Volume center - Revision history

Daniel Castaño at 09:01, 5 March 2018

2018-03-05T09:01:02Z

Daniel Castaño at 08:59, 5 March 2018

2018-03-05T08:59:56Z

Daniel Castaño at 10:43, 26 October 2017

2017-10-26T10:43:07Z

Daniel Castaño: /* Direct space */

2017-02-28T13:27:01Z

Direct space

Daniel Castaño at 13:26, 28 February 2017

2017-02-28T13:26:37Z

Daniel Castaño at 08:00, 28 April 2016

2016-04-28T08:00:43Z

Daniel Castaño at 07:59, 28 April 2016

2016-04-28T07:59:09Z

Daniel Castaño: /* Fourier space */

2016-04-28T07:58:38Z

Fourier space

Daniel Castaño at 07:58, 28 April 2016

2016-04-28T07:58:24Z

Daniel Castaño: Created page with "Category: Geometry Category: Conventions The center of a discretized volume can be defined in different ways. ''Dynamo'' uses the same convention in all its functions..."

2016-04-28T07:16:59Z

Created page with "Category: Geometry Category: Conventions The center of a discretized volume can be defined in different ways. ''Dynamo'' uses the same convention in all its functions..."

New page

[[Category: Geometry]]
[[Category: Conventions]]

The center of a discretized volume can be defined in different ways. ''Dynamo'' uses the same convention in all its functions. For the formulas in this page, remember also that ''Dynamo'' uses the convention of counting pixels with the index 1, (not 0). The pixel "on the corner" with the first coordinates in a volume {{t|V}} is thus denoted as {{t|V[1,1,1]}}.

We use the notation <tt>[ i,j,k]</tt> for pixel indices.

==Direct space==

For a volume of {{t|V}} NxNxN pixels, ''Dynamo'' considers the pixel located at [N/2+1,N/2+1,N/2+1] as the center of {{t|V}}

===Rotation===
Rotation needs defining on a concrete ''point'', not on a pixel (as a pixel is actually a square with an area). The convention in ''Dynamo'' is to define the rotation center point in the middle of the center pixel

Note that the center is defined as ''point'', not as a pixel.

===Placing templates in a Volume===

Sometimes you need to place a template {{t|T}} inside a bigger volume {{t|V}}, by passing a pixel coordinate {{t|r}} inside the bigger volume. In such cases the placement follows by setting the central pixel [N/2+1,N/2+1,N/2+1] at the pixel position {{t|r}} in {{t|V}}, i.e.:

<tt>
V[x-N/2 : x+N/2-1 , y-N/2 : y+N/2-1, z-N/2 : z+N/2-1] = T[:,:,:];
</tt>

==Fourier space==

← Older revision		Revision as of 09:01, 5 March 2018
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	== During particle cropping==		== During particle cropping==

−	Particles are never interpolated when extracted from a volume, i.e., a file that represents a particle extracted from a tomogram will contain on each voxel the same intensity value that the analogous voxel had in the tomogram. If the center of a particle indicated by the table used for cropping is not the center of a voxel, the difference between the coordinate in the table and the center of the voxel will be added in the columns 4:6 (the ''shift'' columns) in the output table ''crop.tbl''.	+	Particles are never interpolated when extracted from a volume, i.e., a file that represents a particle extracted from a tomogram will contain on each voxel the same intensity value that the analogous voxel had in the tomogram. If the center of a particle indicated by the table used for cropping is not the center of a voxel (i.e., not a semiinteger value), the difference between the coordinate in the table and the center of the voxel will be added in the columns 4:6 (the ''shift'' columns) in the output table ''crop.tbl''. In this crop.tbl, the columns 24:26 will always have semiinteger values (i.e., centers of voxels)

← Older revision		Revision as of 08:59, 5 March 2018
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	Note: When computing the [[Fourier Shell Correlation]] of two volumes, the central pixel is left aside from the computation, as this ''shell'' would only contain one component, and the FSC component relative to this shell will always be 1.		Note: When computing the [[Fourier Shell Correlation]] of two volumes, the central pixel is left aside from the computation, as this ''shell'' would only contain one component, and the FSC component relative to this shell will always be 1.
		+
		+	== During particle cropping==
		+
		+	Particles are never interpolated when extracted from a volume, i.e., a file that represents a particle extracted from a tomogram will contain on each voxel the same intensity value that the analogous voxel had in the tomogram. If the center of a particle indicated by the table used for cropping is not the center of a voxel, the difference between the coordinate in the table and the center of the voxel will be added in the columns 4:6 (the ''shift'' columns) in the output table ''crop.tbl''.

@@ Line 9: / Line 9: @@
 We use the notation <tt>[i,j,k]</tt> for pixel indices, and the notation ''(x,y,z)'' for continuous coordinates (or simply ''coordinates''). The origin of coordinates in a volume is the ''corner'' of the first pixel <tt>[1,1,1]</tt>. This way, the coordinate ''(N,N,N)'' is located exactly in the corner oposite to the origin of coordinates.
 ==Direct space==
@@ Line 19: / Line 18: @@
 This might seem inelegant, as the geometrical center of the volume is clearly located at the position ''(N/2,N/2,N/2)'', (which is located in a corner shared by eight pixels). However, ''Dynamo'' skips to the ''(N/2+1/2,N/2+1/2,N/2+1/2)''  convention, as it keeps coherency with the definition of center in  Fourier space (where it is important that the coordinate center falls in the middle of a fourier pixel), and it is also a convention widely used in EM software packages.
 ===Placing templates in a Volume===

← Older revision		Revision as of 08:00, 28 April 2016
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	Note that this implies that the Fourier pixels conveying the Nyquist frequency in a direction are NOT to be found at the most positive indices of the Fourier volume. For instance, you might expect the Nyquist frequency for waves in the x direction to be conveyed in the pixel <tt>[N,N/2+1,N/2+1]</tt>. That's wrong: this pixel is located at a distance (measured between pixel centers) of N/2-1 pixels. The actual Nyquist frequency corresponds to pixel <tt>[0,N/2+1,N/2+1]</tt>, which is at the correct distance. Remember, the theory of Discrete Fourier Transform on ''real'' volumes (as density maps are) tells us that all components of the power spectrum are center symmetric.... other than the Nyquist component, that appears only one for each direction (thus no redundancy)		Note that this implies that the Fourier pixels conveying the Nyquist frequency in a direction are NOT to be found at the most positive indices of the Fourier volume. For instance, you might expect the Nyquist frequency for waves in the x direction to be conveyed in the pixel <tt>[N,N/2+1,N/2+1]</tt>. That's wrong: this pixel is located at a distance (measured between pixel centers) of N/2-1 pixels. The actual Nyquist frequency corresponds to pixel <tt>[0,N/2+1,N/2+1]</tt>, which is at the correct distance. Remember, the theory of Discrete Fourier Transform on ''real'' volumes (as density maps are) tells us that all components of the power spectrum are center symmetric.... other than the Nyquist component, that appears only one for each direction (thus no redundancy)

−		+	Note: When computing the [[Fourier Shell Correlation]] of two volumes, the central pixel is left aside from the computation, as this ''shell'' would only contain one component, and the FSC component relative to this shell will always be 1.
−	When computing the [[Fourier Shell Correlation]] of two volumes, the central pixel is left aside from the computation.

@@ Line 34: / Line 34: @@
 When we depict a power spectrum, the center pixel <tt> [N/2+1,N/2+1,N/2+1]</tt> corresponds to the zero-th Fourier order. The magnitude of this pixel does not represent an oscillation, but just the average intensity of the volume in the current space. Luckily, this is a convention used in many EM packages an also in Matlab.
-Note that this implies that the Fourier pixels conveying the Nyquist frequency in a direction are NOT to be found at the most positive indices of the Fourier volume. For instance, you might expect the Nyquist frequency for waves in the x direction to be conveyed in the pixel <tt>[N,N/2+1,N/2+1]</tt>. That's wrong: this pixel is located at a distance (measured between pixel centers)  of N/2-1 pixels. The actual Nyquist frequency corresponds to pixel <tt>[0,N/2+1,N/2+1]<tt>, which is at the correct distance.  Remember, the theory of Discrete Fourier Transform on ''real'' volumes (as density maps are) tells us that all components of the power spectrum are center symmetric.... other than the Nyquist component, that appears only one for each direction (thus no redundancy)
+Note that this implies that the Fourier pixels conveying the Nyquist frequency in a direction are NOT to be found at the most positive indices of the Fourier volume. For instance, you might expect the Nyquist frequency for waves in the x direction to be conveyed in the pixel <tt>[N,N/2+1,N/2+1]</tt>. That's wrong: this pixel is located at a distance (measured between pixel centers)  of N/2-1 pixels. The actual Nyquist frequency corresponds to pixel <tt>[0,N/2+1,N/2+1]</tt>, which is at the correct distance.  Remember, the theory of Discrete Fourier Transform on ''real'' volumes (as density maps are) tells us that all components of the power spectrum are center symmetric.... other than the Nyquist component, that appears only one for each direction (thus no redundancy)
 When computing the [[Fourier Shell Correlation]] of two volumes, the central pixel is left aside from the computation.

@@ Line 19: / Line 19: @@
 This might seem inelegant, as the geometrical center of the volume is clearly located at the position ''(N/2,N/2,N/2)'', (which is located in a corner shared by eight pixels). However, ''Dynamo'' skips to the ''(N/2+1/2,N/2+1/2,N/2+1/2)''  convention, as it keeps coherency with the definition of center in  Fourier space (where it is important that the coordinate center falls in the middle of a fourier pixel), and it is also a convention widely used in EM software packages.
-===Placing templates in a Volume===
+===Placing templates in a volume===
 Sometimes you need to place a template {{t|T}}  inside a bigger volume {{t|V}}, by passing a pixel coordinate {{t|r}} inside the bigger volume. In such cases the placement follows by setting the central pixel [N/2+1,N/2+1,N/2+1] at the pixel position {{t|r}} in {{t|V}}, i.e.:
@@ Line 26: / Line 26: @@
 V[x-N/2 : x+N/2-1 , y-N/2 : y+N/2-1, z-N/2 : z+N/2-1] = T[:,:,:];
 </tt>
 ==Fourier space==

@@ Line 4: / Line 4: @@
 The center of a discretized volume can be defined in different ways. ''Dynamo'' uses the same convention in all its functions. For the formulas in this page, remember also that ''Dynamo'' uses the convention of counting pixels with the index 1, (not 0). The pixel "on the corner" with the  first coordinates in a volume {{t|V}} is thus denoted as {{t|V[1,1,1]}}.
-We use the notation <tt>[ i,j,k]</tt> for pixel indices.
+==Pixels and continuous coordinates==
 ==Direct space==
-For a volume of {{t|V}} NxNxN pixels, ''Dynamo'' considers the pixel located at [N/2+1,N/2+1,N/2+1] as the center of {{t|V}}
+For a volume of {{t|V}} NxNxN pixels, ''Dynamo'' considers by convention the pixel located at <tt> [N/2+1,N/2+1,N/2+1]</tt> as the central pixel of {{t|V}}.
 ===Rotation===
-Rotation needs defining on a concrete ''point'', not on a pixel (as a pixel is actually a square with an area). The convention in ''Dynamo'' is to define the rotation center point in the middle of the center pixel
+Rotation needs defining on a concrete ''point'', not on a pixel (as a pixel is actually a square with an area). The convention in ''Dynamo'' is to define the rotation center point in the middle of the [[#Direct space|center pixel]], thus at the continuous coordinate ''(N/2+1/2,N/2+1/2,N/2+1/2)''.
 ===Placing templates in a Volume===
@@ Line 26: / Line 31: @@
 ==Fourier space==