demo07 of Im2mesh package

demo07 - Parameter hmax and grad_limit in mesh generation
Cite as: Ma, J., & Li, Y. (2025). Im2mesh: A MATLAB/Octave package for generating finite element mesh based on 2D multi-phase image (2.1.5). Zenodo. https://doi.org/10.5281/zenodo.14847059
Jiexian Ma, mjx0799@gmail.com. Project website.
Table of Contents
Note Setup Import image hmax = 8 hmax = 500 grad_limit = 0.5 grad_limit = 0.25

Note

I suggest familiarizing yourself with Im2mesh_GUI before learning Im2mesh package. With graphical user interface, Im2mesh_GUI will help you better understand the workflow and parameters of Im2mesh package.
Im2mesh_GUI: https://www.mathworks.com/matlabcentral/fileexchange/179684-im2mesh_gui-2d-image-to-finite-element-mesh
If you are using Im2mesh package in MATLAB, you need to install MATLAB Image Processing Toolbox and Mapping Toolbox because Im2mesh package use a few functions in these toolboxes.

Setup

Before we start, please set folder "Im2mesh_Matlab" as your current folder of MATLAB.
clearvars
Set default image size.
x = 250; y = 250; width = 250; height = 250;
set(groot, 'DefaultFigurePosition', [x,y,width,height])
% To reset:
% set(groot, 'DefaultFigurePosition', 'factory')
Function im2mesh use a mesh generator called MESH2D (developed by Darren Engwirda). We can use the following command to add the folder 'mesh2d-master' to the path of MATLAB.
addpath(genpath('mesh2d-master'))

Import image

Let's start demo. Import image Transition.tif.
im = imread('Transition.tif');
if size(im,3) == 3; im = rgb2gray( im ); end
imshow( im,'InitialMagnification','fit' );

hmax = 8

opt = []; % reset opt
opt.grad_limit = 0.25;
opt.hmax = 8;
[ vert, tria, tnum ] = im2mesh( im, opt );
Refine triangulation... ------------------------------------------------------- |ITER.| |CDT1(X)| |CDT2(X)| ------------------------------------------------------- 4 160 284 10 202 882 16 203 989 Refine triangulation... ------------------------------------------------------- |ITER.| |CDT1(X)| |CDT2(X)| ------------------------------------------------------- 10 361 284 13 373 284 20 381 3867 26 381 4493 Smooth triangulation... ------------------------------------------------------- |ITER.| |MOVE(X)| |DTRI(X)| ------------------------------------------------------- 10 113 4457
plotMeshes( vert, tria, tnum )
We can see that the max mesh size is limited by the parameter hmax.

hmax = 500

Let's use a much larger hmax to see how things are changing.
opt = []; % reset opt
opt.grad_limit = 0.25;
opt.hmax = 500;
[ vert, tria, tnum ] = im2mesh( im, opt );
Refine triangulation... ------------------------------------------------------- |ITER.| |CDT1(X)| |CDT2(X)| ------------------------------------------------------- 4 160 284 10 202 882 16 203 989 Refine triangulation... ------------------------------------------------------- |ITER.| |CDT1(X)| |CDT2(X)| ------------------------------------------------------- 7 330 284 10 333 1832 20 338 4107 20 338 4107 Smooth triangulation... ------------------------------------------------------- |ITER.| |MOVE(X)| |DTRI(X)| ------------------------------------------------------- 10 224 4067 20 12 4067
plotMeshes( vert, tria, tnum )
Now, the max mesh size is much larger. Note that hmax is a upper bound for the max mesh size. There exists boundary edge contraints and gradient-limt, so the max mesh size in the generated mesh can not reach hmax.

grad_limit = 0.5

opt = []; % reset opt
opt.grad_limit = 0.5;
opt.hmax = 500;
[ vert, tria, tnum ] = im2mesh( im, opt );
Refine triangulation... ------------------------------------------------------- |ITER.| |CDT1(X)| |CDT2(X)| ------------------------------------------------------- 4 160 284 10 202 882 16 203 989 Refine triangulation... ------------------------------------------------------- |ITER.| |CDT1(X)| |CDT2(X)| ------------------------------------------------------- 5 276 284 10 288 1967 20 294 2549 22 295 2556 Smooth triangulation... ------------------------------------------------------- |ITER.| |MOVE(X)| |DTRI(X)| ------------------------------------------------------- 10 32 2500 20 1 2500 30 1 2500
plotMeshes( vert, tria, tnum )
You can see the mesh is very dense in some region.
Let's zoom in to take a look.
plotMeshes( vert, tria, tnum )
xlim([79 184])
ylim([34 134])
We can evaluate the mesh quality.
tricost( vert, tria, tnum );
The mean value of Q is 0.94. The minimum Q is 0.537.

grad_limit = 0.25

Let's use a smaller grad_limit to see how things are changing.
opt = []; % reset opt
opt.grad_limit = 0.25;
opt.hmax = 500;
[ vert, tria, tnum ] = im2mesh( im, opt );
Refine triangulation... ------------------------------------------------------- |ITER.| |CDT1(X)| |CDT2(X)| ------------------------------------------------------- 4 160 284 10 202 882 16 203 989 Refine triangulation... ------------------------------------------------------- |ITER.| |CDT1(X)| |CDT2(X)| ------------------------------------------------------- 7 330 284 10 333 1832 20 338 4107 20 338 4107 Smooth triangulation... ------------------------------------------------------- |ITER.| |MOVE(X)| |DTRI(X)| ------------------------------------------------------- 10 224 4067 20 12 4067
plotMeshes( vert, tria, tnum )
We can evaluate the mesh quality again.
tricost( vert, tria, tnum );
Now, the new Q mean is 0.964. The minimum Q is 0.7. Very good!
% reset image size
set(groot, 'DefaultFigurePosition', 'factory')
% end of demo