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#### flst_bilinear

Name

flst_bilinear Fast Level Sets Transform of a bilinear interpolated image

Command Synopsis

flst_bilinear [-a min_area] image tree

-a min_area : argument of the grain filter

image : Input fimage

tree : The tree of shapes

Function Summary

void flst_bilinear (pMinArea , pImageInput , pTree )

int *pMinArea ;

Fimage pImageInput ;

Shapes pTree ;

Description

This module computes the FLST (see module flst) of a bilinear interpolated image. As the image is considered as a continuous function, the number of level lines is actually infinite. This module makes the following choice: It computes the shapes associated to level lines passing through data points and those passing through saddle points.

We call data points the centers of pixels, where image values are given. Pixels are considered as squares, whose corners have integral coordinates. Therefore data points are at half-integral coordinates. A dual pixel is a square of corner the data points (X - 0.5, Y - 0.5), (X + 0.5, Y), (X, Y + 0.5) and (X + 0.5, Y + 0.5). Its coordinates are by convention the ones of its center, (X, Y). A dual edgel is a face of a dual pixel. The image is interpreted as a continuous function taking the given values at the data points and being affine when restricted to any dual edgel. More precisely, the image values inside the dual pixel are given by the formula

u(x + X, y + Y) = a(1 - x)(1 - y) +  bx(1 - y) +  c(1 - x)y +  dxy

where a, b, c and d are the respective given values at the data points (X, Y), (X + 1, Y), (X, Y + 1) and (X + 1, Y + 1). It is also the result of the convolution of Dirac weights positioned at data points with the function

max(0, 1 - | x|)max(0, 1 - | y|).

The image is thus considered as continuous, defined on the rectangle R = [0, W]×[0, H], W and H being the width and height of the image. Inside a dual pixel, an isolevel set is composed of one or two branches of a hyperbola, maybe degenerate. For points (x, y) such that x < 0.5 or x > W - 0.5 or y < 0.5 or y > H - 0.5, it gets the same value as the nearest point in the rectangle [0.5, W - 0.5]×[0.5, H - 0.5].

The isolevel sets are the sets u = c, c being a constant. The connected components of isolevel sets are disjoint. If I is such a set, we define the exterior of I as:

• the connected component of R I containing the border of the image if I does not meet this border;
• the connected component of R I meeting the border of the image and of area larger than , provided it exists, otherwise.
Remark: the term `area' here must be understood as the number of included data points. This convention simplifies the algorithm, avoiding the computation of a true area. In the same manner, the field area of Shape is actually the number of data points.

The saturation of I is the complement of the exterior of I, and called the shape associated to I. A shape is thus closed, connected and has a connected complement. The boundary of the saturation of I is called a level line. It is a part of I, so a connected part of an isolevel set of the image. If I meets the border, the line is open, otherwise it is closed.

The shapes have a tree structure induced by the inclusion. S is an ancestor of T if and only if T S. If S and T are two shapes such that none is an ancestor of the other, S T = .

With our interpolation model, the number of isolevel sets at a given level are finite. Thus in a sufficiently small neighborhood U of a level line at level value, the image takes values that are all strictly larger or all strictly lower than value on the intersection of U and the exterior of the level line. In the first case, the field inferior_type is set to true (1), while in the second case it is set to false (0). The other fields of Shape and Shapes have the same meaning as for the module flst, except area, as explained above, and pixels which stores the data points inside the shape, not pixels.

This field is not filled by this module, but can be done by flst_pixels if needed. In the same manner, the field boundary is not filled by this module, but by flstb_boundary.

The shapes included in the tree are the ones whose associated level line contains at least one data point or saddle point. Therefore, all the shapes whose boundary contains a critical point are in the tree, but no other shape (except the root). If other shapes are desired, see module flstb_quantize. The module flst_reconstruct permits to reconstruct the image from the tree, as for the tree extracted by flst.

The min_area parameter, if used, is the minimum number of data points for a shape to be included in the tree. If it is used, the reconstructed image is different from the original image.

NB: The input image is lost after a call to this function

See Also

Version 1.1

Last Modification date : Thu Nov 29 20:23:56 2001

Author

Pascal Monasse

Next: flst_boundary Up: Reference Previous: flstb_dualchain   Contents   Index
mw 2004-05-05