What velue to use for "smooth3(A, 'gaussian', v)?

When using "smooth3(A, 'gaussian', v), unlike the 'box' method, the value v doesn't seem to take affect. Without v, or 5, or 99 it has the same effect. The 'doc smooth3' doesn't provide real information.

Thanks.

Answer 1

[Adam Danz]

Good question @John.

The syntax smooth3(A,'gaussian',V) passes the data A through a gaussian lowpass filter with a default standard deviation of 0.65 (you could specify that in an optional 4th argument). V defines the 3D window size for the filter.

Without seeing your data or the results, I can't explain why various V values produce the same results. If you attach a mat file that contains A and V I could dig deeper.

[John D'Errico]

A different John, of course.

A = randn(20,20,20);
B1 = smooth3(A,'gaussian',3);
B2 = smooth3(A,'gaussian',5);
B3 = smooth3(A,'gaussian',7);

norm(B1 - B2,'fro')

ans = 1.0055

norm(B1 - B3,'fro')

ans = 1.0074

The various smoothed results are certainly not identical. That the different calls are similar in norm of the difference is something I attribute to the fact that my data is itself pure noise. I guess I could have had some real signal in there, but that would take more thought than I think it worth. There IS a difference, which is all I need to show.

[John]

Hi, Adam:

That's almost not noticeable by eye on images with 0.2% of difference.

With:

A = randn(20,20,20);

B1 = smooth3(A,'gaussian',3);

B2 = smooth3(A,'gaussian',9);

B3 = smooth3(A,'gaussian',15);

norm(B1 - B2,'fro')

ans = 1.0156

norm(B1 - B3,'fro')

ans = 1.0156

No difference. That's why I didn't see the difference. Maybe I will try ith wider SD at 2nd value place.

The DOC file should say more about the 'gaussian' option.

[John D'Errico]

@John - please learn to use comments, not answers.

Anyway, consider that a Gaussian smoothing kernel of size 15 is HUGE, in comparison to a an array of size only 20x20x20. Do you see that I did get differences in my example, for SMALL gaussian kernels?

Anyway, the size of the tiny difference in Frobenious norm you indicate is also not relevant. That is again no surprise. Taking the frobenenius norm just averages out any remnants of noise.

A = randn(100,100,100);
B1 = smooth3(A,'gaussian',3);
B2 = smooth3(A,'gaussian',9);
B3 = smooth3(A,'gaussian',15);
format long g
norm(B1 - B2,'fro')

ans = 
          11.5343687718617

norm(B1 - B3,'fro')

ans = 
          11.5343687719941

Again, there ARE Differences to be seen, but the law of large numbers averages those differences out.

LOOK AT THE ARRAYS THEMSELVES TO SEE THAT!

B1(1:3,1:3,1:3)

ans = 
ans(:,:,1) =

        -0.234102312749705         0.382072219543637         0.339308902178634
         0.287843370691201        -0.111414000006913       -0.0625339480640082
         0.129363424476748        -0.390423770739839       0.00426595845787685


ans(:,:,2) =

         0.094990454554044          0.44155534092969         0.575420147677844
        -0.266089109123344       -0.0330899672385768      0.000463778487347755
        -0.632685810434764        -0.612382934354194        -0.312260654482024


ans(:,:,3) =

        -0.435468586043421        -0.211167306940052        -0.064833858178933
        -0.185984173044869         0.345543259401304         0.168109924813588
         0.256538827469533         0.295591962534587        0.0558459484555244

B2(1:3,1:3,1:3)

ans = 
ans(:,:,1) =

         -0.24001109353323         0.373849267934935         0.322000538123073
         0.281420048281016       -0.0982743532723887       -0.0524519085140193
         0.129767507354035         -0.37558919785582       0.00684001729650901


ans(:,:,2) =

        0.0901436665433138         0.428674231870868         0.562958332174248
        -0.259188944716559       -0.0291847019562449         0.011934896561421
        -0.614651337056743        -0.594305007405726        -0.300844876871851


ans(:,:,3) =

        -0.430076302309084        -0.208657293818347       -0.0675240672893176
        -0.182313648494426         0.330234304800294         0.163644475651634
         0.238106150134462         0.278741479390564        0.0512551907513211

Different numbers.

[Adam Danz]

Thanks for that detailed explanation and demo @John D'Errico!