Regular Articles

Image enhancement and segmentation using weighted morphological connected slope filters

[+] Author Affiliations
Jorge D. Mendiola-Santibañez

Universidad Autónoma de Querétaro, Doctorado en Ingeniería, Cerro de las Campanas S/N, CP. 76703, Querétaro, México

Iván R. Terol-Villalobos

CIDETEQ, S.C., Parque Tecnológico Querétaro S/N, San Fandila-Pedro Escobedo, 76703, Querétaro, México

J. Electron. Imaging. 22(2), 023022 (Jun 10, 2013). doi:10.1117/1.JEI.22.2.023022
History: Received September 20, 2012; Revised May 2, 2013; Accepted May 10, 2013
Text Size: A A A

Open Access Open Access

Abstract.  The morphological connected slope filters (MCSFs) are studied as gray level transformations, and two contributions are made on these operators with the purpose of modifying the gradient criterion performance. The proposals consist of: (a) the introduction of three weighting functions and (b) the application of a displacement parameter. The displacement parameter will permit the image segmentation in a certain intensity interval and the contrast improvement at the same time. This characteristic is an important difference among the MCSFs introduced previously, together with the other transformations defined in the current literature utilized uniquely to enhance contrast. Also, an application example of the weighted morphological slope filters is provided. In such an example, white matter is separated from brain magnetic resonance images T1.

In this paper, a transformation capable of producing a multiscale contrast enhancement and a multiscale image segmentation is studied. This morphological connected transformation is obtained from the concepts presented in Refs. 1234. In the current literature, the multiscale contrast enhancement has been studied from different approaches. For example, there are methods based on: (1) histogram,57 (2) edges,79 (3) wavelets,1013 (4) mathematical morphology (MM),14,15 (5) retinex method,16 and (6) other methods.17 With respect to image segmentation, there also have been proposed several multiscale connected transformations, for example, viscous watershed,18 viscous openings,19 levelings,20,21 opening by reconstruction,22,23 and others. Some differences among the transformations proposed in this paper and those provided in the current literature are the following: (a) contrast is modified as a consequence of a segmentation process and (b) no new contours are generated during the processing, because the partition generated by flat zones (regions with a constant intensity level) is utilized to define them. The flat zones will be used to compute the morphological transformations by merging them, but always preserving the partition of the original image.

The proposed transformations are called weighted morphological connected slope filters (WMCSFs). They are based on the morphological connected slope filters (MCSFs) defined in 3 and derived from the morphological slope filters (MSF) introduced in 2. The WMCSFs, MCSFs, and MSFs operate similarly. For example, for enhancing dark components, if a slope given as a threshold parameter is greater than the gradient value at a certain point (denominated proximity criterion or gradient criterion), then the erosion is applied, otherwise the original image is maintained (such behavior corresponds to a contrast mapping24). This process is iterated on the whole image until the stability is reached. The MCSFs show a better behavior than the MSFs because they permit a multiscale treatment, and the performance of the WMCSFs is superior to the MCSFs due to the use of a weighted gradient criterion. Originally, the gradient was weighted by the original image. From this antecedent, two proposals are made on the WMCSFs in this paper: (1) the first contribution consists of the introduction of three functions to weight the gradient: logarithm, exponential, and cubic root, which are related to the human visual perception25 and (2) the second contribution deals with the establishment of a displacement parameter, which has two implications, indicating the intensity level at which the gradient criterion is applied and allowing the selection of an intensity interval to segment the processed image.

Some morphological definitions are necessary previous to the proposals, and for this Sec. 2 presents a background on the transformations defined on the partition as well as a brief summary on the MSFs. The flat zone and adjacency notions are defined in Sec. 2.1. Later, the morphological erosion and dilation on the partition are introduced, which will be useful to outline MCSFs in Sec. 2.2. In Sec. 3.1, a new class of weighting functions with certain displacement is defined, and in particular three weighting functions previously mentioned are used to weight the gradient criterion throughout the paper. In Sec. 3.2, an experiment to detect the best weighting function is shown. In Sec. 3.3, WMCSFs are characterized considering the displacement parameter fixed and the slope variable. The maxima and minima modifications are depicted in Sec. 3.4, where several inclusion properties are provided. The hierarchical or multiscale segmentation is justified in Sec. 3.5 together with the presentation of two experiments, one of which illustrates the visual differences among the results produced by the watershed,26,27 the morphological connected contrast mappings based on top-hat criteria,15 and our proposal. The other illustrates a quantitative difference between the edges of the original image, the contours of the output images generated by the retinex method,16 a method based on a wavelet transformation,11 and the WMCSFs introduced in this paper. An algorithm is provided in Sec. 3.6, where brain magnetic resonance images T1 (BMRI T1) are processed to separate the white matter from the rest of the brain tissue. Conclusions are presented in Sec. 4.

Connectivity and Connected Transformations

Those transformations processing the components of the image under a certain connectivity allow a better formalization and deeper understanding of what happens when the components of the image are treated. Serra28 established connectivity by means of the connected class concept.

Graphic Jump LocationF1 :

Adjacent flat zones. (a) Original partition with five flat zones; (b) flat zone in the point x, Fx(f); (c) flat zone in the point y, Fy(f); (d) two adjacent flat zones, i.e., Fx(f)Fy(f)=γx(Fx(f)Fy(f)); and (e) adjacent flat zones to Fv(f).

Graphic Jump LocationF2 :

Erosion and dilation on the partition. (a) Original image with 16 flat zones; (b) contours of the partition generated by the flat zones of the image in (a); (c) erosion on the partition of the image in (a); (d) dilation on the partition of the image in (a); and (e) Left image: flat zone with gray level of 255 and its four adjacent flat zones. Right image: gray level value of the erosion.

Graphic Jump LocationF3 :

Erosion on the partition size μ=2. (a) Original image (top) and its partition (bottom); (b) erosion on the partition size μ=1 (top) of image in (a) and its partition (bottom); and (c) erosion on the partition size μ=1 (top) of the image in (b), i.e., εμ=2(f,Pf) and its partition (bottom).

Morphological Slope Filters

In MM, the morphological contrast enhancement is based on the contrast mappings notion.24 The contrast mappings concept is developed similarly to the transformation proposed by Kramer and Bruckner,33 expressed as follows: Display Formula

Wδε(f)(x)={δμB(f(x))ifδμB(f)(x)f(x)εμB(f)(x)otherwise,(9)
where δμB(f)(x)={f(y):yμB˘x} and εμB={f(y):yμB˘x} are the morphological dilation and erosion, respectively, at pixel level. B represents the structuring element of size 3×3 pixels, which contains its origin. While B˘ is the transposed set of B respect to its origin B˘={x:xB} and μ is a size parameter.34 The number of elements within a structuring element of size μ is (2μ+1)(2μ+1). From Eq. (9), notice that the morphological erosion or dilation act depending on the values taken from the gradients.

However, the Kramer and Bruckner transformation produces oscillations when it is iterated. As a result, the output image is degraded. In order to avoid this inconvenience, Serra35,36 propounded the use of idempotent primitives, in specific the morphological opening and closing. On the other hand, in Refs. 1 and 2, the primitives erosion and dilation are used in a separate way, generating a new family of contrast mappings which is presented in Definition 7. This contrast transformation is the basis of the MSFs.

Graphic Jump LocationF4 :

Example of morphological slope filters (MSFs) and morphological connected slope filters (MCSFs). (a) Original image; (b) ξϕ=10ε(f)(x); (c) ξϕ=10ε(f,Pf)(x); and (d, e) internal gradients of the images presented in (b) and (c).

In this section new weighting functions using a displacement parameter are introduced. This class of weighting functions has the purpose of modifying the gradient criterion when the displacement parameter changes. An important characteristic of these transformations is that the contrast is improved without varying the slope ϕ.

Weighting Function
Table Grahic Jump Location
Table 1Weighting functions.
Graphic Jump LocationF5 :

(a) Original image (f,Pf) and (b) weighting function χd=150(t)=log10(t150)/log10255 with t=(f,Pf)(x).

The set Sϕ,d represents the support of the points of high contrast, while the set Sϕ,dc represents the support of the points of weak contrast. Figure 6 illustrates the performance of MCSFs using the logarithm function given in Table 1 with ϕ=6. Figure 6(a) shows the original image. The MSFs at pixel level can be seen in Fig. 6(b), whereas Fig. 6(c) displays MCSFs defined on the partition Pf induced by f. In Fig. 6(d), WMSFs at pixel level are computed taking d=0, while in Fig. 6(e) the same transformation is applied, but now considering d=120. In Fig. 6(f) and 6(g), WMCSFs on the partition Pf induced by f with d=0 and d=120 are shown.

Graphic Jump LocationF6 :

Weighted morphological connected slope filters (WMCSFs) using the logarithm function. (a) Original image; (b) ξϕ=6ε(f)(x); (c) ξϕ=6ε(f,Pf)(x); (d) ξϕ=6,log10(t0)/log10(255)ε(f)(x); (e) ξϕ=6,log10(t120)/log10(255)ε(f)(x); (f) ξϕ=6,log10(t0)/log10(255)ε(f,Pf)(x); and (g) ξϕ=6,log10(t120)/log10(255)ε(f,Pf)(x).

The main advantages of WMCSFs with respect to the other MSFs proposed so far are the following:

  1. Only a part of the internal gradient support, which depends on the displacement parameter d, is analyzed.
  2. If the gradient criterion is weighted by an increasing function as those presented in Table 1, the result will be stretching or compressing of the intensity levels of the image contours.
  3. The parameter d allows the selection of the intensity level from which the contrast will be improved.

In order to illustrate the weighted gradient support and the contrast enhancement produced by WMCSFs, the images in Fig. 7 are given. The original image can be seen in Fig. 7(a). The supports Sϕ,d of the regions associated with high contrast for ϕ=10 with d=60 and 120 are presented in Fig. 7(b) and 7(c). The output images processed by WMCSFs are displayed in Fig. 7(d) and 7(e) using the logarithm function defined in Table 1 for ϕ=10 with d=60 and 120, respectively.

Graphic Jump LocationF7 :

Support of WMCSFs using the logarithm function. (a) Original image; (b) Sϕ,d (1/log255log(t60)gradi(f,Pf)(x)10); (c) Sϕ,d (1/log255log(t120)gradi(f,Pf)(x)10); (d) ξϕ=10,log10(t60)/log10(255)ε(f,Pf)(x); and (e) ξϕ=10,log10(t120)/log10(255)ε(f,Pf)(x).

To enhance the regions in certain intervals of intensities d to d1 with dd1, the weighting function must be applied in the following way: Display Formula

χd(t)[u(td)u(td1)],(28)
where u(t) represents the unitary step. In Fig. 8, Eq. (28) is illustrated. The original image corresponds to Fig. 8(a). The output image in Fig. 8(b) is obtained by applying the WMCSF with Display Formula
χd(t)[u(td)u(td1)]=log10(t121)log10(255)[u(t121)u(t203)],
t=(f,Pf)(x), and ϕ=2. Uniquely, the regions within the original image with intensity levels between 121 to 203 are enhanced.

Graphic Jump LocationF8 :

Contrast enhancement in a certain interval by applying WMCSFs using the logarithm function. (a) Original image and (b) ξϕ=2,log10(t121)/log10(255)[u(t121)u(t203)]ε(f,Pf)(x).

Experiment to Detect the Best Weighting Function

Section 3.1 introduced three weighting functions: logarithm, exponential, and cubic root. To know which of them produces the best weighting is not an easy task; however, an experiment can be done to detect the weighting function which preserves more contours, and with this criterion decided, one can ascertain which function to apply. The experiment consists of the following: (1) improve the contrast of the image, and segment it using the transformation ξϕ,χdε(f,Pf)(x) by considering some values for ϕ and d respectively and the three weighting functions; (2) posteriorly, the internal gradient size μ=1 at pixel level, gradiμf(x)=f(x)εμ(f)(x), is computed from the original and the processed images; (3) the morphological index to compare contours at pixel level between the original and output images is applied. For this, the morphological contour preservation index denoted as MCPI is used. This index allows one to obtain a measure to evaluate the quality of the contours of the output image with respect to those of the original image. For obtaining MCPI index, it is necessary to compute the edge preservation parameter (EPP). The EPP is obtained from the convolution (denoted as *) of two kernels of size 3×3 with the morphological internal gradient. These kernels allow one to detect horizontal, Gh, and vertical, Gv, changes at each point in the internal gradient. EPP is expressed as follows: Display Formula

EPP=xDgradi(f)Gh(x)+xDgradi(f)Gv(x)(29)
Display Formula
withGh=[111000111]*gradiμ(f)andGv=[101101101]*gradiμ(f),
where Dgradi(f) represents the definition domain of the internal gradient. The MCPI is obtained as follows: Display Formula
MCPI=EPPprocessedEPPoriginal

EPPprocessed and EPPoriginal correspond to the processed and original images, respectively. Figure 9 (37) illustrates this procedure. The original image is presented in Fig. 9(a). The parameters ϕ=10 and d=50 are considered, and the output images applying the transformation ξϕ,χdε(f,Pf)(x) using the three weighting functions are displayed in Fig. 9(b)9(d). The computed MCPI values for images in Fig. 9(a)9(d) are presented in the table located in Fig. 9(e), and its corresponding graph is presented in Fig. 9(f). The best MCPI value must be nearest to 1. The image in Fig. 9(c) fulfills this condition, and corresponds to the image enhanced with the exponential function. Notice that Fig. 9(c) has less information than the other output images, but its edges are better preserved.

Graphic Jump LocationF9 :

Weighting functions comparison. (a) Original image37; (b) ξϕ=10,log10(t50)/log10(255)ε(f,Pf)(x); (c) ξϕ=10,1/1+9exp0.02663(t50)ε(f,Pf)(x); (d) ξϕ1=10,t503/2553ε(f,Pf)(x); (e) MCPI values; and (f) Graph corresponding to the MCPI values.

WMCSFs Considering d Fixed and ϕ Variable

Let us consider d fixed and ϕ1<ϕ2: Display Formula

ξϕ2,χdε(f,Pf)(x)ξϕ1,ϕ2,χdε(f,Pf)(x)ξϕ1,χdε(f,Pf)(x)
with Display Formula
ξϕ1,ϕ2,χdε(f,Pf)(x)=ξϕ2,χdε[ξϕ1,χdε(f,Pf)(x)].

Notice that Display Formula

ξϕ2,ϕ1,χdε(f,Pf)(x)=ξϕ1,χdε[ξϕ2,χdε(f,Pf)(x)]=ξϕ2,χdε(f,Pf)(x).

For a family of parameters ϕi with i{1,2,,m} such that ϕiϕk for i<k and taking d fixed, Display Formula

ξϕm,χdε(f,Pf)(x)ξϕ1,,ϕm,χdε(f,Pf)(x)ξϕ1,χdε(f,Pf)(x).

On the other hand, if ξϕ1,,ϕm,χdε(f,Pf)(x) contains some high-contrast regions for the levels ϕ1, ϕ2ϕm, the family ξϕ2,,ϕm,χdε(f,Pf)(x) does not contain details of ξϕ1,χdε(f,Pf)(x). This situation is expressed as follows: Display Formula

ξϕ1,,ϕm,χdε(f,Pf)(x)ξϕ2,,ϕm,χdε(f,Pf)(x)ξϕm1,ϕm,χdε(f,Pf)(x)ξϕm,χdε(f,Pf)(x).(30)

To illustrate the behavior of the WMCSFs by varying the parameter ϕ and keeping the displacement parameter d unchanged, an example is presented in Fig. 10. The original image is displayed in Fig. 10(a). The weighting function χd applied in this example is χd=td3/2553 with t=(f,Pf)(x) and d=18 (see Table 1). In Fig. 10(b), the transformation ξϕ1=1,ϕ2=2,,ϕ10=10,t183/2553εn(f,Pf)(x) is applied. The operator ξϕ1=2,ϕ2=4,,ϕ5=10t183/2553εn(f,Pf)(x) is presented in Fig. 10(c), and ξϕ1=10,t183/2553εn(f,Pf)(x) can be seen in Fig. 10(d). Notice that this set of images fulfills the inclusion relation given in Eq. (30).

Graphic Jump LocationF10 :

WMCSFs using the cubic root function. (a) Original image; (b) ξϕ1=1,ϕ2=2,,ϕ10=10,εt183/2553(f,Pf)(x); (c) ξϕ1=2,ϕ2=4,,ϕ5=10t183/2553ε(f,Pf)(x); and (d) ξϕ1=10,t183/2553ε(f,Pf)(x).

Modification of Maxima and Minima

The weighting of the gradient criterion allows the modification of maxima and minima when WMCSFs are applied. A regional maximum M of a grayscale image f is a connected component of pixels with a given value h, and the plateau is at altitude h, such that every pixel in the neighborhood of M has strictly a lower value, whereas a regional minimum is a plateau of uniform altitude without neighboring pixels of lower altitude.23 When the filter ξϕmε(f,Pf)(x) is applied, the regional minima are modified. The ξϕm,χdε(f,Pf)(x) also allows the modification of the regional minima, because the increase of the displacement parameter d produces a decrement of the regional minima. This occurs because when the parameter d is increased, more regions will be merged with the background image (see Definition 8). Given that the support Sϕ,d defines the flat zones with high contrast and taking Eq. (21) in consideration, the next relation can be derived: Display Formula

  d1,d2Z,such thatd1<d2Sϕ,d1{gradi[ξϕ,xd1ε(f,Pf)(x)]}Sϕ,d2{gradi[ξϕ,xd2ε(f,Pf)(x)]}.(31)

Equation (31) is interpreted as follows: the control on the gradient support in the WMCSFs means not only improves the contrast, but also allows the separation of regions with intensities greater than the parameter d. This effect is illustrated in Fig. 11. The exponential function presented in Table 1 involves three parameters that can take the following values A=1, B=9, and K2=0.02663. The computation of these constants is carried out considering that the function χd=0(t)=A/1+Bexpk2t is normalized by the maximum intensity value of the processed image, i.e., for t=255, A/1+Bexpk2t=1 with t=(f,Pf)(x). The original image is presented in Fig. 11(a). Images in Fig. 11(b)11(g) correspond to the support of the internal gradient of the transformation ξϕ=10,1/1+9exp0.02663(tdi)ε(f,Pf)(x) with d1=0<d2=40<d3=80<d4=120<d5=160<d6=200. Notice that the internal gradient support decreases as the displacement parameter increases in such a way that the regional minima in the processed images decrease until they reach the image with zero values when d255.

Graphic Jump LocationF11 :

Support of the WMCSFs using the exponential function. (a) Original image; (b) Sϕ,d1[gradi(ξϕ=10,1/1+9exp0.02663(t0)ε(f,Pf)(x))]; (c) Sϕ,d2[gradi(ξϕ=10,1/1+9exp0.02663(t40)ε(f,Pf)(x))]; (d) Sϕ,d3[gradi(ξϕ=10,1/1+9exp0.02663(t80)ε(f,Pf)(x))]; (e) Sϕ,d4[gradi(ξϕ=10,1/1+9exp0.02663(t120)ε(f,Pf)(x))]; (f) Sϕ,d5[gradi(ξϕ=10,1/1+9exp0.02663(t160)ε(f,Pf)(x))]; and (g) Sϕ,d6[gradi(ξϕ=10,1/1+9exp0.02663(t200)ε(f,Pf)(x))].

Hierarchical Segmentation

When the parameter d in the WMCSFs is increased, according to Eq. (31), fewer regions are preserved. This fact indicates that a family of descriptors is originated as the displacement parameter changes. From a hierarchical point of view, some properties must be fulfilled:38 (a) the transformation must allow a real simplification of the processed image, i.e., some information must be lost from one scale to the following, (b) no new structures must be created, i.e., the transformations must not create new maxima and minima, and (c) causality, coarser scales can be originated by what happens on finer scales.

In particular, WMCSFs verify the requirements (a), (b), and (c). The following inclusion relations among the supports of the internal gradient express the characteristics of a hierarchical process.

  1. When the displacement parameter d is fixed and the parameter ϕ is increased: Display Formula
    Sϕ0,d{gradi[ξϕ0,χdε(f,Pf)(x)]}Sϕm1,d{gradi[ξϕm1,χdε(f,Pf)(x)]}Sϕm,d{gradi[ξϕm,χdε(f,Pf))(x]}.
  2. When the displacement parameter d is increased and ϕ is unchanged: Display Formula
    Sϕ,d0{gradi[ξϕ,χd0ε(f,Pf)](x)}Sϕ,dm1{gradi[ξϕ,χdm1ε(f,Pf)](x)}Sϕ,dm1{gradi[ξϕ,χdm1ε(f,Pf)](x)}.
    Other ways of simplification can be found in the luminance. The luminance in a coarse scale is always smaller than in a fine scale and when the filter ξϕ,χdε(f,Pf)(x) is applied, it does not matter whether the parameter ϕ or d varies.
  3. When the displacement parameter d is fixed and ϕ is increased: Display Formula
    ξϕ0,χdε(f,Pf)(x)ξϕ1,χdε(f,Pf)(x)ξϕm,χdε(f,Pf)(x)
  4. When the displacement parameter d is increased and ϕ is unchanged: Display Formula
    ξϕ,χd0ε(f,Pf)(x)ξϕ,χd1ε(f,Pf)(x)ξϕ,χdmε(f,Pf)(x)

WMCSFs do not create new maxima or minima given that these transformations are connected. On the other hand, the coarser scales produced by WMCSFs are obtained from the sequential process. An example to illustrate the hierarchical segmentation is shown in Fig. 12. The original image is located in the Fig. 12(a). In Fig. 12(b)12(g), the WMCSFs using χd=1/1+9exp0.02663(tdi) with di=0,40,80,120,160,200 and considering ϕ=10 are shown. In this example, the segmentation is controlled by parameter d.

Graphic Jump LocationF12 :

Hierarchical segmentation by applying WMCSFs using the exponential function. (a) Original image; (b) ξϕ=10,1/1+9exp0.02663(t0)ε(f,Pf)(x); (c) ξϕ=10,1/1+9exp0.02663(t40)ε(f,Pf)(x); (d) ξϕ=10,1/1+9exp0.02663(t80)ε(f,Pf)(x); (e) ξϕ=10,1/1+9exp0.02663(t120)ε(f,Pf)(x); (f) ξϕ=10,1/1+9exp0.02663(t160)ε(f,Pf)(x); and (g) ξϕ=10,1/1+9exp0.02663(t200)ε(f,Pf)(x).

Other morphological transformations have been proposed in the current literature to produce hierarchical segmentations, for example, watershed26,27 and the contrast mappings based on connected top-hats15 to mention a few. Figure 13 illustrates the segmentation produced by the WMCSFs and those produced by the multiscale transformations aforementioned. The original image appears in Fig. 13(a). In Fig. 13(b), the segmented images by the WMCSFs are displayed. The transformation ξϕ=2,([u(td)u(td1)]/1+9exp0.02663(td))ε(f,Pf)(x) considering three different intervals, [d=1,d1=80],[d=81,d1=160], and [d=161,d1=255], is applied. In Fig. 13(c), the result of the watershed is overlapped to the original image. The control on the minima is made by applying the alternate filter opening–closing by reconstruction γ˜μφ˜μ(f)(x)28,39 with μ=0,1,5. Opening and closing by reconstructions merge the maxima and minima, respectively. Whereas in Fig. 13(d), the morphological connected contrast mappings based on top-hat criterion are presented. The parameter used for this set of images are one scale with μ=16 (left), one scale with μ=32 (center), and two scales with μ=16,32 (right). The hierarchical segmentations presented in Fig. 13 show different performances. WMCSFs segment from an intensity level imposed by the displacement parameter, while the watershed uses the minima of the image, and the contrast mapping based on top-hat criterion operates considering the size of the structures contained within the image.

Graphic Jump LocationF13 :

Hierarchical segmentations. (a) Original image;37 (b) WMCSFs maintaining ϕ and varying the displacement parameters d and d1; (c) watershed obtained by controlling the minima; and (d) contrast mappings based on connected top-hat at different scales.

Also, the WMCSF using the exponential function is compared quantitatively with respect to the retinex multiscale method40 implemented in 16, and the multiscale method to enhance edges using wavelets reported in 11. The original image is exhibited in Fig. 14(a). Output images produced by the retinex method can be seen in Fig. 14(b)14(e). The images obtained from our proposal are presented in Fig. 14(f)14(i), while the images obtained from the wavelet transform are displayed in Fig. 14(j)14(m). In order to detect the method presenting better contour preservation, the MCPI is computed. The values and the graph of the MCPI are displayed in Fig. 14(n) and 14(o). According to the graph in Fig. 14(o), the three images presenting a better edge preservation are the images in Fig. 14(f), 14(h), and 14(i) (which are obtained with our proposal) because their MCPI values are nearest to 1. Notice that in the output images generated by retinex method, the changes for each scale given in terms of the iteration number cannot be perceived, whereas the output images obtained from the methodology introduced in 11 produces positive and negative values for the MCPI.

Graphic Jump LocationF14 :

Comparison of multiscale methods. (a) Original image; (b)–(e) Retinex with iteration 2,4,6, and 8; (f)–(i) ξϕ1=1,ϕ2=2,,ϕ20=20,[u(td)u(t255)]/1+9exp0.02663(td)ε(f,Pf)(x) with d=25, 50, 75, and 200; (j) wavelet transformation using the parameters N=3 with β1=1, β2=3.5, and β3=2; (k) wavelet transformation using the parameters N=3 with β1=1, β2=4.5, and β3=2; (l) wavelet transformation using the parameters N=3 with β1=1, β2=6, and β3=2; (m) wavelet transformation using the parameters N=3 and β1=1, β2=8, and β3=2; (n) MCPI values for the images in (a)–(m); and (o) graph of the MCPI values presented in (n).

Application Example of WMCSFs

The application presented in this section consists of detecting and segmenting the white matter in axial sections of the brain. The analyzed sections belong to the BMRI T1 bank of the Institute of Neurobiology, UNAM, Campus Juriquilla, Querétaro, México. In Fig. 15(a), sections of different brains are presented. The proposed algorithm to segment white matter is divided into the following steps, which are illustrated in Fig. 15(c)15(e). The original image is the picture in Fig. 15(b), and the weighting exponential function is selected due to the experiment provided in Sec. 3.2.

  1. The transformation ξϕ=2,4,6,8,10,χd=100δ(f,Pf)(x) is applied [see Fig. 15(c)]. This operator improves the contrast of white regions. The parameter d is selected with the value of 100, because the white matter begins to appear in a thresholding process between intensity levels 80 to 255. However, given that there are other different components to the white matter in this interval, the parameter d=100 is selected. The ϕ parameter takes the values 2, 4, 6, 8, and 10. Notice from Eq. (30) that intermediate results are obtained during the enhancement process. In addition, weak intensity slopes are reinforced.
  2. A thresholding between sections 120 and 255 is carried out on the image in step (1) [see Fig. 15(d)].
  3. A mask is placed between the binary image obtained in step(2) and the original one to allow the separation of the white matter [see Fig. 15(e)].

Graphic Jump LocationF15 :

White matter segmentation by applying WMCSFs using the exponential function. (a) Original images; (b) original image used to illustrate the proposed algorithm; (c) ξϕ=2,4,8,10,χd=100δ(f,Pf)(x) is applied to image in (b); (d) threshold of the image in (c); (e) mask between the images in (b) and (d); (f) output images where the white matter has been segmented by applying the proposed algorithm; and (g) white matter separated directly by applying a threshold.

The set of images in which the white matter has been separated from the other brain tissues are displayed in Fig. 15(f). The white matter in this example is separated with a thresholding, because it has been merged as a connected component during the segmentation process. If the white matter is separated solely by thresholding, the results are not good. An example of this situation is presented in Fig. 15(g). The disadvantage of the algorithm proposed above is that the threshold parameters are applied empirically.

The WMCSFs introduced in this paper use weighting functions and the displacement parameter for improving the contrast and segmenting regions of interest. Three weighting functions were proposed that must fulfill the criterion of being increased. The displacement parameter d allows that the gradient criterion works from a specific intensity level. This concept was illustrated with an application example in which the white matter was separated from the rest of a brain section, BMRI T1. For this practical example, the exponential function was selected due to the experiment done for selecting the best weighting function. On the other hand, WMCSFs inherit the properties of sequential MCSFs; however, the new family of WMCSFs exhibits a superior performance since the output images processed with these transformations are not only enhanced, but also the segmentation process can be controlled through the displacement parameter originating a multiscale segmentation at certain intensity interval. Additional to this, no new structures were created during the processing because they are connected transformations. According to the examples presented, WMCSFs using the logarithm, exponential, and cubic root functions, introduced herein, present a better performance than WMCSFs given in 3, where the weighting function was the input image.

Terol-Villalobos  I. R., “Morphological slope filters,” Proc. SPIE. 2588, , 712 –722 (1995). 0277-786X CrossRef
Terol-Villalobos  I. R., “Nonincreasing filters using morphological gradient criteria,” Opt. Eng.. 35, (11 ), 3172 –3182 (1996). 0091-3286 CrossRef
Terol-Villalobos  I. R., “Morphological image enhancement and segmentation,” Chapter 4, in Advances in Imaging and Electron Physics. , Hawkes  P. W., Ed., Vol. 118, pp. 207 –273,  Elsevier  (2001).
Terol-Villalobos  I. R., Cruz-Mandujano  A. J., “Contrast enhancement and image segmentation using a class of morphological nonincreasing filters,” J. Electron. Imag.. 7, (3 ), 641 –654 (1998). 1017-9909 CrossRef
Chen  Z. et al., “Gray-level grouping (GLG): an automatic method for optimized image contrast enhancement-part I: the basic method,” IEEE Trans. Image Process.. 15, (8 ), 2290 –2302 (2006). 1057-7149 CrossRef
Jin  Y., Fayad  L. M., Laine  A. F., “Contrast enhancement by multiscale adaptive histogram equalization,” Proc. SPIE. 4478, , 206 –213 (2001). 0277-786X CrossRef
Panetta  K. A., Wharton  E. J., Agaian  S. S., “Human visual system-based image enhancement and logarithmic contrast measure,” IEEE Trans. Syst. Man Cybern. B Cybern.. 38, (1 ), 174 –188 (2008). 1083-4419 CrossRef
Jian  L. et al., “Contrast enhancement of medical images using multiscale edge representation,” Proc. SPIE. 2242, , 711 –719 (1994). 0277-786X CrossRef
Keeling  S. L., Stollberger  R., “Nonlinear anisotropic diffusion filtering for multiscale edge enhancement,” Inverse Probl.. 18, (1 ), 175 –190 (2002). 0266-5611 CrossRef
Numan  U., Asari  V. K., Zia-ur  R., “Fast and robust wavelet-based dynamic range compression with local contrast enhancement,” Proc. SPIE. 6978, , 697805  (2008). 0277-786X CrossRef
Rosito-Jung  C., Scharcanski  J., “Adaptive image denoising and edge enhancement in scale-space using the wavelet transform,” Pattern Recogn. Lett.. 24, (7 ), 965 –971 (2003). 0167-8655 CrossRef
Saad  A., “Visual enhancement of digital ultrasound images: wavelet versus Gauss-Laplace contrast pyramid,” Int. J. Comput. Assist. Radiol. Surg.. 2, (2 ), 117 –125 (2007). 1861-6410 CrossRef
Tang  J., Sun  Q., Agyepong  K., “An image enhancement algorithm based on a contrast measure in the wavelet domain for screening mammograms,” in  IEEE Int. Conf. Image Processing, 2007, ICIP 2007 ,  San Antonio, TX , Vol. 5, pp. V–29 –V–32 (2007).
Mukhopadhaya  S., Chanda  B., “A multiscale morphological approach to local contrast enhancement,” Signal Process.. 80, (4 ), 685 –696 (2000). 0165-1684 CrossRef
Terol-Villalobos  I. R., “Morphological connected contrast mappings based on top-hat criteria: a multiscale contrast approach,” Opt. Eng.. 43, (7 ), 1577 –1595 (2004). 0091-3286 CrossRef
Funt  B. V., Ciurea  F., McCann  J. J., “Retinex in matlab,” in  Proceedings of the IS&T/SID Eighth Color Imaging Conference: Color Science and Engineering Systems, Technologies, and Applications ,  Scottsdale, Arizona , pp. 112 –121 (2000).
Fattal  R., Agrawala  M., Rusinkiewicz  S., “Multiscale shape and detail enhancement from multi-light image collections,” ACM Trans. Graph.. 26, (3 ) (2007). 0730-0301 CrossRef
Vachier  C., Meyer  F., “The viscous watershed transform,” J. Math. Imag. Vis.. 22, (2–3 ), 251 –267 (2005). 0924-9907 CrossRef
Santillán  I. et al., “Morphological connected filtering on viscous lattices,” J. Math. Imag. Vis.. 36, (3 ), 254 –269 (2010). 0924-9907 CrossRef
Meyer  F., “From connected operators to levelings,” in Proceedings of the fourth international symposium on Mathematical morphology and its applications to image and signal processing (ISMM '98). , Heijmans  H. J. A. M., Roerdink  J. B. T. M., Eds., pp. 191 –198,  Kluwer Academic Publishers ,  Norwell, MA  (1998).
Meyer  F., “The levelings,” Proceedings of the fourth international symposium on Mathematical morphology and its applications to image and signal processing (ISMM '98). , Heijmans  H. J. A. M., Roerdink  J. B. T. M., Eds., pp. 199 –206,  Kluwer Academic Publishers ,  Norwell, MA  (1998).
Salembier  P., Serra  J., “Flat zones filtering, connected operators and filters by reconstruction,” IEEE Trans. Image Process.. 4, (8 ), 1153 –1160 (1995). 1057-7149 CrossRef
Vincent  L., “Morphological grayscale reconstruction in image analysis: applications and efficient algorithms,” IEEE Trans. Image Process.. 2, (2 ), 176 –201 (1993). 1057-7149 CrossRef
Meyer  F., Serra  J., “Contrast and activity lattice,” Signal Process.. 16, (4 ), 303 –317 (1989). 0165-1684 CrossRef
Jain  A. K., Fundamentals of Digital Image Processing. ,  Prentice Hall ,  Englewood Cliffs, NJ  (1989).
Beucher  S., “Segmentation D'Images Et Morphologie Mathematique Centre de Morphologie Mathématique,” Ph.D. Thesis, ENSMP (1990).
Vincent  L., Soille  P., “Watershed in digital spaces: an efficient algorithm based on immersion simulations,” IEEE Trans. Pattern Anal. Machine Intell.. 13, (6 ), 583 –598 (1991). 0162-8828 CrossRef
Serra  J., Image Analysis and Mathematical Morphology, Chapter 2 in Mathematical Morphology for Boolean Lattices. , Serra  J., Ed., Vol. II, pp. 51 –57,  Academic Press ,  New York  (1988).
Mendiola-Santibáñez  J. D. et al., “Morphological contrast measure and contrast enhancement: one application to the segmentation of brain MRI,” Signal Process.. 87, (9 ), 2125 –2150 (2007). 0165-1684 CrossRef
Crespo  J. et al., “The flat zone approach: a general low-level region merging segmentation method,” Signal Process.. 62, (1 ), 37 –60 (1997). 0165-1684 CrossRef
Crespo  J., Serra  J., Schafer  R. W., “Theoretical aspects of morphological filters by reconstruction,” Signal Process.. 47, (2 ), 201 –225 (1995). 0165-1684 CrossRef
Rivest  J. F., Soille  P., Beucher  S., “Morphological gradient,” J. Electron Imag.. 2, (4 ), 326 –336 (1993). 1017-9909 CrossRef
Kramer  H. P., Bruckner  J. B., “Iterations of a non-linear transformation for enhancement of digital images,” Pattern Recogn.. 7, (1–2 ), 53 –58 (1975). 0031-3203 CrossRef
Serra  J., Mathematical Morphology. , Vol. I,  Academic Press ,  London  (1982).
Serra  J., “Toggle mappings,” Technical Report N-18/88/MM, Centre de Morphologie Mathematique ENSMP, Fontainebleau France (1988).
Serra  J., “Toggle mappings,” in From Pixels to Features (Proc. COST 13, Bonas, 1988). , Simon  J. C., Ed., pp. 61 –72,  Elsevier ,  North Holland, Amsterdam  (1989).
Martin  D. et al., “A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics,” in  Computer Vision, 2001. ICCV 2001. Proceedings. Eighth IEEE International Conference on ,  Vancouver, BC , Vol. 2, pp. 416 –423 (2001).
Meyer  F., Maragos  P., “Nonlinear scale-space representation with morphological levelings,” J. Vis. Commun. Image Represent.. 11, (2 ), 245 –265 (2000). 1047-3203 CrossRef
Serra  J., Vincent  L., “An overview of morphological filtering,” Circ. Syst. Signal Process.. 11, (1 ), 47 –108 (1992). 0278-081X CrossRef
Frankle  J., McCann  J., “Method and apparatus for lightness imaging,” US Patent #4,384,336 (17  May 1983).

Grahic Jump LocationImage not available.

Jorge D. Mendiola-Santibañez received his BS degree in electronic engineering from the Benemérita Universidad Autónoma de Puebla, México, and MS degree in electronics from INAOE, México. He received his PhD degree from the Universidad Autónoma de Querétaro (UAQ), México. He is currently a professor/researcher at the Universidad Autonoma de Querétaro. His research interests include morphological image processing and computer vision.

Grahic Jump LocationImage not available.

Iván R. Terol-Villalobos received his BSc degree from Instituto Politécnico Nacional (México), and MSc degree in electrical engineering from CINVESTAV (México). He received his PhD degree from the Centre de Morphologie Mathématique, Ecole des Mines de Paris (France). He is currently a researcher for CIDETEQ (Querétaro, México). Presently, his main current research interests include morphological image processing and computer vision.

© The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.

Citation

Jorge D. Mendiola-Santibañez and Iván R. Terol-Villalobos
"Image enhancement and segmentation using weighted morphological connected slope filters", J. Electron. Imaging. 22(2), 023022 (Jun 10, 2013). ; http://dx.doi.org/10.1117/1.JEI.22.2.023022


Figures

Graphic Jump LocationF1 :

Adjacent flat zones. (a) Original partition with five flat zones; (b) flat zone in the point x, Fx(f); (c) flat zone in the point y, Fy(f); (d) two adjacent flat zones, i.e., Fx(f)Fy(f)=γx(Fx(f)Fy(f)); and (e) adjacent flat zones to Fv(f).

Graphic Jump LocationF2 :

Erosion and dilation on the partition. (a) Original image with 16 flat zones; (b) contours of the partition generated by the flat zones of the image in (a); (c) erosion on the partition of the image in (a); (d) dilation on the partition of the image in (a); and (e) Left image: flat zone with gray level of 255 and its four adjacent flat zones. Right image: gray level value of the erosion.

Graphic Jump LocationF3 :

Erosion on the partition size μ=2. (a) Original image (top) and its partition (bottom); (b) erosion on the partition size μ=1 (top) of image in (a) and its partition (bottom); and (c) erosion on the partition size μ=1 (top) of the image in (b), i.e., εμ=2(f,Pf) and its partition (bottom).

Graphic Jump LocationF4 :

Example of morphological slope filters (MSFs) and morphological connected slope filters (MCSFs). (a) Original image; (b) ξϕ=10ε(f)(x); (c) ξϕ=10ε(f,Pf)(x); and (d, e) internal gradients of the images presented in (b) and (c).

Graphic Jump LocationF5 :

(a) Original image (f,Pf) and (b) weighting function χd=150(t)=log10(t150)/log10255 with t=(f,Pf)(x).

Graphic Jump LocationF6 :

Weighted morphological connected slope filters (WMCSFs) using the logarithm function. (a) Original image; (b) ξϕ=6ε(f)(x); (c) ξϕ=6ε(f,Pf)(x); (d) ξϕ=6,log10(t0)/log10(255)ε(f)(x); (e) ξϕ=6,log10(t120)/log10(255)ε(f)(x); (f) ξϕ=6,log10(t0)/log10(255)ε(f,Pf)(x); and (g) ξϕ=6,log10(t120)/log10(255)ε(f,Pf)(x).

Graphic Jump LocationF7 :

Support of WMCSFs using the logarithm function. (a) Original image; (b) Sϕ,d (1/log255log(t60)gradi(f,Pf)(x)10); (c) Sϕ,d (1/log255log(t120)gradi(f,Pf)(x)10); (d) ξϕ=10,log10(t60)/log10(255)ε(f,Pf)(x); and (e) ξϕ=10,log10(t120)/log10(255)ε(f,Pf)(x).

Graphic Jump LocationF8 :

Contrast enhancement in a certain interval by applying WMCSFs using the logarithm function. (a) Original image and (b) ξϕ=2,log10(t121)/log10(255)[u(t121)u(t203)]ε(f,Pf)(x).

Graphic Jump LocationF9 :

Weighting functions comparison. (a) Original image37; (b) ξϕ=10,log10(t50)/log10(255)ε(f,Pf)(x); (c) ξϕ=10,1/1+9exp0.02663(t50)ε(f,Pf)(x); (d) ξϕ1=10,t503/2553ε(f,Pf)(x); (e) MCPI values; and (f) Graph corresponding to the MCPI values.

Graphic Jump LocationF10 :

WMCSFs using the cubic root function. (a) Original image; (b) ξϕ1=1,ϕ2=2,,ϕ10=10,εt183/2553(f,Pf)(x); (c) ξϕ1=2,ϕ2=4,,ϕ5=10t183/2553ε(f,Pf)(x); and (d) ξϕ1=10,t183/2553ε(f,Pf)(x).

Graphic Jump LocationF11 :

Support of the WMCSFs using the exponential function. (a) Original image; (b) Sϕ,d1[gradi(ξϕ=10,1/1+9exp0.02663(t0)ε(f,Pf)(x))]; (c) Sϕ,d2[gradi(ξϕ=10,1/1+9exp0.02663(t40)ε(f,Pf)(x))]; (d) Sϕ,d3[gradi(ξϕ=10,1/1+9exp0.02663(t80)ε(f,Pf)(x))]; (e) Sϕ,d4[gradi(ξϕ=10,1/1+9exp0.02663(t120)ε(f,Pf)(x))]; (f) Sϕ,d5[gradi(ξϕ=10,1/1+9exp0.02663(t160)ε(f,Pf)(x))]; and (g) Sϕ,d6[gradi(ξϕ=10,1/1+9exp0.02663(t200)ε(f,Pf)(x))].

Graphic Jump LocationF12 :

Hierarchical segmentation by applying WMCSFs using the exponential function. (a) Original image; (b) ξϕ=10,1/1+9exp0.02663(t0)ε(f,Pf)(x); (c) ξϕ=10,1/1+9exp0.02663(t40)ε(f,Pf)(x); (d) ξϕ=10,1/1+9exp0.02663(t80)ε(f,Pf)(x); (e) ξϕ=10,1/1+9exp0.02663(t120)ε(f,Pf)(x); (f) ξϕ=10,1/1+9exp0.02663(t160)ε(f,Pf)(x); and (g) ξϕ=10,1/1+9exp0.02663(t200)ε(f,Pf)(x).

Graphic Jump LocationF13 :

Hierarchical segmentations. (a) Original image;37 (b) WMCSFs maintaining ϕ and varying the displacement parameters d and d1; (c) watershed obtained by controlling the minima; and (d) contrast mappings based on connected top-hat at different scales.

Graphic Jump LocationF14 :

Comparison of multiscale methods. (a) Original image; (b)–(e) Retinex with iteration 2,4,6, and 8; (f)–(i) ξϕ1=1,ϕ2=2,,ϕ20=20,[u(td)u(t255)]/1+9exp0.02663(td)ε(f,Pf)(x) with d=25, 50, 75, and 200; (j) wavelet transformation using the parameters N=3 with β1=1, β2=3.5, and β3=2; (k) wavelet transformation using the parameters N=3 with β1=1, β2=4.5, and β3=2; (l) wavelet transformation using the parameters N=3 with β1=1, β2=6, and β3=2; (m) wavelet transformation using the parameters N=3 and β1=1, β2=8, and β3=2; (n) MCPI values for the images in (a)–(m); and (o) graph of the MCPI values presented in (n).

Graphic Jump LocationF15 :

White matter segmentation by applying WMCSFs using the exponential function. (a) Original images; (b) original image used to illustrate the proposed algorithm; (c) ξϕ=2,4,8,10,χd=100δ(f,Pf)(x) is applied to image in (b); (d) threshold of the image in (c); (e) mask between the images in (b) and (d); (f) output images where the white matter has been segmented by applying the proposed algorithm; and (g) white matter separated directly by applying a threshold.

Tables

Table Grahic Jump Location
Table 1Weighting functions.

References

Terol-Villalobos  I. R., “Morphological slope filters,” Proc. SPIE. 2588, , 712 –722 (1995). 0277-786X CrossRef
Terol-Villalobos  I. R., “Nonincreasing filters using morphological gradient criteria,” Opt. Eng.. 35, (11 ), 3172 –3182 (1996). 0091-3286 CrossRef
Terol-Villalobos  I. R., “Morphological image enhancement and segmentation,” Chapter 4, in Advances in Imaging and Electron Physics. , Hawkes  P. W., Ed., Vol. 118, pp. 207 –273,  Elsevier  (2001).
Terol-Villalobos  I. R., Cruz-Mandujano  A. J., “Contrast enhancement and image segmentation using a class of morphological nonincreasing filters,” J. Electron. Imag.. 7, (3 ), 641 –654 (1998). 1017-9909 CrossRef
Chen  Z. et al., “Gray-level grouping (GLG): an automatic method for optimized image contrast enhancement-part I: the basic method,” IEEE Trans. Image Process.. 15, (8 ), 2290 –2302 (2006). 1057-7149 CrossRef
Jin  Y., Fayad  L. M., Laine  A. F., “Contrast enhancement by multiscale adaptive histogram equalization,” Proc. SPIE. 4478, , 206 –213 (2001). 0277-786X CrossRef
Panetta  K. A., Wharton  E. J., Agaian  S. S., “Human visual system-based image enhancement and logarithmic contrast measure,” IEEE Trans. Syst. Man Cybern. B Cybern.. 38, (1 ), 174 –188 (2008). 1083-4419 CrossRef
Jian  L. et al., “Contrast enhancement of medical images using multiscale edge representation,” Proc. SPIE. 2242, , 711 –719 (1994). 0277-786X CrossRef
Keeling  S. L., Stollberger  R., “Nonlinear anisotropic diffusion filtering for multiscale edge enhancement,” Inverse Probl.. 18, (1 ), 175 –190 (2002). 0266-5611 CrossRef
Numan  U., Asari  V. K., Zia-ur  R., “Fast and robust wavelet-based dynamic range compression with local contrast enhancement,” Proc. SPIE. 6978, , 697805  (2008). 0277-786X CrossRef
Rosito-Jung  C., Scharcanski  J., “Adaptive image denoising and edge enhancement in scale-space using the wavelet transform,” Pattern Recogn. Lett.. 24, (7 ), 965 –971 (2003). 0167-8655 CrossRef
Saad  A., “Visual enhancement of digital ultrasound images: wavelet versus Gauss-Laplace contrast pyramid,” Int. J. Comput. Assist. Radiol. Surg.. 2, (2 ), 117 –125 (2007). 1861-6410 CrossRef
Tang  J., Sun  Q., Agyepong  K., “An image enhancement algorithm based on a contrast measure in the wavelet domain for screening mammograms,” in  IEEE Int. Conf. Image Processing, 2007, ICIP 2007 ,  San Antonio, TX , Vol. 5, pp. V–29 –V–32 (2007).
Mukhopadhaya  S., Chanda  B., “A multiscale morphological approach to local contrast enhancement,” Signal Process.. 80, (4 ), 685 –696 (2000). 0165-1684 CrossRef
Terol-Villalobos  I. R., “Morphological connected contrast mappings based on top-hat criteria: a multiscale contrast approach,” Opt. Eng.. 43, (7 ), 1577 –1595 (2004). 0091-3286 CrossRef
Funt  B. V., Ciurea  F., McCann  J. J., “Retinex in matlab,” in  Proceedings of the IS&T/SID Eighth Color Imaging Conference: Color Science and Engineering Systems, Technologies, and Applications ,  Scottsdale, Arizona , pp. 112 –121 (2000).
Fattal  R., Agrawala  M., Rusinkiewicz  S., “Multiscale shape and detail enhancement from multi-light image collections,” ACM Trans. Graph.. 26, (3 ) (2007). 0730-0301 CrossRef
Vachier  C., Meyer  F., “The viscous watershed transform,” J. Math. Imag. Vis.. 22, (2–3 ), 251 –267 (2005). 0924-9907 CrossRef
Santillán  I. et al., “Morphological connected filtering on viscous lattices,” J. Math. Imag. Vis.. 36, (3 ), 254 –269 (2010). 0924-9907 CrossRef
Meyer  F., “From connected operators to levelings,” in Proceedings of the fourth international symposium on Mathematical morphology and its applications to image and signal processing (ISMM '98). , Heijmans  H. J. A. M., Roerdink  J. B. T. M., Eds., pp. 191 –198,  Kluwer Academic Publishers ,  Norwell, MA  (1998).
Meyer  F., “The levelings,” Proceedings of the fourth international symposium on Mathematical morphology and its applications to image and signal processing (ISMM '98). , Heijmans  H. J. A. M., Roerdink  J. B. T. M., Eds., pp. 199 –206,  Kluwer Academic Publishers ,  Norwell, MA  (1998).
Salembier  P., Serra  J., “Flat zones filtering, connected operators and filters by reconstruction,” IEEE Trans. Image Process.. 4, (8 ), 1153 –1160 (1995). 1057-7149 CrossRef
Vincent  L., “Morphological grayscale reconstruction in image analysis: applications and efficient algorithms,” IEEE Trans. Image Process.. 2, (2 ), 176 –201 (1993). 1057-7149 CrossRef
Meyer  F., Serra  J., “Contrast and activity lattice,” Signal Process.. 16, (4 ), 303 –317 (1989). 0165-1684 CrossRef
Jain  A. K., Fundamentals of Digital Image Processing. ,  Prentice Hall ,  Englewood Cliffs, NJ  (1989).
Beucher  S., “Segmentation D'Images Et Morphologie Mathematique Centre de Morphologie Mathématique,” Ph.D. Thesis, ENSMP (1990).
Vincent  L., Soille  P., “Watershed in digital spaces: an efficient algorithm based on immersion simulations,” IEEE Trans. Pattern Anal. Machine Intell.. 13, (6 ), 583 –598 (1991). 0162-8828 CrossRef
Serra  J., Image Analysis and Mathematical Morphology, Chapter 2 in Mathematical Morphology for Boolean Lattices. , Serra  J., Ed., Vol. II, pp. 51 –57,  Academic Press ,  New York  (1988).
Mendiola-Santibáñez  J. D. et al., “Morphological contrast measure and contrast enhancement: one application to the segmentation of brain MRI,” Signal Process.. 87, (9 ), 2125 –2150 (2007). 0165-1684 CrossRef
Crespo  J. et al., “The flat zone approach: a general low-level region merging segmentation method,” Signal Process.. 62, (1 ), 37 –60 (1997). 0165-1684 CrossRef
Crespo  J., Serra  J., Schafer  R. W., “Theoretical aspects of morphological filters by reconstruction,” Signal Process.. 47, (2 ), 201 –225 (1995). 0165-1684 CrossRef
Rivest  J. F., Soille  P., Beucher  S., “Morphological gradient,” J. Electron Imag.. 2, (4 ), 326 –336 (1993). 1017-9909 CrossRef
Kramer  H. P., Bruckner  J. B., “Iterations of a non-linear transformation for enhancement of digital images,” Pattern Recogn.. 7, (1–2 ), 53 –58 (1975). 0031-3203 CrossRef
Serra  J., Mathematical Morphology. , Vol. I,  Academic Press ,  London  (1982).
Serra  J., “Toggle mappings,” Technical Report N-18/88/MM, Centre de Morphologie Mathematique ENSMP, Fontainebleau France (1988).
Serra  J., “Toggle mappings,” in From Pixels to Features (Proc. COST 13, Bonas, 1988). , Simon  J. C., Ed., pp. 61 –72,  Elsevier ,  North Holland, Amsterdam  (1989).
Martin  D. et al., “A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics,” in  Computer Vision, 2001. ICCV 2001. Proceedings. Eighth IEEE International Conference on ,  Vancouver, BC , Vol. 2, pp. 416 –423 (2001).
Meyer  F., Maragos  P., “Nonlinear scale-space representation with morphological levelings,” J. Vis. Commun. Image Represent.. 11, (2 ), 245 –265 (2000). 1047-3203 CrossRef
Serra  J., Vincent  L., “An overview of morphological filtering,” Circ. Syst. Signal Process.. 11, (1 ), 47 –108 (1992). 0278-081X CrossRef
Frankle  J., McCann  J., “Method and apparatus for lightness imaging,” US Patent #4,384,336 (17  May 1983).

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging & repositioning the boxes below.

Related Book Chapters

Topic Collections

PubMed Articles
Shape-driven 3D segmentation using spherical wavelets. Med Image Comput Comput Assist Interv 2006;9(Pt 1):66-74.
Advertisement
  • Don't have an account?
  • Subscribe to the SPIE Digital Library
  • Create a FREE account to sign up for Digital Library content alerts and gain access to institutional subscriptions remotely.
Access This Article
Sign in or Create a personal account to Buy this article ($20 for members, $25 for non-members).
Access This Proceeding
Sign in or Create a personal account to Buy this article ($15 for members, $18 for non-members).
Access This Chapter

Access to SPIE eBooks is limited to subscribing institutions and is not available as part of a personal subscription. Print or electronic versions of individual SPIE books may be purchased via SPIE.org.