Fourier Shell Correlation - Revision history

Daniel Castaño at 06:40, 27 April 2016

2016-04-27T06:40:15Z

Daniel Castaño at 08:45, 25 April 2016

2016-04-25T08:45:57Z

Daniel Castaño at 08:39, 25 April 2016

2016-04-25T08:39:49Z

Daniel Castaño at 08:34, 25 April 2016

2016-04-25T08:34:47Z

Daniel Castaño at 08:33, 25 April 2016

2016-04-25T08:33:07Z

Daniel Castaño at 08:14, 25 April 2016

2016-04-25T08:14:55Z

Daniel Castaño at 08:13, 25 April 2016

2016-04-25T08:13:43Z

Daniel Castaño at 08:13, 25 April 2016

2016-04-25T08:13:12Z

Daniel Castaño at 08:11, 25 April 2016

2016-04-25T08:11:14Z

Daniel Castaño at 08:05, 25 April 2016

2016-04-25T08:05:14Z

@@ Line 14: / Line 14: @@
 #for each shell index i=1:N/2
 #: compute a discretized shell n(i) as a volume with entries 0 or 1
-#::  a fourier pixel with a distance ''k'' from the center is in shell n(i) if   i-1<=|k|<i
+#::  a fourier pixel with a distance ''k'' from the center is in shell n(i) if  i<=|k|<i+1
 #: locate elements of A and B  inside shell n(i)
 #:: A<sub>i</sub> = A|n(i) ; a vector with as many entries as non-zero elements in n(i)
@@ Line 22: / Line 22: @@
 #:: ''*''  represents pixelwise multiplication, |.| is the vector norm.
 </tt>
-With this definition, the first element in a FSC would measure the similarity of A and B restricted to the first shell A<sub>1</sub> and B<sub>1</sub>. This shell is just the single central Fourier pixel in both cases: the zero-th component of the Fourier transform. This will always be 1 (being the correlation of two scalars). In the other extreme of resolution, FSC(N/2) would be measuring the similarity of those Fourier modes in A and B that are ''strictly worse'' that the resolution 1/(N/2), but ''better'' than the resolution 1/(N/2-1).
+With this definition, the first element in a FSC would measure the similarity of A and B restricted to a first shell A<sub>1</sub> and B<sub>1</sub>.  With the definition above, this first shell skips the central Fourier mode, i.e. the zero-th component of the Fourier transform (i.e, the total intensity of particle ''a'' in direct space). The FSC of the central Fourier model of two volumes will always be 1 (being the correlation of two scalars), and it's not represented.

@@ Line 22: / Line 22: @@
 #:: ''*''  represents pixelwise multiplication, |.| is the vector norm.
 </tt>
-With this definition, the first element in a FSC would measure the similarity of A and B restricted to the first shell A<sub>1</sub> and B<sub>1</sub>. This shell is just the single central Fourier pixel in both cases: the zero-th component of the Fourier transform. In the other extreme of resolution, FSC(N/2) would be measuring the similarity of those Fourier modes in A and B that are ''strictly worse'' that the resolution 1/(N/2), but ''better'' than the resolution 1/(N/2-1).
+With this definition, the first element in a FSC would measure the similarity of A and B restricted to the first shell A<sub>1</sub> and B<sub>1</sub>. This shell is just the single central Fourier pixel in both cases: the zero-th component of the Fourier transform. This will always be 1 (being the correlation of two scalars). In the other extreme of resolution, FSC(N/2) would be measuring the similarity of those Fourier modes in A and B that are ''strictly worse'' that the resolution 1/(N/2), but ''better'' than the resolution 1/(N/2-1).
 Don't get confused about the notation. FSC(i)|<sub>i=1</sub> corresponding to the Fourier component of order zero (and not one) just means that we start counting the components of the FSC vector with index i=1 (up to i=N/2). The first fourier component A(0) is traditionally defined as corresponding to

@@ Line 24: / Line 24: @@
 With this definition, the first element in a FSC would measure the similarity of A and B restricted to the first shell A<sub>1</sub> and B<sub>1</sub>. This shell is just the single central Fourier pixel in both cases: the zero-th component of the Fourier transform. In the other extreme of resolution, FSC(N/2) would be measuring the similarity of those Fourier modes in A and B that are ''strictly worse'' that the resolution 1/(N/2), but ''better'' than the resolution 1/(N/2-1).
-Don't get confused about the notation. FSC(i)|<sub>i=1</sub> corresponding to the Fourier component of order zero (and not one) just means that we start counting the components of the FSC vector with index i=1 (up to i=N/2). The first fourier component A(0) has fixed definition, and it corresponds to
+Don't get confused about the notation. FSC(i)|<sub>i=1</sub> corresponding to the Fourier component of order zero (and not one) just means that we start counting the components of the FSC vector with index i=1 (up to i=N/2). The first fourier component A(0) is traditionally defined as corresponding to
 <tt>
+              N
-A(k) =  sum  a(n)*exp(-j*2*pi*(k-1)*(n-1)/N), 1 <= k <= N.
+ A(k) =  sum  a(n)*exp(-j*2*pi*(k)*(n-1)/N), 0 <= k <= N-1.
              n=1
 </tt>
 for k = 0 (i.e, the total intensity of particle ''a'' in Fourier space).

@@ Line 24: / Line 24: @@
 With this definition, the first element in a FSC would measure the similarity of A and B restricted to the first shell A<sub>1</sub> and B<sub>1</sub>. This shell is just the single central Fourier pixel in both cases: the zero-th component of the Fourier transform. In the other extreme of resolution, FSC(N/2) would be measuring the similarity of those Fourier modes in A and B that are ''strictly worse'' that the resolution 1/(N/2), but ''better'' than the resolution 1/(N/2-1).
-Don't get confused about the notation. FSC(i)|<sub>i=1</sub> corresponding to the Fourier component of order zero (and not one) just means that we start counting the components of the FSC vector with index i=1 (up to i=N/2). The first fourier component A(0) corresponds to
+Don't get confused about the notation. FSC(i)|<sub>i=1</sub> corresponding to the Fourier component of order zero (and not one) just means that we start counting the components of the FSC vector with index i=1 (up to i=N/2). The first fourier component A(0) has fixed definition, and it corresponds to
 <tt>
-−
+             N
-       A(k) =     sum  a(n)*exp(-j*2*pi*(k-1)*(n-1)/N), 1 <= k <= N.
+A(k) =  sum  a(n)*exp(-j*2*pi*(k-1)*(n-1)/N), 1 <= k <= N.
-                    n=1
+            n=1
 </tt>
-for k = 0;
+for k = 0 (i.e, the total intensity of particle ''a'' in Fourier space).

@@ Line 22: / Line 22: @@
 #:: ''*''  represents pixelwise multiplication, |.| is the vector norm.
 </tt>
-With this definition, the first element in a FSC would measure the similarity of A and B restricted to the first shell A<sub>1</sub> and B<sub>1</sub>. This shell is just the single central Fourier pixel in both cases. In the other extreme of resolution, FSC(N/2) would be measuring the similarity of those Fourier modes in A and B that are ''strictly worse'' that the resolution 1/(N/2), but ''better'' than the resolution 1/(N/2-1).
+With this definition, the first element in a FSC would measure the similarity of A and B restricted to the first shell A<sub>1</sub> and B<sub>1</sub>. This shell is just the single central Fourier pixel in both cases: the zero-th component of the Fourier transform. In the other extreme of resolution, FSC(N/2) would be measuring the similarity of those Fourier modes in A and B that are ''strictly worse'' that the resolution 1/(N/2), but ''better'' than the resolution 1/(N/2-1).

@@ Line 20: / Line 20: @@
 #: compute cross correlation of vectors A<sub>i</sub>, B<sub>i</sub>:
 #:: FSC(i) = A<sub>i</sub> * B<sub>i</sub> / |A<sub>i</sub>| |B<sub>i</sub>|
-#:: ''*''  represents pixelwise multiplication, | . | is the vector norm.
+#:: ''*''  represents pixelwise multiplication, |.| is the vector norm.
 </tt>
 With this definition, the first element in a FSC would measure the similarity of A and B restricted to the first shell A<sub>1</sub> and B<sub>1</sub>. This shell is just the single central Fourier pixel in both cases. In the other extreme of resolution, FSC(N/2) would be measuring the similarity of Fouier modes that are strictly worse that the resolution 1/(N/2), but better than the resoultion 1/(N/2-1).

@@ Line 9: / Line 9: @@
 The command line that computes the FSC is {{docfunction|dynamo_fsc|dfsc}}. In its default settings, the FSC for two volumes with a sidelength of  ''N'' pixels would be a vector with a length of "N"/2 pixels.
 <tt>
-;compute A as fourier transform of real space volume a
+#compute A as fourier transform of real space volume a
-;compute B as fourier transform of real space volume a
+#compute B as fourier transform of real space volume a
-;for each i=1:N/2
+#for each i=1:N/2
-: compute a discretized shell n(i)
+#: compute a discretized shell n(i)
-::  a fourier pixel with a distance ''k'' from the center is in n(i) if   i-1<=|k|<i
+#::  a fourier pixel with a distance ''k'' from the center is in n(i) if   i-1<=|k|<i
-: locate elements of A and B  inside shell n(i)
+#: locate elements of A and B  inside shell n(i)
-:: A_i = A|n_i ; A vector with a
+#:: A<sub>i</sub> = A|n(i) ; a vector with as many entries as non-zero elements in n(i)
-:: B_i =
+#:: B<sub>i</sub> = B|n(i) ;
-: compute cross correlation of
+#: compute cross correlation of
-:: FSC(i) = A_i * B_i / |A_i| |B_i|
+#:: FSC(i) = A<sub>i</sub> * B_i / |A_i| |B_i|
 </tt>
 With this definition, the first element in a FSC would measure the similarity of the two first shells A_1,B_1.