A super-Gaussian discrete cosine transform filter for the ... 5/Issue 3/IJESIT201603_01.pdf · International Journal of Engineering Science and Innovative Technology (IJESIT) Volume

ISSN: 2319-5967

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Abstract— This paper presents a novel method using discrete transforms to segment exudates and blood vessels in

retinographies. Illumination correction is previously done based on a homomorphic filter due to uneven illumination in

retinographies. To distinguish foreground objects from the background, we propose a family of super-Gaussian filters in

the discrete cosine transform domain and we analyze the difference between Butterworth and super-Gaussian band-pass

filters. The filters are applied on the green channel since it has the relevant information to segment pathologies. To detect

exudates in the filtered image, a gamma correction is first applied to enhance foreground object. Then, Otsu’s global

thresholding method is used, after which, a masking operation over the effective area of retinographies is performed to

obtain final segmentation. For blood vessels, the negative of the filtered image is first calculated, and then a median filter is

applied to reduce noise and artifacts followed by gamma correction. Again, Otsu’s global thresholding method is applied

for image binarization. Next a morphological closing operation is employed and a logical masking operation gives the

resulting segmentation. Illustrative examples taken from a worldwide free clinical database are included to demonstrate the

capability of the proposed segmentation method.

Index Terms— Butterworth filter, discrete cosine transform, image segmentation, super-Gaussian filter.

I. INTRODUCTION

Millions of people across the world live with varying degrees of irreversible vision loss because they have an

untreatable, degenerative eye disorder, which affects the retina [1]. Under these conditions, the delicate layer of

tissue that lines the inside back of the eye is damaged, affecting its ability to send light signals to the brain. When

the blood vessels are damaged by high blood sugar levels and initially become defective, later they may become

blocked. The defective vessels can lead to hemorrhages (spots of bleeding), fluid, and exudates (fats) to escape

from the blood vessels over the retina. The blocked vessels can starve the retina from oxygen (ischemia), leading

to the growth of new abnormal vessels in the retina.

Actually, there are several screening eye exams that help to find any illness, among the exams are the amsler grid,

auto fluorescence, dilated eye exam, fundoscopy or ophthalmoscopy, eye fundus photography, fluorescence

angiography, optical coherence tomography, and tonometry [2], [3]. In recent researches, diverse techniques such

as the Hough transform, mathematical morphology, illumination correction, and histogram equalization [4]-[9]

have been applied to eye fundus photography to find blood vessels, exudates, or hemorrhages. Other works

enhance the image using Gaussian filters as well as the watershed transform [10]-[12]. In this paper, we use the

databases known as DIARETDB0 and DIARETDB1 (Standard Diabetic Retinopathy Database Calibration Level

{0,1}) from Lappeenranta University in Finland [13], where each image in these databases has a size of 1152 ×

1500 pixels. Knowing that the area of the optic disk is 2.47 mm2

(radius = 0.887 mm), the corresponding

approximate spatial resolution is about 4.746 µm per pixel.

The purpose of this research work is to extract exudates and blood vessels in eye fundus color images. Basically,

our proposal consists of the following steps. First, a binary image mask is created to obtain the boundary of the eye

fundus in the acquired color image by clipping the effective area taken by the camera. Second, a homomorphic

filter in the discrete cosine transform (DCT) [14] or Fourier transform domain is applied to homogenize image

illumination, after which a super-Gaussian band-pass filter in the frequency DCT domain is used to distinguish

between foreground objects and eye fundus image background. Then, two procedures are proposed for the

different types of pathologies mentioned earlier. The first procedure that determines exudates employs gamma

correction to enhance contrast, Otsu’s global thresholding method serves to binarize the image, and a logical

operation between the binary mask and the thresholded image is realized to get the segmented image.

The second procedure is for blood vessels. First the negative of the filtered image is obtained, then a median filter

A super-Gaussian discrete cosine transform filter

for the detection of pathologies in retino graphies Luis David Lara-Rodríguez and Gonzalo Urcid

Optics Department, National Institute of Astrophysics, Optics, and Electronics (INAOE)

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is applied to reduce noise and artifacts, again a gamma correction is applied to enhance contrast, and image

thresholding (Otsu’s method) is performed using global statistics to obtain the desired object regions including

their edges. Final segmentation of blood vessels is obtained by morphological closing followed by a logical

operation between the binary mask and the thresholded image. Our paper is organized as follows: Section II

explains in detail the different image processing steps involved in the proposed frequency DCT filtering based

method for the segmentation of the aforementioned pathologies including several representative examples.

Section III presents the segmentation results obtained as well as the proposed segmentation algorithm in pseudo

code format. We close the paper with Section IV of conclusion and some pertinent comments.

II. SEGMENTATION OF EXUDATES AND BLOOD VESSELS

A. Theoretical Background

A binary mask is designed to delete the effective area taken by the eye fundus camera, an example of a mask is

shown in Fig. 1. The desired mask is obtained by extracting the red, green and blue channels, then a quotient

between the red and green channel is calculated. To the resulting image a median filter is applied to reduce noise

and Otsu’s global thresholding method is used to get the final mask.

Fig. 1. a), b) eye fundus color images of right and left eyes respectively; c), d) associated binary masks.

In a segmentation process, it is possible to discriminate objects of interest from the background by dividing the

image in regions that satisfy certain conditions [15]. In general, due to the presence of non-uniform illumination in

eye fundus images, we propose the use of a Fourier or DCT homomorphic filter to homogenize illumination. The

discrete cosine transform (DCT) is a finite sequence of data points in terms of a sum of cosine functions

oscillating at different frequencies and amplitudes. DCT’s are important in numerous applications in science and

engineering. Formally, the DCT is a linear and invertible operator that can be identified with an array of M × N

elements. Mathematically, the two dimensional DCT is given by,

1

0

1

0 2

)12(cos

2

)12(cos,

,2,

M

x

N

y N

vx

M

uxyxf

MN

vuvuC

,

(1)

where is the two-dimensional Kronecker’s delta. If u = v = 0, then δ = 1; if u,v ≠ 0, then δ = 0. On the other

hand, recall that the two dimensional discrete Fourier transform (DFT) is given by,

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1 1

2 / /

0 0

, ,M N

i ux M vy N

x y

F u v f x y e

,

(2)

Where, is an input image of size M × N , u = 0,1,…, M − 1, and v = 0,1,…, N – 1, for both transforms. If

is multiplied by the transform is centered and its DFT is computed with a fast Fourier

transform (FFT) algorithm. The filtered image, denoted by , is given by,

, IDFT , ,g x y F u v H u v , or (3)

, IDCT , ,g x y C u v H u v (4)

Where, in (3), IDFT is the inverse discrete Fourier transform and is the DFT of the input image .

In (4), is the DCT of the input image . Note that the inverse discrete cosine transform (IDCT) has

the same expression as (1) with u and v changed by x and y, respectively. Besides, is a specific

homomorphic filter in the Fourier or DCT domain. For numerical computation, the functions C, F, H, and g are

matrices of the same size as the given image. A high-pass Butterworth homomorphic filter (HF) is defined by [15],

4

HF H L L

0

,,

n

D u vH u v

D

,

(5)

Where, the filter order n determines the slope of the function between the low (γL < 1) and high (γH > 1) gamma

bounds. In (5), the particular values used for DFT filtering of the eye fundus color images are set to γL = 0.75, γH =

1.75, n =2, and D0 = 10 is the chosen cutoff spatial frequency. In similar fashion, the values used for DCT filtering

of the eye fundus color images are set to γL = 0.75, γH = 1.75, n = 2, and D0 = 20 is the cutoff spatial frequency. An

example of illumination correction for an RGB eye fundus color image is shown in Fig. 2.

Fig. 2. a) Eye fundus color image, b) illumination corrected color image using DFT, c) illumination corrected color

image using DCT.

We can observe in Fig. 2 that the homomorphic filtered image using the DCT looks better since illumination is

more uniform than the DFT filtered image. For that reason we work with the DCT instead of the DFT filtered

image. Once the illumination is corrected, filtering with the discrete cosine transform is used to intensify the

foreground objects against the surrounding background of the corresponding green channel.

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We introduce a super-Gaussian function as a novel filter for processing images in the frequency domain [16]. The

advantages of this type of filter is explained next in comparison with a Butterworth filter. Table I gives the

defining expressions of the types of super-Gaussian filter frequency functions, i.e., low-pass, high-pass, band-pass

and band- reject that can be used for image processing.

Table I. Mathematical expressions of the four types of Butterworth and super-Gaussian filters.

Low-pass

1

2

BLP 1 /n

oH r D D

(6a)

2

02

SGLP

nr

DH r e

(6b)

High-pass

1

2

BHP 1 /n

oH r D D

(7a)

2

02

SGHP 1

nr

DH r e

(7b)

Band-pass

12

2 2

BBP 0, 1 /n

H r W D D WD

(8a)

22 2

0

SGBP ,

nr D

rWH r W e

(8b)

Band-reject BBR BBP, 1 ,H r W H r W (9a) SGBR SGBP, 1 ,H r W H r W

(9b)

In equations (6a) through (8b), is the filter order, W is the band-pass width, is

the Euclidean distance from the center of the filter (the origin), and D0 is the cutoff spatial frequency in the DCT

domain. An analysis between a super-Gaussian band-pass filter and a Butterworth band-pass filter in terms of

slope and bandwidth rates is discussed next. Specifically, we determine the ratio between slope variations of both

filters to find out which one has a major slope in or near the cutoff frequency. Also, we find the ratio between

band-pass widths variations of both filters to find which filter has the narrower bandwidth or is more selective.

Therefore, partial derivation of both filter expressions, first with respect to r (radial distance) and second with

respect to W (band-width) is displayed in the following equations. Note that subscript SGBP refers to a band-pass

super-Gaussian filter and subscript BBP is an abbreviation for a band-pass Butterworth filter.

The slope variation for the BBP and SGBP filters, in (8a) and (8b), is determined as follows,

2 22 2 2 2

0 02 12 2 2 2

0 0SGBP SGBP 2

2 12 2 2 2

0 0

BBP BBP2 22 22 2

02 2

0

, , 2 ,

21, 1 , .

1

n nr D r Dn

rW rW

n

nn n

r D r DH r W e H r W n e

r rW r W

nW r D rW r DH r W H r W

rrW r D rWr D

Hence, the corresponding slopes ratio, denoted by ρS(r,W), is found to be,

22 2

022 22 2SGBP

0

4BBP

, .

nr D

n n rW

S n

Hr D rW e

rr WH rW

r

(10)

Using (10), we now prove that in a neighborhood of the cutoff frequency D0, the slope of a SGBP filter is greater

than or equal to the slope of a BBP filter with the same order while keeping a fixed bandwidth. First, let r = D0,

then, ρ(D0) = 1; second, consider r = r1 ≈ D0 then,

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222

122 2 2 SGBP BBP

1 0 1 4

1

0 , 1 or .

nn

nn

S n

a rW H Hr D a r W

r rrW

Similarly, from (8a) and (8b), the bandwidth variation for the SGBP and BBP filters is given, respectively by,

2

2 202

2 2

SGBP 0

22 2

BBP

2 2 2 2

0 0

2 and

21 .

nr Dn

rW

n n

H r Dne

W W rW

H n rW rW

W W r D r D

From the previous two expressions, the bandwidths ratio ρW(r,W) results in the following expression,

22 2

022 22 2SGBP

0

4BBP

,

nr D

n n rW

W n

Hr D rW e

Wr WH rW

W

(11)

Next, based on (11), we verify that within the band limits DL and DH, the band-pass width of a SGBP filter is

strictly less than the band-pass width of a BBP filter of the same order. Recall that, DL = D0 – W/2 and DH = D0+

W/2; also, if we let r = DL or r = DH, then we have,

2 22 20 0

2 22 2 2 2 2

112 2

2 2 22 2

4 4

2224 44

4 4

0

,

1 .

n n

L

L L

nn

n nn

L o H o

D D D

n nD W Dn n

L L

W L n n

L L

nn nL

L SGBPBBP

n n

L L

D D D D a

a D W e a D W eD W

D W D W

e a D W e D W HH

W WD W D W

From the ratio analysis based on (10) and (11) between filter types, we can observe that the super-Gaussian

band-pass filter exhibits a higher slope and a more selective band-pass width (delimiting fall-off frequencies).

Therefore, it should be clear that a super-Gaussian filter with tunable order n establishes a balanced frequency

response between those displayed by a simple Gaussian and Butterworth filters. Figure 3 shows the 1D profiles of

super-Gaussian high and low-pass filters along positive frequencies, computed using the values (in pixels), D0 =

105, n = 1,..,5, and W = 35, over the interval r = 0,...,511. Analogously, Fig. 4 displays the 1D profiles for

super-Gaussian band-reject and band-pass filters.

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Fig. 3. a) SGHP filters, b) SGLP filters; D0 = 105 and n = 1,…,5 over the pixel subrange r = 0,…,350.

Fig. 4. a) SGBR filter, b): SGBP filter; D0 = 105, n = 1,…,5, and W = 35 over the pixel subrange r = 0,…,200.

Furthermore, for a visual comparison, Fig. 5 shows several Butterworth band-pass and super-Gaussian band-pass

filter profiles of different orders in the DCT domain, with parameters values set to D0= 200 (cutoff frequency) and

W = 150 (bandwidth) for both filters.

As can be seen in the given graphs, the band width value is better delimited with a super-Gaussian filter since its

curve is steeper than the corresponding curve of a Butterworth filter, where the values of D, D0, W are given in

pixels. In the discussion that follows, we use a super-Gaussian band-pass (SGBP) filter given by (8b) in Table I.

In the present study, the design parameter values of a super-Gaussian band-pass filter are set to n = 2, W = 275,

and D0 = 100. The filtered image, after inversion with the IDCT, is given by the equivalent spatial expression,

SGBP HF SGBP, , , , .Gg x y I x y h x y h x y

(12)

In (12), * denotes convolution. The DCT filtering stage is the same for segmenting exudates as well as blood

vessels and the specific steps to segment each type of pathology are described in the following subsections.

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Fig. 5. a) Butterworth and b) super-Gaussian band-pass filter 1D profiles; c) Butterworth and d) super-Gaussian

band-pass filter 3D contours for n = 2.

B. Segmentation of Exudates

For this type of pathology, a 3 × 3 median filter is first applied over the whole image with the purpose to

preserve exudates edges; then, a gamma correction with value 2 is applied to to enhance

contrast, and Otsu’s method, based on global and local statistics [15], is used to compute a global threshold value

denoted by LOtsu. Specifically, the binary output image is determined as

SGBP EOtsu

E

if

otherwise

0 g ,,

1

x y LB x y

,

(13)

Where, the E sub index label refers to exudates. The segmented image is found by masking the previous image

with the initial binary mask M, i.e., SE = BE ∧ M, where ∧ is the logical AND or minimum matrix operation (cf.

Case construct in Algorithm DTBS, Sec. III). An illustrative example of exudates segmentation is displayed in

Fig. 6.

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Fig. 6. a) Eye fundus color image with hard and soft exudates (database: diaretdb0−v1−1/image007) with enhanced

illumination, b) super-Gaussian band-pass filtered enhanced image, c) binary image showing segmented hard and soft

exudates.

C. Segmentation of Blood Vessels

In order to emphasize blood vessels, the negative image, of the previous

filtered image is first computed, where L is the maximum value of the corresponding grayscale dynamic range. In

our case, L = 256, since we are using the usual 8-bit coded grayscale. Then, as in the case of exudates, a 3 × 3

median filter is passed over image to preserve edges, followed by a gamma correction of value 3 applied to

. In similar fashion, thresholding is performed using Otsu’s method to obtain a binary image

based on the global threshold value LOtsu. Specifically, the binary output image is computed as,

NSGBP VOtsu

V

if

otherwise

0 g ,,

1

x y LB x y

,

(14)

where, the V sub index label refers to blood vessels. Once the binary image in (14) is computed, a morphological

closing operation is used to connect object edges in the corresponding regions of , i. e.,

, where S is a structuring element selected as an isotropic square of size 3 × 3 on-pixels

[17]. In other words, image BV (x,y) is first eroded and then dilated by S thus first eliminating artifacts or spikes

smaller than S and then restoring the overall objects shapes. Finally, the segmented image SV containing the blood

vessels is obtained by masking the previous image. Hence, SV = BV ∧ M , where ∧ is the logical AND or minimum

matrix operation (cf. corresponding Case construct in Algorithm DTBS, Sec. III). An representative example of

blood vessels segmentation is depicted in Fig. 7. We remark that using the same technique as described in this

subsection, it is possible to segment other pathologies such as aneurysms, which are localized blood filled balls of

variable size in the wall of a blood vessel [18].

III. IMAGE SEGMENTATION RESULTS

To test our proposed segmentation technique, we use six eyes fundus color images taken from the free domain

databases DIARETDB0 and DIARETDB1. The first database consists of 130 eye fundus color images, of which 20

are normal and 110 contain signs of diabetic retinopathy such as hard exudates, soft exudates, micro aneuyrysms,

hemorrhages, and neovascularization. The second database consists of 89 eye fundus color images, from which 84

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images contain at least mild non-proliferate signs of diabetic retinopathy. The other five images are considered

normal since these do not contain any signs of diabetic retinopathy. The aforementioned clinical characteristics

are based on the expert’s knowledge who participated in the overall evaluation of both databases.

Fig. 7. a) Eye fundus color image with blood vessels and aneurysms (database:diaretdb0−v1−1/image005) with

enhanced illumination, b) super-Gaussian band-pass filtered and contrast enhanced image, c) segmented binary image

showing segmented pathologies.

The eye fundus color images were captured using the same 50 degree field-of-view digital fundus camera with

varying imaging settings [19]. As mentioned earlier, each image has a size of 1152 × 1500 pixels, with an

approximate spatial resolution of 4.756 micrometers per pixel. We remark that the global threshold value obtained

by applying Otsu’s method is different for each image. Figures 8 to 13 show illustrative examples of exudates and

blood vessels segmentations. Table II gives the segmentation parameter numerical values (gamma correction and

Otsu’s global threshold) used for exudates and blood vessels. In the exudates segmented images, we can observe

that the optic disk has a similar color as exudates; however, our technique does not segment the optic disk. In Figs.

10 to 12, in addition to segmented blood vessels, aneurysms are also distinguished in the segmentation and

correspond to small amorphous regions not connected to the blood vessels. Algorithm DTBS gives in

pseudo-code format the sequence of instructions to implement of our filtering based segmentation technique.

Table II. Gamma correction and threshold values for segmenting exudates and blood vessels.

Exudates (γ = 2) Blood Vessels (γ = 3)

Figure No. : Pathology type Figure No. : Pathology type

5 : hard & soft exudates 0.20 6 : blood vessels & aneurysms 0.32

7 : hard exudates 0.19 9 : blood vessels & aneurysms 0.27

8 : hard exudates 0.16 10 : blood vessels & aneurysms 0.25

11 : blood vessels 0.29

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Fig. 8. a) Eye fundus color image with hard exudates (database: diaretdb1−v02−1/image015), b) binary image with

segmented hard exudates.

Fig. 9. a) Eye fundus color image with hard exudates (database: diaretdb1−v02−1/image019), b) binary image with

segmented hard exudates.

Fig. 10. a) Eye fundus color image with blood vessels and aneurysms (database: diaretdb1−v02−1/image033),

b) binary image with segmented blood vessels and aneurysms.

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Fig. 11. a) Eye fundus color image with blood vessels and aneurysms (database: diaretdb0−v01−1/image054),

b) binary image with segmented blood vessels and aneurysms.

Fig. 12. a) Eye fundus color image with blood vessels (database: diaretdb1−v01−1/image093), b) binary image with

segmented blood vessels.

hom hom

0 0 L H 0

hom hom hom

0 0 L H

ho

- Discrete Transform Based Segmentation for Exudates and Blood Vessels

, ,

G

D n D n W M S

H HF D n

g IDCT DCT I H

Algorithm DTBS

Input parameters , , , , , ,

Illumination correction

, , ,

m

SGBP 0

SGBP SGBP

SGBP SGBP

E SGBP EOtsu

NSGBP

1

H HBP D n W

g IDCT DCT g H

g GammaCorrection MedFilter g

S Binarize Minimum g L M L

g Ne

DCT Filtering

, ,

Case : Exudates Segmentation

,

, ,

Case : Blood Vessels Segmentation

SGBP

NSGBP NSGBP

V NSGBP VOtsu

1

gative g

g GammaCorrection MedFilter g

S Binarize Minimum g L M L S

,

, ,

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Soft exudates as shown in Fig. 6, are nerve fiber layer infarcts or pre-capillary arterial occlusions. On the other

hand, hard exudates as illustrated in Figs. 6, 8, and 9, represent the accumulation of lipid in or under the retina

secondary to vascular leakage; this is due to the aqueous portion of the fluid that is absorbed more quickly than the

lipid component. Thus, the lipid that builds up in or under the retina becomes visible as yellowish deposits. In

Figs. 7 and 10 to 12, we can observe the segmentation of blood vessels (temporal arcades) whereas aneurysms,

that look like small islands and are easily seen where the macula is located, are displayed in Figs. 7, 10, and 11.

The confusion matrix contains information about actual and predicted classifications realized by a pattern

recognition system [20]. Performance of such systems is commonly evaluated using the data in the matrix shown

in Table III.

Table III. Confusion matrix.

Positive test True positive

TP

False positive

FP

Negative test False negative

FN

True negative

TN

Total TP + FN FP + TN

The confusion matrix has four entries which are, the number TP of correctly detected objects that a test group

sample gives positive, the number TN of incorrectly detected objects that a test group sample gives negative, the

number FP of correctly detected objects classified erroneously (negative), and the number FN of incorrectly

detected objects classified as positive.

The confusion matrix help us to find the true positive rate (TPR), also known as sensitivity, which refers to the

proportion of objects that give positive test results, and the true negative rate (TNR), also known as specificity,

which refers to the proportion of objects that give negative test results, (14) gives these proportions,

and TP TN

TPR TNRTP FN TN FP

(15)

To obtain the sensitivity and specificity rates of our test image examples, twenty eye fundus color images hand

marked of exudates and blood vessels were used as ground-truth. In Table IV, the values of sensitivity and

specificity of the method proposed (DTBS) are compared with other methods taken from the technical literature,

both for the segmentation of exudates and blood vessels.

Table IV. Sensitivity and specificity rates for the segmentation of exudates and blood vessels in eye fundus color images

between various segmentation techniques versus the proposed method DTBS.

Exudates sensitivity and specificity Blood vessels sensitivity and specificity

Method proposed by Method proposed by

Jaafar [4] 0.8930 0.9930 Saleh [9] 0.8423 0.9658

Garaibeh [6] 0.9210 0.9910 Niemeijer [22] 0.6898 0.9696

Welfer [7] 0.7048 0.9884 Oloumi [23] 0.8579 0.9000

Kande [21] 0.8600 0.9800 Staal [24] 0.7194 0.7793

DTBS (in this work) 0.9412 0.9910 DTBS (in this work) 0.8517 0.9832

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IV. CONCLUSION

In this work, we have introduced a discrete cosine transform based filtering approach to the problem of extracting

exudates and blood vessels (including aneurysms) in eye fundus color images. A Fourier transform or DCT

homomorphic Butterworth type filter is proposed for illumination correction of input images and a binary mask of

the effective area of the retinography is constructed as a quotient between the red and green channels.

An important step of the proposed method is the DCT spatial frequency domain processing using a

super-Gaussian band-pass filter with carefully selected parameters. This novel type of filtering achieves an

adequate contrast of the foreground objects against the background. We choose the DCT over the Fourier

transform since its computation deals only with real values instead of complex numbers; in addition, the results

obtained with the corresponding homomorphic DCT based filter are visually better. For blood vessels, a negative

of the filtered image is first obtained, then a median filter is applied for both pathologies to reduce background

noise and artifacts, and a gamma correction is applied to enhance the resulting image contrast. Then, a binary

image is determined using Otsu’s statistical method. However, before the final masking operation is realized, an

intermediate operation is used in the case of detecting blood vessels. In particular, a closing morphological

operation is required with an appropriate structuring element. Several illustrative examples are provided to

demonstrate the segmentation results obtained with the proposed method. Based on the commonly used confusion

matrix in the medical clinical realm, notice that high sensitivity and specificity rate values have been obtained in

our segmentation examples in comparison with other techniques described in the technical literature.

Future work contemplates two aspects. First, to extend the number of tests using the DTBS method on other

clinical databases available such as DRIVE [25] and STARE [26]. Second, to introduce the necessary additions to

our base algorithm in order to include the segmentation of other known pathologies such as hemorrhages or retinal

neovascularization, considering for example, the image samples taken from the database used here (DIARETDB).

A further enhancement of our DCT based proposal will consider the design and implementation of an automatic

pattern recognition system for classifying eye fundus pathologies as a tool for assisted medical diagnosis, capable

of distinguishing, e.g, diabetic retinopathy from maculopathy.

ACKNOWLEDGMENT

Luis D. Lara-Rodríguez thanks the National Council of Science and Technology (CONACYT) for doctoral

scholarship CVU 332238. Gonzalo Urcid is grateful with the National Research System (SNI-CONACYT) for

financial support through grant No. 22036.

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ISSN: 2319-5967



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AUTHOR BIOGRAPHY

Luis David Lara-Rodríguez received his B.E. (2007) from Puebla Institute of Technology (ITP), Mexico, and his

M.Sc. (2011) from the Benemeritus Autonomous University of Puebla (BUAP), Mexico. He is a Ph.D. student in the

Optics Department at the National Institute of Astrophysics, Optics, and Electronics (INAOE) in Tonantzintla,

Mexico. His present research interests include 3D modeling, digital image processing and analysis, optical systems

simulations, and computer programming.

Gonzalo Urcid received his B.E. (1982) and M.Sc. (1985) both from the University of the Americas in Puebla,

Mexico, and his Ph.D. (1999) in optics from the National Institute of Astrophysics, Optics, and Electronics (INAOE)

in Tonantzintla, Mexico. He holds the appointment since 2001 of National Researcher from the Mexican National

Council of Science and Technology (SNI-CONACYT), and currently is an Associate Professor in the Optics

Department at INAOE. His present research interests include applied mathematics, digital processing of multichannel

images, artificial neural networks based on lattice algebra, and pattern recognition.

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