Towards a multivariate probabilistic morphologyfor colour images
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Caliman, Alexandru
Ivanovici, Mihai
Richard, Noël
Fernandez-Maloigne, Christine
The mathematical morphology for colour images faces the delicate issue of defining a total order in a vectorial space. There are various approaches based on partial or total orders defined for color images. We propose a probabilistic approach, that uses principal component analysis (PCA), for the computation of the convergence colours, i.e. the extrema of a set. Then we define two pseudo-morphological operations, the dilation and the erosion, applying the Chebyshev’s inequality on the first eigenvector of the image colour data. As an application, we use our approach to extract the Beucher colour gradient. We discuss the advantages and disadvantages of our approach, we comment our results and then we conclude this paper.
 Good afternoon. My name is Ayfandil Kottemar. And I'd like to present you a new method for color mathematical morphology using a probabilistic approach. The mathematical morphology needs a totally ordered lattice structure to be applied on. For color images, however, this total ordering is difficult to be obtained due to the vectorial nature of the data. There have been proposed several approaches for vectorial data ordering. But for color images, there are still problems. For instance, for the conditional ordering, the conditional ordering is not linear from the human perception point of view, as it may be seen in this image where the conditional ordering for the RGB color space is presented. Another problem would be for marginal ordering, for which we may obtain false colors, which are not suitable for a full vectorial approach. The mathematical morphology embeds two aspects. So the first aspect is that we have to find a feature ordering. The features may be gray level, color data, or MRI, or multispectral. And the second aspect is that it analyzes the special arrangement of the pixels in a local neighborhood given by the structuring element. For the feature ordering, there have been proposed several approaches for color and multispectral images. There are a lot of conditional and lexicographic ordering. Recently, there have been proposed distance functions in a feature space and using a convergence coordinate. And also, there have been proposed another approach using a classification scheme on distribution of the data. Our question would be, could we directly use this distribution of the data in our mathematical morphology directly instead of using a classification scheme? We try to answer to this question. And the way in which we use the data distribution is by applying the principal component analysis. So for a color image with a generic color space C1, C2, C3, after applying the PCA, we obtain different components named the principal component, which are the correlated. And the first component has the largest variance. Since it has the largest variance, we may assume that it contains the main gradient in the image. So we may use it to find two global extremes for our image. The way in which we use this component is by regarding it as a random variable and applying the Chebyshevian equality on it. So this is the Chebyshevian equality. It defines a confidence interval, which depends on the mean of the data, the variance, and one parameter k. So we took that interval and we defined its extremes as our extremes coordinates, as it may be seen in these relations. For instance, for two-dimensional data, we compute the PCA in order to determine the principal components. And on the first component, the one which has the greatest variance, we apply Chebyshevian equality. So the green arrows are indicated, the two extreme points obtained by applying the Chebyshevian equality with k parameter equals 2. And here we have the two extremes with k equals 1. Our two extremes now are, in fact, the global maximum and the global minimum, which may be used in our morphological operations like dilations and erosion. The problem now is to find a rule to match each of these extreme points to a certain morphological operation. We used the energy rule, as defined here. For our natural images, we didn't find the two extreme points with the same energy, so it worked. In our case, for k equal 2, if this point has a greater energy than this one, this would be the global maximum, and the other would be the global minimum. After defining the two global extremes, we may proceed to the last step, which is the mathematical morphology step, which is, in fact, using a structuring element and analyze the local variance of the data. We only used flex structuring element case, which means that we have to extract the maximum and the minimum values from a neighborhood given by the structuring element. We compute the two extremes locally using only the local data instead of old data, with the same method as in the global approach. But instead of using the energy criteria, now we have an information, the two global extremes, which we may use, and we project the two extremes on that line given by the two global extremes. We also make a translation of the projections into the middle of the segment in order to avoid situations like this where the projections would be outside the segment, and then we use the distance criteria in order to discover the local minimum and maximum values. Of course, we may find situations in which the two directions would be perpendicular and the projections would be the same. Then we may use the second or the third component given by the PCA for the local data. About the color space, we used the RGB color space for two reasons. First, we envisaged using, accelerating this algorithm on hardware, like an FPGA or DSP platform, for real-time applications. And the second one is that in certain situations, which I will describe later, we have some problems and we have to limit the extreme values inside the RGB cube. We use this three-dimensional representation of the color space, cubic representation. Some results, here are some erosions and dilations with different element structure sizes, with the two global extremes here. I think to be noticed here is that both the erosions and the dilations are blurrier than the original image. This happens for two reasons. The first one is that the mean and the shape of the local data distribution do not change dramatically between two consecutive local data sets. And the second reason is that we also used the same local key parameter. This is another set of results, the same effect of blurring, but here we have other extreme points, global extreme points. So they are different for each image, depending on the image distribution. As an application, we propose the morphological gradient, which in our case is a color image, and may be used to feature extraction. However, it is only a local contrast gradient, which depends on the norm and the color directions of the local data and the initial color distribution. For the key parameters, the key parameter in the Chebyshev inequality is too large, or two extremes would be outside the color gamut. So the key parameter has to be decreased in order to keep these extreme points inside the color space. But depending on the image distribution, we may reach to k values, which are smaller than one, which from the geometrical point of view is not a problem, since we have our two extremes closer to the mean. But from the Chebyshev inequality point of view, it is a problem since we have probabilities which are greater than one. So we still have to solve that problem. For instance, for the LENA image here, we have the two extremes which are inside the color gamut. But for the MIRROR image, we may see that one extreme is outside the color gamut using k equal one. So here it has to be decreased, so it will be less than one. The local k parameter effect is, in fact, the local k parameter is an adjustment between a linear and a nonlinear filter. For instance, if k is close to zero, we get a linear filter effect. We compute the average of the color distribution. If k is large, we obtain a nonlinear effect. In fact, it is our two defined morphological operations. We also present the duality of our approach. So the two extreme values are at the same distance from the data mean. So we may regard them as the complement of each other. And also the operations would be the complement of each other. So as an example here, if we start from this image and we apply a dilation, we will obtain the same result as obtaining after a negation, erosion, and another negation. These two images are identical. In conclusion, we have proposed a new probabilistic approach for color mathematical morphology. Our approach has three steps. The first one is applying the PCA in order to determine the first principle component. Then we apply the Chebushev inequality on that component, and then we perform the local approach by a morphological step. We also presented the K parameter, which is an adjustment between linear and nonlinear effect, and the duality for the two basic morphological operations. And we may conclude that our approach may represent an alternative for the choice of the two convergence points, or even for the hue origin in a lexical graphical approach. In perspective, we want to analyze this effect in other color spaces. Now we just wanted to propose the mathematical method. And we also want to validate this approach by proposing new applications like fractal dimension estimation using covering blanket, or pattern spectrum analysis, or nonlinear filters. Thank you very much. APPLAUSE So we have time for two or three questions. Have you compared your color gradient with the gradient we can get on using the same idea on marginal color? Well, we didn't make such a comparison because our purpose was just to propose the two morphological operations, and then maybe to see the effect. Our purpose was not to come with a new application, which is better than other, just with two basic morphological operations for color image. Maybe in our future work we'll do that. I wondered why you didn't consider also taking as your extrema the extrema of the actual sample. In fact, now we work on a new approach which uses exactly that extreme point. But for this approach, we wanted to use, let's say, a theoretical approach using the Cebu-Shevina quality. That's why. Because... We thought that starting from a theoretical base would lead us to better results than just proposing some... I don't know. Empirical. Okay, thanks. Any other questions? So thanks, our speaker again. Thank you.
 Good afternoon.  My name is Ayfandil Kottemar.  And I'd like to present you a new method for color  mathematical morphology using a probabilistic approach.  The mathematical morphology needs a totally ordered  lattice structure to be applied on.  For color images, however, this total ordering  is difficult to be obtained due to the vectorial  nature of the data.  There have been proposed several approaches  for vectorial data ordering.  But for color images, there are still problems.  For instance, for the conditional ordering,  the conditional ordering is not linear from the human  perception point of view, as it may be seen in this image  where the conditional ordering for the RGB color space  is presented.  Another problem would be for marginal ordering,  for which we may obtain false colors, which are not  suitable for a full vectorial approach.  The mathematical morphology embeds two aspects.  So the first aspect is that we have  to find a feature ordering.  The features may be gray level, color data, or MRI,  or multispectral.  And the second aspect is that it analyzes  the special arrangement of the pixels  in a local neighborhood given by the structuring element.  For the feature ordering, there have  been proposed several approaches for color  and multispectral images.  There are a lot of conditional and lexicographic ordering.  Recently, there have been proposed distance functions  in a feature space and using a convergence coordinate.  And also, there have been proposed another approach  using a classification scheme on distribution of the data.  Our question would be, could we directly  use this distribution of the data in our mathematical  morphology directly instead of using a classification scheme?  We try to answer to this question.  And the way in which we use the data distribution  is by applying the principal component analysis.  So for a color image with a generic color space C1, C2, C3,  after applying the PCA, we obtain different components  named the principal component, which are the correlated.  And the first component has the largest variance.  Since it has the largest variance,  we may assume that it contains the main gradient in the image.  So we may use it to find two global extremes for our image.  The way in which we use this component  is by regarding it as a random variable  and applying the Chebyshevian equality on it.  So this is the Chebyshevian equality.  It defines a confidence interval, which  depends on the mean of the data, the variance,  and one parameter k.  So we took that interval and we defined its extremes  as our extremes coordinates, as it may be seen in these relations.  For instance, for two-dimensional data,  we compute the PCA in order to determine the principal components.  And on the first component, the one which has the greatest variance,  we apply Chebyshevian equality.  So the green arrows are indicated,  the two extreme points obtained by applying the Chebyshevian  equality with k parameter equals 2.  And here we have the two extremes with k equals 1.  Our two extremes now are, in fact,  the global maximum and the global minimum,  which may be used in our morphological operations  like dilations and erosion.  The problem now is to find a rule to match each of these extreme points  to a certain morphological operation.  We used the energy rule, as defined here.  For our natural images, we didn't find the two extreme points  with the same energy, so it worked.  In our case, for k equal 2, if this point has a greater energy  than this one, this would be the global maximum,  and the other would be the global minimum.  After defining the two global extremes,  we may proceed to the last step,  which is the mathematical morphology step,  which is, in fact, using a structuring element  and analyze the local variance of the data.  We only used flex structuring element case,  which means that we have to extract the maximum and the minimum values  from a neighborhood given by the structuring element.  We compute the two extremes locally using only the local data  instead of old data,  with the same method as in the global approach.  But instead of using the energy criteria,  now we have an information, the two global extremes,  which we may use, and we project the two extremes on that line  given by the two global extremes.  We also make a translation of the projections  into the middle of the segment in order to avoid situations  like this where the projections would be outside the segment,  and then we use the distance criteria  in order to discover the local minimum and maximum values.  Of course, we may find situations  in which the two directions would be perpendicular  and the projections would be the same.  Then we may use the second or the third component  given by the PCA for the local data.  About the color space, we used the RGB color space for two reasons.  First, we envisaged using, accelerating this algorithm on hardware,  like an FPGA or DSP platform,  for real-time applications.  And the second one is that in certain situations,  which I will describe later, we have some problems  and we have to limit the extreme values inside the RGB cube.  We use this three-dimensional representation  of the color space, cubic representation.  Some results, here are some erosions and dilations  with different element structure sizes,  with the two global extremes here.  I think to be noticed here is that both the erosions  and the dilations are blurrier than the original image.  This happens for two reasons.  The first one is that the mean and the shape  of the local data distribution do not change dramatically  between two consecutive local data sets.  And the second reason is that we also used  the same local key parameter.  This is another set of results, the same effect of blurring,  but here we have other extreme points, global extreme points.  So they are different for each image,  depending on the image distribution.  As an application, we propose the morphological gradient,  which in our case is a color image,  and may be used to feature extraction.  However, it is only a local contrast gradient,  which depends on the norm and the color directions  of the local data and the initial color distribution.  For the key parameters, the key parameter in the Chebyshev inequality  is too large, or two extremes would be outside the color gamut.  So the key parameter has to be decreased  in order to keep these extreme points inside the color space.  But depending on the image distribution,  we may reach to k values, which are smaller than one,  which from the geometrical point of view is not a problem,  since we have our two extremes closer to the mean.  But from the Chebyshev inequality point of view,  it is a problem since we have probabilities  which are greater than one.  So we still have to solve that problem.  For instance, for the LENA image here,  we have the two extremes which are inside the color gamut.  But for the MIRROR image,  we may see that one extreme is outside the color gamut  using k equal one.  So here it has to be decreased, so it will be less than one.  The local k parameter effect is, in fact,  the local k parameter is an adjustment between  a linear and a nonlinear filter.  For instance, if k is close to zero,  we get a linear filter effect.  We compute the average of the color distribution.  If k is large, we obtain a nonlinear effect.  In fact, it is our two defined morphological operations.  We also present the duality of our approach.  So the two extreme values are at the same distance  from the data mean.  So we may regard them as the complement of each other.  And also the operations would be the complement of each other.  So as an example here, if we start from this image  and we apply a dilation, we will obtain the same result  as obtaining after a negation, erosion, and another negation.  These two images are identical.  In conclusion, we have proposed a new probabilistic approach  for color mathematical morphology.  Our approach has three steps.  The first one is applying the PCA in order  to determine the first principle component.  Then we apply the Chebushev inequality on that component,  and then we perform the local approach  by a morphological step.  We also presented the K parameter,  which is an adjustment between linear and nonlinear effect,  and the duality for the two basic morphological operations.  And we may conclude that our approach may represent  an alternative for the choice of the two convergence points,  or even for the hue origin in a lexical graphical approach.  In perspective, we want to analyze this effect  in other color spaces.  Now we just wanted to propose the mathematical method.  And we also want to validate this approach  by proposing new applications like fractal dimension estimation  using covering blanket, or pattern spectrum analysis,  or nonlinear filters.  Thank you very much.  APPLAUSE  So we have time for two or three questions.  Have you compared your color gradient with the gradient  we can get on using the same idea on marginal color?  Well, we didn't make such a comparison  because our purpose was just to propose  the two morphological operations,  and then maybe to see the effect.  Our purpose was not to come with a new application,  which is better than other,  just with two basic morphological operations for color image.  Maybe in our future work we'll do that.  I wondered why you didn't consider also  taking as your extrema the extrema of the actual sample.  In fact, now we work on a new approach  which uses exactly that extreme point.  But for this approach, we wanted to use, let's say,  a theoretical approach using the Cebu-Shevina quality.  That's why.  Because...  We thought that starting from a theoretical base  would lead us to better results than just proposing some...  I don't know.  Empirical.  Okay, thanks.  Any other questions?  So thanks, our speaker again.  Thank you.
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