IVUS CONTOURS DETECTION BY USING

SNAKES AND WAVELET TRANSFORM

Mohamed Ali HAMDI

Mohamedali.hamdi@insat.u-carthage.tn

National Institute of Applied Sciences and Technology

Carthage University

Tunisia

Abstract –This paper presents a new approach for computer aided detection of intima border detection and Media-Adventitia border Detection in the intra-vascular ultrasound (IVUS) image. The proposed segmentation method is done in two stages. The first stage is a Pre-processing and noise treatment, in the second stage the ability of these filtered images in detecting region of interest is done using Active Contours method.Results shows that the proposed approach gives a satisfactory detection performance

Keywords: IVUS, wavelet Transform, Histogram, snakes

I- introduction

The goal of medical imaging is to create intelligible visual representation information with a medical character. This problem is more generally in the context of the scientific and technical image: the objective is to be able to represent in a relatively simple format a large amount of information from a multitude of measures acquired according to a well-defined mode. The resulting image can be processed by computer o obtain for example: A three-dimensional reconstruction of an organ or tissue a film or an animation showing the volution

or movements of an organ over time, quantitative imagery that represents the values measured for some biological parameters in a given volume.

In a broader sense, the field of medical imaging encompasses all techniques for storing and manipulating this information. So, there is a standard for computer management of data from imaging medical: the DICOM standard Depending on the techniques used, medical imaging examinations allow to obtain information on the anatomy of the organs (their size,

their volume, their location, the shape of a possible lesion, etc.) or their functioning (their physiology, their metabolism, etc.). In the first in the case of structural imaging, the second is functional imaging.

Among the most commonly used structural imaging methods in medicine, we can mention the methods based on:

- X-rays (conventional radiology, digital radiology, tomodensitombe or CT-scan, angiography, etc.),

- Nuclear magnetic resonance (MRI),

- Ultrasound (ecographic methods),

- Light rays (opticalmethods).

Functional imaging methods are also very varied. They group together

- Nuclear medicine techniques (PET, TEMP) based on the emission of positrons or gamma rays by radioactive tracers which, after injection, are concentrated in areas of intense metabolic activity,

- Electrophysiological techniques that measure changes inthe electrochemical state of the tissues (especially in relation to the activity nerve)

- The techniques resulting from the so-called functional MRI,

- Thermographic or infra-red spectroscopy measurements.

Cardiovascular disease is the leading source of death worldwide. They are responsible each year for the death of more than 17 million people, or 30% of the world's mortality, according to the World Health Organization (WHO), whose ¾ are held in low- and middle-income countries. Cardiovascular pathologies should keep this list for several more years. 25 million deaths are expected in 2020. Most of them are the clinical consequence of the complications occurring at the level of atheromatous lesions of the arteries. Atherosclerosis is an arterial disease that is at the root of many serious cardiovascular diseases. [1][2][3]It concerns lesions of the walls of all the arteries which are responsible for acute assignments, in particular ischemic heart disease due to atherosclerosis of the coronary arteries (myocardial infarction MI), stroke (stroke) due to atherosclerosis of the encephalic arteries, arteritis due to atherosclerosis of the arteries of the lower limbs, or acute limb ischemia.

Atherosclerosis is a variable association of rearrangements of the intimal walls of medium and large arteries. It is actually a local accumulation of lipoprotein complexes, complex carbohydrates, blood and blood products, fibrous tissue and calcareous deposits, inflammatory cells; all accompanied by changes in the media (WHO, Washington 1954). The clinical complications of atherosclerosis are numerous; figure 7 shows the main ones are classically four in number: vascular obstruction or stenosis, rupture and thrombosis, hemorrhage, and embolism.

The composition of an atherosclerotic plaque is complex and its morphology thus varies gradually over time in an erratic and therefore unpredictable manner. In order to really understand the severity and the different evolutionary behaviors of an atherosclerotic plaque, it is very interesting to use endovascular and therefore interventional exploration techniques. Ultrasound endovascular or intravascular ultrasound (IVUS)[4][5]is an endovascular imaging technique that provides accurate quantitative and qualitative information on obstructive or vulnerable plaque. It may also be indicated in the therapeutic follow-up of the pathology as an alternative to or in addition to X-ray angiography whenever the angiographic data are insufficient or ambiguous. In contrast to angiography and angioscopy, IVUS imaging is able to image a cross-section of the arteries and has demonstrated specific features that identify and measure vascular wall and atherosclerotic lesion size[6][7][8]. However, the 3D shape of the artery is unknown and the different regions of interest are difficult to delineate on the IVUS images. The first objective of this work is therefore to develop a segmentation method based on the snakes model for detecting vascular wall contours of IVUS images of arteries as well as minimizing the user intervention needed for initialization of the segmentation method by adopting a new method

II- Different aspects of image processing

Mathematical tools to perform image processing are many and varied. We will present some of them along the studied issues. In particular we will place ourselves in a deterministic frame work even if the image considered may be tainted by random noise. For some stochastic mathematical methods we refer for example to:

- Basic operations on images (lightening, contrast) by transforming each level of grayin another according to criteria related to the histogram of the image.

- filtering helps to clean noisy images and / or isolate some details. The main tools of filtering are convolution, Fourier transformation (frequency filters) ,wavelet, curvelet and contourlet transform.

- The segmentation makes it possible to determine the outlines and / or the regions in an image.

- The restoration (or deconvolution) allows to find an image from noisy or fuzzy data. We will talk mainly about variational methods.

III- Pre-processing and noise treatment

We propose the following method to improve the quality of the image explain in Figure 1

Return the filtered image to the correct size

Reconstruction of the filtered image by Transformed into

Wavelet Fast Reverse (IFWT)

Threshold of the coefficients

Calculation of wavelet coefficients w by Fast Wavelet Transform (FWT)

Reflection of the image to have an image of dimensions of the powers of 2

Choosing a wavelet base and a threshold ε

Denoising by nonlinear filtering :

Wavelet filtering

Cropping Transformation

Histogram of an image

Dynamic cropping – contrast

Medical image

Figure 1: preprocessing proposed

1- Dynamic cropping – contrast

This is a transformation that changes the dynamics of the levels of gray in order to improve the visual appearance of the image. At one level of gray f of the original image corresponds to the level t (f) in the transformed image. Each pixel is subjected to a treatment depending only on its value. The transformation t (f) can be carried out in real time on the image in acquisition course using a transcoding table. Transformations are related to the histogram of the image:

2- Histogram of an image

The histogram of an imageis the histogram of the data series corresponding to the levels

gray pixels. It is a discrete function defined by:

∀ p∈ {0 , …… .,255 } hp: Numberofpixels h avingpforgrayscale

We can define a continuous version h of the histogram by making an interpolation (eg piecewise linear) hp values so that:∀ p∈ {0 , …… .,255 } hp=h( p).

The histogram gives an indication of the dynamics of the image (distribution grayscale) but is in no way a feature of the image.

3- Cropping Transformation

We give ourselves an image presenting a concentrated histogram in the interval [a, b]. The values a, b correspond to extreme gray levels present in this image. Dynamics reframing is about expanding the dynamic of the image transformed to the total extent [0, 255]. The transformation Cropping is therefore an affine application which is written, the result obtained in figure 5:

t ( f )={255 f −ab−a

0 iff <a255 iff >b

fora ≤ f ≤b (1)

Variations for contrast enhancement, The types of correction given below make it possible to accentuate the contrast in a precise level range.

t ( f )={ ba

for 0 ≤ f ≤ a

¿(255−b ) f +255 (b−a ) fora ≤ f ≤255

255−a

(2)

4- Equalization of the histogramThe histogram of an image is rarely flat which translates entropy not maximum [26][27]. Equalization transformation is constructed of such so that the histogram of the transformed image is as flat as possible.This technique improves the contrast and allows increasing artificially the clarity of an image thanks to a better distribution of relative intensities [28][29][30].Consider the continuous histogram→ h( f ) , thereforef '=t ( f )the histogramequalized is f ' → h'¿¿) must approach the ideal result.Two elementary surfaces in correspondence in the initial histograms and equalized have the same number of points which allows to write:

f '=t ( f )=255N ∫

0

f

h ( s) ds (3)

Here, N is the total number of pixels in the image and h / Nis the normalized histogram (between 0 and 1). By replacing the continuous integration with a sum, we obtain the following discrete equalization transformation:

f '=t ( f )=255N ∑

i=0

f

h i (4)

Figure 5 shows the result of the equalization of histogram.

5- Non-linear filtering noiseIn any digital image, the gray or color values observed present an uncertainty. This uncertainty is due to the uncertainties of the count photons arriving on each sensor. The measured color values are disturbed because the sensors receive parasitic photons and undergo electrostatic fluctuations during their charges and discharges. When a sensor receives a lot of photons coming from a well-lit scene; the parasites are negligible compared to the flow of true photons. But even in a photo of sufficient exposure, the dark pixels receive very little photons and are therefore noisy. Visually, we usually distinguish two types of image noise that accumulate:- The chrominance noise, which is the colored component of the noisy pixels: it is visible in the form of random color spots,- The luminance noise, which is the luminous component of the pixels noisy: it is visible in the form of darker or lighter spots giving a grainy appearance to the image.

6- Noise and blur modelingImages obtained by an artificial acquisition process which often degraded by disturbances generically called noise. This noise is in most cases due to the acquisition conditions:

- Calibration of the device: despite increasingly sophisticated technologies, there are always minimal measurement or manipulation errors, disturbances by the environment: this is the case of medical images for example (radiography, ultrasound) where the information is noisy by the crossing of the observed tissues.We call signal to noise ratio (SNR), of a given signal x tainted with a noise b over a range of finite time I the quantity:

SNR=20 log10(‖x‖‖b‖) (5)

7- Wavelet filteringThe wavelet transform uses a family of translations and expansions of the same function: the mother wavelet [13]. The translation and expansion factors are the two parameters of this transform. A wavelet transform allows the sine and the cosine to decompose a signal. The wavelets are localized in time. The localized character of the wavelet is expressed by the fact that the function is non-zero over a finite interval, and zero everywhere else.The wavelet can be dilated and translated. Thus from the mother wavelet, we generate the daughter wavelets also called wavelet atoms defined by the expression (6):Where s is the scale parameter (expansion) and u is the translation parameter. Thus, by

varying these two parameters, it is possible to completely cover the time-frequency plane and thus obtain a complete and redundant representation of the signal to be analyzed. In contrast to short-term Fourier analysis, wavelet analysis can vary temporal and frequency resolutions. In expression 6, the scale parameter s plays the role of the inverse of the frequency. Indeed, decreasing s, the temporal support of s is reduced and thus covers a range of high frequencies. By increasing the value of s, the temporal support of u, s is increased and makes it possible to cover a range of low frequencies. Figure 2 shows the time-frequency tiling for the wavelet transform.

Figure 2: Time-frequency for the wavelet transformtime

frequency

(6)ψu,s ( t )= 1√s

ψ ( t−us

)

8- Fast decomposition algorithm of MALLAT:The Mallat algorithm, [20][21]is practically the most used in the case of orthogonal and bi-orthogonal multi-resolution analyzes. This algorithm allows, starting from the approximation to the 2-j-1 resolution, to represent the approximation and the details at the 2-j resolution by projection on the space of the approximations Vj and the space

The details Wj are translated by the expressions:[22][23] h and g being the quadrature mirror filters, respectively low pass and high pass.

Mallat's algorithm then successively decomposes an approximation of the signal:

f in a rough approximation plus the details. This decomposition is obtained by projecting the signal on the approximations space and on the details space by filtering and decimation by a factor of 2At reconstruction, each 2-j-1 approximation is inversely computed from A2

-jf and D2-jf (2-j resolution approximation and details) by filtering and

interpolating by 2:The two-dimensional transition is expressed by the tensor product of filters h and g:

h[n] h[m]=hh(n,m)

h[n] g[m]=hg(n,m)

g[n] h[m]=gh(n,m)

g[n] g[m]=gg(n,m)

Using the separability of the wavelet bases, Mallat's algorithm in 2D can be described

by:

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

A2− j f =∑

kA

2− j−1 f .h (k−2 n )=A2− j−1 f∗h (2n )

A2− j−1 f =∑

kh(k−2 n) A

2−j f +∑k

g( k−2 n) D2− j f

D2− j f =∑

kA

2− j−1 f . g(k−2 n)=A2− j−1 f ∗g(2 n)

A2− j f =∑

k∑

lhh(k−2n , l−2 m) A

2− j−1 f =A2− j−1 f ∗hh(2 n ,2 m)

The Figure 3 shows the decomposition scheme of an image by the Mallat’s algorithm

[24][25]:

Figure 3: Functional diagram of the Mallat synthesis algorithm

Additive noise adds to the basic signal a great many details that will translate into

multiresolution analysis by wavelet coefficients to small scales. To denoise, we can keep only

the coefficients wavelets that correspond to important scales. In practice, we thresholds the

coefficients with a cutoff function that cancels the coefficients below a certain threshold ε,

then we reconstruct the signal from the coefficients thus kept. Two types of corresponding

2

2

2

2

2

2

(16)

(17)

D32-j f

A2-j-1 f

A2-j f

D12-j f

D22-j f

+

+

+

h

g

h

h

g

g

D2− j3 f =∑

k∑

lgg( k−2 n ,l−2m) A2− j−1 f =A2− j−1 f ∗gg (2n, 2m)

D2− j2 f =∑

k∑

lgh(k−2 n ,l−2m ) A2− j−1 f=A2− j−1 f∗gh(2 n ,2m)

D2− j1 f =∑

k∑

lhg (k−2n , l−2m) A2− j−1 f =A2− j−1 f ∗hg (2n ,2 m)

thresholding can be used with two cutoff functions: the hard thresholding is to apply to the

coefficients the function.

We present below (Figure 4) the denoising with the bases of wavelets from Haar and

Daubechies respectively. The choice of the threshold is often delicate. In practice, it is taken

proportional to the noise level, ε = 3σ for the hard thresholding and ε = 1.5 σ for the soft

thresholding for a more precise estimation of the threshold according to the size of the image.

Fig. 4 Denoising for a noisy IVUS (Figures produced with Matlab) :(A)Original image . (B) Haar wavelt. (C) Daubechies wavelet

Fig. 5 Denoising for a noisy image with σ = 50 (Figures produced with Matlab) :(A)Original image . (B) Equalization of histogram. (C) Cropping Transformation . (D) Reconstruction of

the filtered image by Transformed into Wavelet Fast Reverse (IFWT)

IV- Segmentation, Active Contours Method

Segmentation isolates parts of the image that present a strong correlation with the objects contained in this image, usually for the purpose of post-processing. The fields of application are numerous: medicine, geophysics, geology, etc. In the medical field, image segmentation is extremely complicated. Indeed, for each organ (brain, heart, etc ...)[9][10][11][12], the approach is different: the segmentation tool must therefore be able to adapt to a particular organ, according to a particular acquisition modality (scanners, Radiography, Magnetic Resonance Imaging, ...) and for a sequence particular data. The goal is the quantification of information, by volumetry: the volume of a tumor in the brain, the study of ventricular heart cavity, etc. This is where the segmentation of the image is used. In mathematical imaging; two types of segmentation are considered: The segmentation by contours which allows delimiting the different regions by their borders. It is essentially this type of segmentation that we are going to present. Segmentation by regions that characterizes regions an image with a homogeneous structure and that uses most of time to statistical tools.

In this section we are interested in the method of active contours (snakes) introduced by Kass, Witkin and Terzopoulos, a method that integrates the notion of regularity of contour points by introducing a functional interpreted in terms of energy for mechanical properties that she represents. This method makes it possible to evolve in time and space the representation of the model towards the solution of the problem of minimization introduced in the modeling. These methods of active contours call upon the notion of an elastic body undergoing external constraints. [13][14]The shape taken by the elastic is linked to a minimization of energy composed of two terms:

- A term of internal energy and

- An external energy term.

The local minimum obtained by minimizing this non-convex functional is related to the initial condition that defines a search neighborhood of the minimum. If one puts the active contour and pε [a, b] and evolution parameter τ ε [0, T] such that C:

The evolution of the contour is given by the following equation [15][16][17]:

(18)

(19)

C(P,τ) =C0(P) such as C0 is the initial edge

The deformation of the contour is done thanks to a force F which one will break it down in the reference of Frenet:

We only consider the normal component, and the equation of evolution of an active contour is:

1- Segmentation of IVUS images

In the literature of imagers the difference between segmentation and contour detection are only the methods used but the results almost the same, the segmentation of IVUS images, the areas of interest to be detected are the light and the vascular wall. So to start the segmentation method, the contours that correspond to the external elastic limb and the light-intimal interface must be localized. The IVUS [18][19]segmentation algorithms that are performed will be validated with images of coronary arteries. Most of the algorithms have been implemented for the detection of a single contour of the vascular wall, but the majority has never segmented the two contours. Our contribution proposes the segmentation of the two contours and this approach is summarized in the flowchart following, in the figure 6:

(20)

(21)

Figure 6.Method proposed

the contour is initialized in the form of a circle with the same center and a diameter greater than that detected, at the start of the algorithm the first contour is disposed inside the light interface -intima after it will expand to fill the forms better, in the same way the second contour is placed on the outside of the external elastic limiter will also retract to marry to the best of the forms

Figure 7: IVUS image

Figure 8: intima border detection

Figure 9: Media-Adventitia border Detection

Figure 10: A) the catheter circle ( green color ) and the Lumen (red color ); B) position of the Media/Adventitia edges(red color)

Figure 11: Six Samples from obtained results

2- Discussion

Figure 8 illustrates intima border detection, figure 9 shows the result obtained as a Media-Adventitia border Detection and figure 10 illustrates the difference between the position of the Media and the adventitia edges

The goal of this work was to demonstrate the IVUS segmentation potential of the method proposed, and the usefulness of region of the interest in the determination of the vessel wall anatomical structures.

The combined algorithm with wavelet preprocessing was first presented to detect a mixture of distributions in intravascular ultrasound data. The mixture parameters were estimated with an initialization of edges and maximum number of iterations the algorithm at the beginning of each segmentation. Figure 11 showed that mixture detection is a robust and stable process with standard deviations for several runs of the algorithm on different pixel subsets of an IVUS. Only small differences were thus observed between different our method estimations of the mixture parameters on the IVUS datasets. As expected, because of instrument settings and echogenicity specific to the different plaque structures. IVUS image processing is a difficult but important task. Image series contain highly relevant clinical information but are sometimes of poor quality and subject to shadowing and catheter artifacts. This method has demonstrated the efficiency of snakes segmentation using Wavelet as a preprocessing, of intravascular ultrasound images.

V- Conclusion

IVUS images present certain difficulties that complicate automated image processing: catheter artefacts, calcified shadows, collateral vessels and blood ultrasound speckle. A segmentation method in IVUS imaging of femoral arteries has therefore been developed. Segmentation is based on morphological snakes. The different contours of the vascular wall are detected simultaneously in the IVUS images.

The segmentation includes a pretreatment phase for filtering the wavelet transform images, followed by an initialization procedure based on the detection of the catheter circle. This procedure is a fully automatic method since the initial contours of the LEE and the initial contours of the light are detected fully automatically. Interactive initialization makes it possible to compensate for the lack of information in certain sequences when large calcifications or collaterals complicate the identification of the outer contour of the wall. In this way, no sequence was excluded from the study.

Compliance with Ethical Standards

Competing Interests

The author declare that there is no conflict of interest regarding the publication of this paper.

References

[1] Lee JT, White RA. Basics of intravascular ultrasound: an essential tool for the endovascular surgeon. SemVascSurg 2004;17(2):110–8.

[2] Klingensmith JD, Shekhar R, Vince DG. Evaluation of three-dimensional segmentation algorithms for the identification of luminal andmedial–adventitia borders in intravascular ultrasound images. IEEE Trans Med Imaging 2000;19(10).

[3] Nissen SE, Yock P. Intravascular ultrasound: Novel pathophysiological insights and current clinical applications. Circulation 2001;103: 604–616

[4] Mintz G, Nissen S, Anderson W, Bailey S, Erbel R, Fitzgerald P, Pinto F, Rosenfield K, Siegel R, Tuzcu E, Yock P. American College of Cardiology Clinical Expert Consensus Document on Standards for Acquisition, Measurement and Reporting of Intravascular Ultra-sound Studies (IVUS). A report of the American College of Car-diology Task Force on Clinical Expert Consensus Documents. J Am CollCardiol 2001;37:1478 –1492.

[5] Nissen SE. Shortcomings of coronary angiography and their implications in clinical practice. ClevelClin J Med 1999;66(8): 479 – 85

[6] Mintz GS, Painter JA, Pichard AD, Kent KM, Satler LF, Popma JJ, Chuang YC, Bucher TA, Sokolowicz LE, Leon MB. Atherosclerosis in angiographically normal coronary artery reference segments: an intravascular ultrasound study with clinical correlations. J Am CollCardiol 1995;25(7):1479 – 85.

[7] Tuzcu EM, Berkalp B, De Franco AC, Ellis SG, Goormastic M, Whitlow PL, Franco I, Raymond RE, Nissen SE. The dilemma of diagnosing coronary calcification: angiography versus intravascular ultrasound. J Am CollCardiol 1996;27(4):832 – 8.

[8] Hausmann D, Johnson JA, Sudhir K, Mullen WL, Friedrich G, Fitzgerald PJ, Chou TM, Ports TA, Kane JP, Malloy MJ, Yock PG. Angiographically silent atherosclerosis detected by intravascular ultrasound in patients with familial hypercholesterolemia and familial combined hyperlipidemia: correlation with high density lipoproteins. J Am CollCardiol 1996;27(7):1562 – 70.

[9] Hoffmann R, Mintz GS, Popma JJ, Satler LF, Kent KM, Pichard AD, Leon MB. Overestimation of acute lumen gain and late lumen loss by quantitative coronary angiography (compared with intravascular ultrasound) in stented lesions. Am J Cardiol 1997;80(10):1277 – 81.

[10] Moraes, M.C . An automatic media-adventitia border segmentation approach for IVUS images . IEEE Computing in Cardiology. 2010 .p 389 - 392

[11] Unal, G. Bucher, S. ;Carlier, S. ; Slabaugh, G. ; Tong Fang ; Tanaka, K. . Shape-Driven Segmentation of the Arterial Wall in Intravascular Ultrasound Images. Information Technology in Biomedicine, IEEE Transactions on .2008. p 335 – 347

[12] Katouzian, A. Angelini, E. ; Carlier, S. ; Suri, J. ; Navab, N. ; Laine, A. .A State of The Art Review on Segmentation Algorithms in Intravascular Ultrasound (IVUS) Images . Information Technology in Biomedicine, IEEE Transactions on .2012 Feb .p1

[13] Cardinal MHR, Meunier J, Soulez G, Maurice R, Therasse E, Cloutier G. “Intravascular Ultrasound Image Segmentation: A Three-Dimensional Fast-Marching Method Based on Gray Level Distributions” IEEE Trans. Med. Imag. 2006 May;vol.25(5):590–601.

[14] Giannoglou GD, Chatzizisis YS, Koutkias V, Kompatsiaris I, Papadogiorgaki M, Mezaris V, Parissi E, Diamantopoulos P, Strintzis MG, Maglaveras N, Parcharidis GE, Louridas GE. “A novel active contour model for fully automated segmentation of intravascular ultrasound images: In vivovalidation in human coronary arteries” Computers Biol. Med. 2007 Sep;37:1292–1302.

[15] Brusseau E, de Korte C, Mastik F, Schaar J, van der Steen A. “Fully automatic luminal contour segmentation in intracoronary ultrasound imaging - A statistical approach” IEEE Trans. Med. Imag.2004;vol.23:554–566.

[16]Giannoglou GD, Chatzizisis YS, Sianos G, Tsikaderis D, Matakos A, Koutkias P, Diamantopoulos N, Maglaveras GE, Parcharidis GE, Louridas V. “In-vivo validation of spatially correct three-dimensional reconstruction of human coronary arteries by integrating intravascular ultrasound and biplane angiography” Coron Artery Dis. 2006;vol.17(6):533–43.

[17] L. Alvarez, L. Baumela, P. Marquez-Neila and P. Henriquez. "Morphological Snakes", CVPR 2010, San Francisco, U.S.A http://www.sciweavers.org/publications/morphological-snakes

[18]V. Caselles, R. Kimmel, and G. Sapiro. "Geodesic active contours", International Journal of Computer Vision, 22(1):61–79, 1997.

[19]F. Catté, F. Dibos, and G. Koepfler. "A morphological scheme for mean curvature motion and applications to anisotropic diffusion and motion of level sets", SIAM Journal on Numerical Analysis, 32(6):1895–1909, 1995.

[20] YounesAkbari et al, Wavelet-based gender detection on off-line handwritten documents using probabilistic finite state automata, Image and Vision Computing, Volume 59, March 2017, Pages 17-30

[21] IssamDagher, SaadIssa, Subband effect of the wavelet fuzzy C-means features in texture classification, Image and Vision Computing, Volume 30, Issue 11, November 2012, Pages 896-905

[22] CláudioRositoJung,Combining wavelets and watersheds for robust multiscale image segmentation, Image and Vision Computing, Volume 25, Issue 1, January 2007, Pages 24-33

[23]Hang Yu Ee-Chien Chang Zhiyong Huang ZhijianZheng, Fast rendering of foveated volumes in wavelet-based representation, Visual Comput (2005) 21: 735–744 DOI 10.1007/s00371-005-0331-1

[24]Ubirata A. Ign ˜ acio ´ Claudio R. Jung, Block-based image inpainting in the wavelet domain, Visual Comput (2007) 23: 733–741DOI 10.1007/s00371-007-0139-2

[25] Huawei Wang · Kai Tang, Biorthogonal wavelet construction for hybrid quad/triangle meshes, Vis Comput (2009) 25: 349–366 DOI 10.1007/s00371-008-0300-6

[26] Sangwoo Lee · Younghui Kim · Jungjin Lee · Kyehyun Kim · Kyunghan Lee · Junyong Noh, Depth manipulation using disparity histogram analysis for stereoscopic 3D, Vis Comput (2014) 30:455–465 DOI 10.1007/s00371-013-0868-3

[27] BahmanSadeghi · Kamal Jamshidi · Abbas Vafaei · S. AmirhassanMonadjemi, 2DIGH: a polar invariant local image descriptor based on joint histogram, Vis Comput DOI 10.1007/s00371-017-1433-2

[28] S. Biasotti · A. Cerri · M. Aono · A. Ben Hamza · V. Garro · A. Giachetti · D. Giorgi · A. Godil ,Retrieval and classification methods for textured 3D models: a comparative study, Vis Comput (2016) 32:217–241 DOI 10.1007/s00371-015-1146-3

[29] Daniel Simões Lopes · Pedro F. Parreira · Ana R. Mendes · Vasco M. Pires, Explicit design of transfer functions for volume-rendered images by combining histograms, thumbnails, and sketch-based interaction, Vis Comput DOI 10.1007/s00371-017-1448-8

[30] Zhiyong Li · Shuang He · MervatHashem, Robust object tracking via multi-feature adaptive fusion based on stability: contrast analysis, Vis Comput (2015) 31:1319–1337 DOI 10.1007/s00371-014-1014-6