OOF2: The Manual

Smooth (Smooth)
Prev  6.5.2. Subclasses  Next

Name

Smooth (Smooth) — Move nodes to the average position of their neighbors.

Synopsis

Smooth(targets, criterion, T, iteration)

Details

  • Base class: SkeletonModifier

  • Parameters:

    targets
    Which nodes to move. Type: An object of the FiddleNodesTargets class.
    criterion
    Acceptance criterion Type: An object of the SkelModCriterion class.
    T
    Failed moves will be accepted if T>0 and exp(-diffE/T) > r, where diffE is the energy gained and r is a random number between 0 and 1. Type: A real number.
    iteration
    Iteration method. Type: An object of the IterationManager class.

Description

Smooth moves Nodes to the average positions of their neighbors in order to smooth out gradients in Node density in a Skeleton. Nodes are moved one by one in random order.

Typically, it takes about 3 to 5 iterations of Smooth to get a good result. The procedure for a single iteration is:

  1. Collect target nodes according to the given targets parameter. The collected Nodes are re-ordered randomly to avoid possible artifacts due to the original ordering of Nodes. This re-ordering is repeated at every iteration.

  2. Give a Node a single chance to move to the average position of its neighbors. For this purpose, neighbors are defined as Nodes that share a Segment with the Node being moved.

    Figure 6.104(a) shows a Node (the red dot) that is about to move to the average position of its neighbors (the green dots).

  3. Accept the Node move only if it satisfies the specified acceptance criterion. Moved Nodes that fail to meet the criterion will spring back to their original positions, as shown in Figure 6.66(b).

  4. If the move is unacceptable according to the acceptance criterion, OOF2 may still accept the move if the smoothing is being done at a non-zero temperature. The parameter T sets the effective temperature of the annealing process. Unacceptable moves are accepted with a probability

    P=exp⁡(−Ediff/T)
    		(6.164)

    where Ediff is the difference between the effective energies of the new and old Element configurations.

  5. Return to step 2 until all of the Nodes in the list have been examined.

Figure 6.104. Schematics of Smooth

Schematics of Smooth

(a) A node (red dot) moves to the average position (black dot) of its four neighboring nodes (green dots). The move is affecting the four neighboring elements (yellow).

(b) The node has moved to the average position. If the move doesn't satisfy the given criterion (computed on the four affected elements), it will move back to its original position.


Smoothing usually converges quickly and each iteration is not very expensive, so it's reasonable to perform a fixed small number of iterations, for example by setting iterations= FixedIteration(iterations=5). It is also possible to use conditional iteration, which can repeatedly smooth the Skeleton until the energy drop or acceptance rate per iteration becomes too small.

Figure 6.105 shows a quadrilteral Skeleton that has been refined twice by bisection of the inhomogeneous Elements, and the result of smoothing it. Figure 6.106 compares ways of preserving Element homogeneity while smoothing.

Figure 6.105. Smoothing a Skeleton

Smoothing a Skeleton

(a) A Microstructure and Skeleton. The Skeleton has been refined twice by bisection.

(b) The result of smoothing the Skeleton with Smooth(targets=AllNodes(),criterion=AverageEnergy(alpha=0.3),T=0,iteration=FixedIteration(iterations=5)). Because alpha is small, Smooth produced well shaped but inhomogeneous Elements.


Figure 6.106. Smoothing without Losing Homogeneity

Smoothing without Losing Homogeneity

(a) A Skeleton created by refining the Skeleton in Figure 6.105(a) at TransitionPoints and rationalizing it three times. Its homogeneity index is 0.9946, meaning that only 0.54% of the Microstructure is misrepresented by the Skeleton. The shape energy of the Element highlighted in red is 0.40.

(b) For comparison, this shows the Skeleton in (a) after smoothing with alpha=0.3 for 5 iterations, the same operation used in Figure 6.105. Many Elements have become less homogeneous — 1.08% of the Microstructure is misrepresented, twice what it was in (a) — but shapes have improved. The shape energy of the red element has decreased to 0.13.

(c) The Skeleton in (a) after smoothing with alpha=1.0 for 5 iterations. Smoothing has improved element shapes away from the boundaries, but not very much near them (the shape energy of the red Element has increased slightly to 0.54). Smoothing did not reduce homogeneity. Nodes that were on pixel boundaries are still on pixel boundaries, and only 0.45% of the Microstructure is misrepresented.

(d) The Skeleton in (a) after pinning the Nodes that lie on its internal boundaries and then smoothing it with alpha=0.3 for 5 iterations. The red Element's shape energy is not very good, 0.53, but in general the shapes of the Elements in the less constrained yellow and orange regions have improved over (a). The misrepresented fraction is 0.54%, the same as it was in (a), which is not surprising because none of the Nodes on the boundaries have moved.


Statistics for each step of the smoothing process are printed in the OOF2 Message window. For example, here is the output from creating Figure 6.106(b): 

    Iteration 1: E = 4.7648e+01, deltaE=-3.4691e+00 ( 7.281%), Acceptance Rate = 67.2%
    Iteration 2: E = 3.1804e+01, deltaE=-7.5127e-01 ( 2.362%), Acceptance Rate = 45.5%
    Iteration 3: E = 2.8475e+01, deltaE=-2.5280e-01 ( 0.888%), Acceptance Rate = 24.9%
    Iteration 4: E = 2.7365e+01, deltaE=-4.0957e-02 ( 0.150%), Acceptance Rate = 11.1%
    Iteration 5: E = 2.7201e+01, deltaE=-7.4279e-03 ( 0.027%), Acceptance Rate =  7.9%
    Smoothing done:  deltaE = -2.0476e+01 (42.975%)

The listing shows the iteration number, the total energy (E) of the Skeleton, the absolute change in energy (deltaE) during the iteration, the percentage change in energy, and the move acceptance rate. If the change in energy or the acceptance rate gets too small, the smoothing process is not being effective at improving the Skeleton. Notice that the energy change per step is decreasing exponentially, indicating rapid convergence.

See Also

Prev  Up  Next
Selected Segments (SkeletonSegmentSelectionDisplay)  Home  All Nodes (SnapAll)