mathematical morphology in polar(-logarithmic) coordinates...

Mathematical morphology in (log-)polar coordinates: Shape analysis and segmentation 1

29eme journee ISS France

Mathematical morphology in

polar(-logarithmic) coordinates for the

analysis of round-objects.

Shape analysis and segmentation.

Jesus Angulo

[email protected] ; http://cmm.ensmp.fr/∼angulo

Centre de Morphologie Mathematique - Ecole des Mines de Paris

February 2nd, 2006

J. Angulo, CMM-EMP 29eme journnee ISS France (Feb. 2006)


Motivation

• Application of mathematical morphology to round-objects

(i.e. which contain a some kind of radial symmetry, or in

general, which have “a center”)

• Difficulty to take advantage of radial/angular properties in

Cartesian coordinates (definition of neighborhood, adapted

structuring elements, etc.)

• Examples (from biomedical microscopy):



Aim

• To change the geometric representation of the image:

The polar-logarithmic representation (or a general polar

coordinates system) presents many advantages for these objects

• To adapt classical morphological operators to this representation



Plan

1. (Log-)Pol coordinates

2. Cyclic morphology

3. Application: Erythrocyte shape analysis

4. Generalised distance function and global minimal path

algorithms

5. Application: Model-based spot segmentation by minimal paths

6. Conclusions and perspectives



Plan





algorithms





(Log-)Pol coordinates (1/7)

Definition

The logarithmic polar transformation converts the cartesian image

function f(x, y) : E → T (T ⊂ Z, E ⊂ Z2) into another log-polar

image function f ′(ρlog, θ) : Eρlog ,θ → T , where the angular

coordinates are placed on the vertical axis and the radial coordinates

are placed on the vertical one. More precisely, with respect to a

central point (xc, yc):

ρ =√

(x− xc)2 + (y − yc)2 → ρlog = log(ρ) 0 ≤ ρlog ≤ R;

θ = arctan(

y−yc

x−xc

)

; 0 ≤ θ < 2π

The support is the space Eρlog,θ, (ρlog, θ) ∈ (Z× Zp).




Definition

A relation is established where the points at the top of the image

(θ = 0) are neighbors to the ones an the bottom (θ = p− 1, period p

equivalent to 2π): the image can be seen as a strip where the

superior and inferior borders are joined.

Example:




Implementation

log and arctan are continuous functions to fit in a digital grid.

• The transformations are adjusted to the boundary points (size of

support space).

• For each point, a bi-linear interpolation is used to avoid the

effect of discretisation.




Properties

Rotation: Rotations in the original Cartesian image become vertical

cyclic shifts in the transformed log-pol.




Properties

Scaling: The changes of size in the original image become horizontal

shifts in the log-pol transformed image.




Properties

The polar transformation preserves the property of rotation but the

changes of scale involve a combined scaling/horizontal shift effect.




Choice of a center

The choice of the center (xc, yc) is crucial in the transformation.

We propose to use the maxima of the distance function (ultimate

erosion) to compute the center of binary objects.

For gray level images, it is proposed to compute the maximum of a

gray-weighted generalised distance function to the border of the

image.



Plan





algorithms





Cyclic morphology (1/10)

Motivation

Introduce the angular periodicity:

The aim is to preserve the invariance with respect to the rotation in

the Cartesian space:




Implementation

In order to implement the new neighborhood relation and to be able

to use morphological operators, two possibilities can be considered:

• Modify the neighborhood relation and the code of the basic

operators (erosion, dilation, etc.) by adding the operator

“module of the size of the image in the direction of the periodic

coordinate”

• Extend the image along its angular direction by adding the top

part of the image on the bottom and the bottom part on the top.

The size of the vertical component from each part should be

bigger than the size from the vertical component of the

structuring element in order to avoid a possible edge effect.




Implementation

After having cycled the image, morphological operators can be

applied as usual and only the image corresponding to the initial mask

is kept.

Using this approach, all the existing code is recyclable!




Meaning of structuring elements

The use of classical structuring elements in the log-pol image is

equivalent to the use of “radial - angular” structuring elements in the

original image, e.g. a vertical structuring element corresponds to an

arc in the original image or a square corresponds to a circular sector.

Example of horizontal structuring element:




Meaning of structuring elements

Other examples:




Tools

Circular filtering: Method for extracting inclusions or extrusions

from the contour of a relatively rounded shape with simple openings

or closings.

The proportion of the vertical size from the structuring element with

respect to the whole vertical size represents the angle affected in the

original Cartesian image.

With respect to a classical extraction in Cartesian coordinates, the

choice of size is not as critical, making this a very advantageous

point. It is due to the large zone plate in the openings/closings

spectrum that is always found after a determined value (until the

complete elimination of the object).




Tools

Circular filtering

Example of opening with a vertical structuring element of size 20% of

the whole image (i.e. 72◦).




Tools

Radial skeleton: Interesting for objects without holes.

• Radial inner skeleton is the skeleton obtained by an homotopic

thinning from the log-pol transformation of an objet. The invert

transformation to Cartesian coordinates from the branches of the

radial inner skeleton has radial sense and tend to converge to the

center (ρ = 0).

• Radial outer skeleton is obtained by an homotopic thinning from

the negative image of the log-pol transformation of an object.

The invert branches tend to diverge to an hypothetical

circumference in the infinity (ρ −→∞).




Tools

Radial skeleton: Illustrative example,




Tools

Radial skeleton: More examples,



Plan





algorithms





Application: Erythrocyte shape analysis (1/4)

Aim: In hematology, the morphological analysis of erythrocytes

(size, shape, color, center,...) is fundamental: anomalies and

variations from the typical red blood cell are associated to anemia or

other pathologies.

The analysis of shape categories is particularly interesting:

Normal “Mushroom” “Spicule” “Echinocyte” “Bitten”




“Spicule” and “Echinocyte” erytrocytes: Extraction of

extrusions using circular filtering and selection/classification by

radial skeleton.




“Mushroom” erytrocytes




“Bitten” erytrocytes



Plan





algorithms





Generalized dist. function and minimal paths (1/7)

Motivation

Inner (spot center) marker and outer (bounding box) marker

watershed-based: problems of segmentation for low intensity spots or

for spots on strong noisy background; and on the other hand,

difficulty to define a right segmentation/quantification for doughnut

and egg-like spots

Original Filtered Segmented




Motivation

Spot segmentation can be approached in a more flexible and

understandable way when working in polar coordinates, but the same

weaknesses of the watershed on the low or noisy gradients are still

underlying.

Typical Doughnut-like

Extract interesting morphological features from projections of the

spot in polar coordinates.



Generalized dist. function and minimal paths (3/)

Watershed transformation segmenting left/right sides:

Extracting a continuous track (=“crest-line”) going from the top to

the bottom of the image by means of a constrained watershed using

as markers the right side and the left side of the image,



Analysis the watershed line (by Vincent, 98):

• It fails when SNR is low = sensitivity of watershed line to noise;

• The watershed between two markers A and B depends on the

position of the saddle points (for all the paths joining A to B

with minimal elevation, the highest pixels along those paths are

the saddle points) between the markers, and their location is one

of the main factors determining the location of the line;

• The criteria used to build the watershed are based on grey levels,

and the length of watershed lines is irrelevant.

Length constraints can be introduced by using global minimal paths

algorithms. This approach is also useful to detect “disconnected”

crest-line between two markers.




Generalized distance function, GDF: Modification of the classic

two-pass sequential distance function algorithm so that: (1) edge cost

is taken into account; (2) raster and anti-raster scans are iterated

until stability.

Let us to denote by N+(p) (resp., N−(p)) the neighbors of pixel p

scanned before p (resp., after p) in a raster scan, for a 8-connected

grid (neighborhood graph):

In this graph, each edge between two neighboring pixels p and q of an

image f has associated the cost value Cf (p, q) = f(p) + f(q) (or any

other increasing function, such as max(f(p), f(q)) or min(f(p), f(q))).



More specifically, the algorithm proceeds as follows,

Algorithm: GDF to set X in image f

• Initialise result image d: d(p) = 0 if p ∈ X and d(p) = +∞

otherwise;

• Iterate until stability:

– Scan image in raster order → For each pixel p, do:

d(p)← min{d(p), min{d(q) + Cf (p, q), q ∈ N+(p)}}

– Scan image in anti-raster order → For each pixel p, do:

d(p)← min{d(p), min{d(q) + Cf (p, q), q ∈ N−(p)}}

The algorithm typically converges in two or three iterations

(relatively efficient).




Global minimal paths, GMP:

Each path P in the 8-connect graph has associated a cost Cf (P ),

equal to the sum of the cost of its successive edges. We can now

define the distance df (p, q) between two pixels p and q in the image f

as: df (p, q) = min{Cf (P ), P path between p and q}.

For the simple problem of finding a path of minimal cost (or global

minimal path) going from the top row U to the bottom row D of the

image, we use the following result: a pixel p belongs to such minimal

path if and only if df (p, U) + df (p, D) = df (U, D).



To extract such path, we can therefore proceed as follows:

Algorithm: Up/Down GMP in image f

• Compute GDF to set U in image f : for each pixel p, compute

df (p, U);

• Compute GDF to set D in image f : df (p, D);

• Sum these two distance functions,

df (U, D)(p) = df (p, U) + df (p, D);

• Find umin, the minimal value of df (U, D) and threshold the result

in order to keep only the pixels whose values in df (U, D) is equal to

umin

From an algoritmic point of view, the problem is reduced to

computing two gray-weighted generalised distance transforms.




Global minimal paths, GMP Illustrative example:

f distU(f) distD(f)

distUD(f) Iso(distUD(f)) min(distUD(f))




Examples of up/down global minimal path:

The main limitation of the approach lies on the degree of horizontal

curvature (see example (d)) but it is very robust against to the noise

(see example (c)).



Plan





algorithms





Application: Model-based spot segmentation (1/6)

Flow chart of algorithm: Starting from the spot in polar

coordinates, the aim is to segment its contour using the GMP

technique.

The segmentation algorithm must be able to yield a spot

segmentation in several regions according to the spot structure




Filtering in polar coordinates: Anisotropic effect in polar

coordinates by applying two separable directional filtering

(unidimensional filtering).

Usually for the polar image of spots the vertical (according to the

angular coordinate) filtering has a size nρ which is notably higher

than the size nθ horizontal filtering (radial direction).




Circular minimal path to close contour: In polar coordinates in

the Up/Down GMP the initial radial value ρup (for θ = 2π) and the

final one ρdown (for θ = 0) are equal.

Typical problem when the center of spot is shifted with respect to

(xc, yc).




Circular minimal path to close contour: To apply the Up/Down

GMP to the cycled image.

In fact, even if ρup 6= ρdown, but |ρup − ρdown| ≤ ∆ρ (∆ρ being a

small value), the contour can be “closed” by dilation.




Example of spot segmentation:

Original (x10) Ref. Segment.




Example of spot segmentation:

Original (x10) Minimal Paths



Plan





algorithms





Conclusions and perspectives

• Key issue: to obtain operators that are adapted to the nature of

the objects to be analyzed, not by “deforming” them, but by

transforming the image itself.

• The conversion into polar-logarithmic coordinates as well as the

derived cyclic morphology appears to be a field that may provide

satisfying results in image analysis applied to round objects or

spheroid-shaped 3D-object models.

• We have developed a methodology to describe in detail and

classify the shape of cells.

• We have developed a model-based evolved methodology for

segmenting the spots in fluorescence-marked microarray images,

allowing an automatic adaptation to all the situations.



References

• M.A. Luengo-Oroz, J. Angulo, G.Flandrin, J. Klossa. Mathematical

morphology in polar-logarithmic coordinates. In Proc. of the 2nd

Iberian Conference on Pattern Recognition and Images Analysis

(IbPRIA’04), Estoril, Portugal, Springer LNCS 3523, p. 199–206.

• J. Angulo. Automated spot classification in cDNA images using

mathematical morphology. Internal Note N-19/05/MM, Ecole des

Mines de Paris, January 2005, 28 p.

• J. Angulo, F. Meyer. Spot segmentation in cDNA images by

computing minimal paths in polar coordinates. Internal Note

N-20/05/MM, Ecole des Mines de Paris, June 2005, 73 p.

• A. Rosenfeld and J. Pfaltz. Distance functions on digital pictures.

Pattern Recognition, 1:33–61, 1968.



• L. Vincent. Minimal Path Algorithms for the Robust Detection of

Linear Features in Gray Images. In Proc. of International Symposium

on Mathematical Morphology (ISMM’98), Amsterdam, Kluwer, pp.

331–338, June 1998.

• C. Sun, S. Pallottino. Circular shortest paths in images. Pattern

Recognition, 36, 709–719, 2003.

• B. Appleton, C. Sun. Circular shortest paths by branch and bound.

Pattern Recognition, 36, 2513–2520, 2003.

• B. Appleton, H. Talbot. Globally Optimal Geodesic Active Contours.

Journal of Mathematical Imaging and Vision, 23, 67– 86, 2005.


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