In USFFT method, the coefficients are obtained by irregularly sampling of the Fourier samples of an image. In wrapping method, the coefficients are determined by a sequence of translation and the wrap around technique. Both these methods give same output, but the wrapping based method has better and faster computation time [

The oversampled curvelet coefficients is defined by,

where, the superscripts “D” and “O” mention “Digital” and “oversampled”.

R_{j}_{,Ɩ} is a rectangle of size R_{1,j} × R_{2,j} and containing the parallelogram P_{j}_{,Ɩ}. If we assume that R_{1,j} and R_{2,j} divide the image of size n, then the coefficients C_{D}_{,O}(j,_{ }Ɩ, k) is obtained by the discrete convolution of a curvelet and the signal f(t_{1},t_{2}). Then the coefficients are down sampled regularly in the manner that one selects only one out of each n/R_{1,j} × n/R_{2,j} pixel. The dimensions R_{1,j} and R_{2,j} of the rectangle are large. In wrapping approach, R_{1,j} and R_{2,j} are replaced by L_{1,j} and L_{2,j}, the original dimensions of the parallelogram P_{j}_{,Ɩ}. By copying the data by wrap-around or periodicity, we can fit P_{j}_{,Ɩ} into a rectangle with the same dimensions. This is just like relabeling of the frequency samples of the form,

The two dimensional inverse FFT of the wrapped array is of the form,

Note that the relabeling of wrapping does not affect the phase factors. Therefore, we can rewrite the above equation as

Let consider the mother curvelet at scale “j” and angle “Ɩ” of the form,

And _{ }specifies its periodization over the unit square [0, 1]^{2},

The coefficients in the east and west quadrants are given by

This is a discrete circular convolution if and only if L_{1,j} and L_{2,j} both divide n.

Steps for curvelet based on wrapping method

Step 1: Fast Fourier transform (FFT) is applied to input image.

Step 2: Curvelet is obtained at given orientation n and scale s.

Step 3: Divide the FFT into digitally small sets.

Step 4: Each set is translated to origin for every set.

Step 5: Wrap the parallelogram.

Step 6: Apply inverse FFT.

Steps for inverse curvelet based on wrapping method

Step 1: Inverse FFT of the array is obtained for each curvelet coefficient array.

Step 2: Inverse process is applied to unwrap the rectangular support to the original orientation shape.

Step 3: Original position is obtained by translating the shape.

Step 4: The whole curvelet array is stored.

Step 5: All the translated and stored curvelet array are added

Step 6: Reconstructed image is obtained by taking Inverse FFT.

Image compression algorithmic flow is given below.

Step 1: Image is read from the user.

Step 2: The three level decomposition is used here, discrete curvelet wrapping technique is used to obtain curvelet coefficients.

Step 3: The curvelet coefficient values are quantized so that the approximate values are obtained from this.

Step 4: After quantization process, thresholding is applied. The thresholding value is initially set. The coefficient value below the threshold is neglected and above is maintained.

Step 5: Curvelet coefficients above the threshold values are calculated.

Step 6: Apply arithmetic encoding technique to encode the coefficients.

Step 7: Inverse Curvelet Transform is achieved by applying Inverse wrapping based algorithm.

Step 8: Reconstructed image is obtained from the Inverse Curvelet Transform.

Step 9: Finally, the performance parameters are calculated.

Flowchart for the algorithmic approach is shown in

Since 1980, Vector Quantization (VQ) is a popular technique. In VQ, inputs are sampled into set of well-defined vectors by the use of some distortion measures. Multimedia images and many other formats of images are used

in varied applications these days. Storage of multimedia data needs bigger storage. So in order to reduce memory image compression plays a vital role. In quantization large set of inputs are mapped into small set of possible outcomes. Inputs are individual numbers in scalar quantization. Scalar quantization refers to the process of rounding off to the nearest value. The vector input is considered for vector quantization. In VQ, the modeling of probability density functions is done through allocation of prototype vectors. In this method, the large set of vectors is divided into a set which has almost the same number of points [

Varieties of techniques are employed for compression having complexities ranging at different degrees. The final step of any compression system is entropy encoding which represents data in a compact manner. This encoding process may complement the outputs for prior stages. Of all entropy encoding techniques, arithmetic coding is the most effective, versatile and an elegant process [

Peak Signal to Noise Ratio is a ratio of the highest signal power and the unwanted noise power. The real image is the signal and the error in re-establishment is the noise. The large Peak Signal to Noise Ratio is used for an enhanced compression. The PSNR is inversely proportional to compression ratio. In order to get actual compression, the PSNR and compression ratio should be balanced. The PSNR can be measured using following equation:

Mean Square Error is the amount of difference among the real image and the compressed image. MSE is the collective square of difference among the compressed image and the real image. The Mean Square Error should be as low as possible in order to get smaller distortion and large output value. MSE can be measured using following equation:

where,

f(i,j) is the pixel in the input image,

F(i,j) is the pixel in the reconstructed image,

M × N is the size of input image.

Compression ratio means the ratio of amount of bits needed to denote the real image to the amount of bits needed to denote the compressed image. If the compression ratio is high then the quality is negotiated. Compression methods with no loss of information will have smaller compression ratio than the lossy compression methods.

Peak Signal to Noise Ratio and Compression Ratio are the important parameters which are used to determine the quality of any compression algorithm.

Simulation results are shown in Figures 5(a)-(g). We have calculated PSNR and Compression ratio for various medical images using proposed algorithm and it has been compared with wavelet and Discrete Cosine Transform results. PSNR and compression ratio values are calculated by using Equation (9) and Equation (11). Comparison of PSNR values obtained for various medical images by proposed algorithm with wavelet and DCT algorithms is shown in

In this paper, we have proposed a new method for image compression depending on wrapping based fast discrete Curvelet Transform. The coefficients are obtained using fast discrete Curvelet Transform. In proposed method, we have used vector quantization to quantize the coefficients. Arithmetic coding technique is used for encoding the quantized coefficients of fast discrete Curvelet Transform. The proposed scheme has been tested on various medical images and the result demonstrates significant improvement in PSNR and compression ratio. Our method can further be enhanced for compressing real time videos with some necessary modifications. This is our future work.

Input Image | Technique | PSNR | CR |
---|---|---|---|

MRI cerebellum image | Proposed method | 40.32 | 74.3 |

DCT | 19.28 | 7.8 | |

Wavelet | 20.8 | 24.2 | |

MRI cerebrum image | Proposed method | 35.27 | 69.3 |

DCT | 17.3 | 9.12 | |

Wavelet | 22.5 | 19.6 | |

MRI brain image | Proposed method | 40.32 | 74.3 |

DCT | 18.6 | 12.65 | |

Wavelet | 24.7 | 25.2 | |

CT abdomenimage | Proposed method | 37.55 | 63.3 |

DCT | 14 | 10.2 | |

Wavelet | 21.2 | 24.5 | |

CT brain image | Proposed method | 35.75 | 55.61 |

DCT | 12.6 | 8.3 | |

Wavelet | 20.3 | 17.8 | |

Ultrasound image of a fetus | Proposed method | 41.80 | 86.63 |

DCT | 13.5 | 12.34 | |

Wavelet | 25.2 | 29.3 | |

Ultrasound liver image | Proposed method | 39.7 | 70.32 |

DCT | 14.3 | 9.36 | |

Wavelet | 19.7 | 12.34 |

P. Anandan,R. S. Sabeenian, (2016) Medical Image Compression Using Wrapping Based Fast Discrete Curvelet Transform and Arithmetic Coding. Circuits and Systems,07,2059-2069. doi: 10.4236/cs.2016.78179