Differences between revisions 53 and 54

Contents

What is a Transform?
The Transform object in Python
2D degenerate use of the Transform object in Python
A more in depth look at the dictionary constructor
A more in depth look at set_params
1. If unspecified, old parameters are retained
2. For rotation and translation, setting one is equivalent to setting them all

What is a Transform?

We use the Transform class for storing/managing Euler angles,translations, scales and x mirroring. At any time a Transform object $$Tr$$ defines a group of 4 transformations of a rigid body that are applied in a specific order, namely

$$ Tr \equiv M T S R $$

Where $$M$$ is a mirroring operation about the x-axis, $$T$$ is a translation, $$S$$ is a uniform, positive, non zero scaling operation and $$R$$ is a rotation. The Transform object stores these transformations internally in a 3x4 matrix, which appears as

$$ Tr = [sMR,M\mathbf{t}]$$

Where $$s$$ is the constant scaling factor, $$M$$ is the identity or the x-mirroring matrix, $$R$$ is a $$3x3$$ rotation matrix and $$\mathbf{t}=(tx,ty,tz)^T$$ is the translation.

The Transform object in Python

Constructing a Transform

There a four ways to construct a Transform object in Python

   1 t = Transform() # default constructor, t is the identity
   2 t = Transform({"type":"eman","az":10,"alt":150,"scale":2.0,"mirror":True,"tx":3.4}) # construction using a dictionary
   3 p = Transform({"type":"eman","az":10,"mirror":True,"tx":3.4}) # construction using a dictionary
   4 q = Transform([1,0,0,0,0,1,0,0,0,0,1,0]) # specify the 12 entries of the matrix explicitly, row wise (3 rows of 4)
   5 s = Transform(t) # copy construction - s is precisely the same as t

For more information on default parameters and what happens when some parameters are not set see a more in depth look at the dictionary constructor

Setting/getting rotations

   1 t = Transform()
   2 t.set_rotation({"type":"spider","phi":32,"theta":12,"psi":-100})
   3 spider_rot = t.get_rotation("spider")
   4 eman_rot = t.get_rotation("eman")
   5 s = Transform(eman_rot) # works fine

Setting/getting scale

   1 t = Transform()
   2 t.set_scale(2.0)
   3 scale = t.get_scale()
   4 s = Transform({"scale":scale}) # set scale as part of construction

Setting/getting mirror

   1 t = Transform()
   2 #  Can be set using integers
   3 t.set_mirror(1) # means True
   4 t.set_mirror(0) # means False
   5 t.set_mirror(False)
   6 t.set_mirror(True)
   7 mirror = t.get_mirror()
   8 s = Transform({"mirror":mirror}) # set mirror as part of construction

Setting/getting translation

   1 t = Transform()
   2 t.set_trans(1,2,3) # method 1
   3 t.set_trans(Vec3f(1,2,3)) # method 2
   4 t.set_trans([1,2,3]) # method 3 - the tuple is converted to a Vec3f automatically
   5 v = t.get_trans()
   6 s = Transform("tx":v[0],"ty":v[1],"tz":v[2]) # set translation as part of construction

Setting/getting parameters

You can tell a Transform deduce any of its parameters from a dictionary. Similarly you can get the parameters of a Transform as a dictionary

   1 t = Transform()
   2 t.set_params({"type":"eman","az":10,"alt":150,"scale":2.0,"mirror":True,"tx":3.4})
   3 d = t.get_params("eman") # must specify the euler convention
   4 s = Transform(d) # s is the same as t
   5 d = t.get_params("spider")
   6 s = Transform(d) # s is the same as t
   7 d = t.get_params("matrix")
   8 s = Transform(d) # s is the same as t

For more information the behaviour of set_params see a more in depth look at set_params

A Transform multiplied by a Vec3f

The transformation of a 3D vector $$ v = (v_x,v_y,v_z)^T $$ by a Transform is defined as

$$ Tr \mathbf{v} = [sMR,M\mathbf{t}] (v_x,v_y,v_z,1)^T = sMR\mathbf{v}+ M\mathbf{t} $$

This can be done in Python using the following

   1 t = Transform()
   2 t.set_params({"type":"eman","az":10,"alt":150,"scale":2.0,"mirror":True,"tx":3.4})
   3 v = Vec3f(3,4,10)
   4 v_transformed = t*v
   5 v_transformed = t.transform(v) # can also do it this way if you prefer

Transforming a 3D image

Transforming a 3D image is achieved using a single function call

   1 t = Transform()
   2 t.set_params({"type":"eman","az":10,"alt":150,"scale":2.0,"mirror":True,"tx":3.4})
   3 t.set_trans( ... )
   4 e = test_image_3d()
   5 e.transform(t)

You can alternatively use the EMData processing framework, for example:

   1 t = Transform()
   2 t.set_params( ... )
   3 t.set_trans( ... )
   4 f = e.process("math.transform",{"transform":t} # memory efficient
   5 e.proceess_inplace("math.transform",{"transform":t})

In fact e.transform(t) actually calls e.proceess_inplace("math.transform",{"transform":t}) internally on the C++ side.

2D degenerate use of the Transform object in Python

The Transform object can be used as though it were a 2D transformation matrix. In this case the interface for setting/getting scale and mirror is unchanged. However setting the rotation should be done using the euler type "2d", and there are some accommodations for setting and getting 2d translations and parameters.

Setting/getting 2D rotations

Use the "2d" euler type, and the angle is specified using "alpha"

   1 t = Transform()
   2 t.set_rotation({"type":"2d","alpha":32})
   3 a = t.get_rotation("2d")
   4 s = Transform(a) # works fine

Setting/getting 2D translation

The interface is more or less identical to the 3D case

   1 t = Transform()
   2 t.set_trans(1,2) # method 1
   3 t.set_trans(Vec2f(1,2)) # method 2
   4 v = t.get_trans_2d()
   5 s = Transform("tx":v[0],"ty",v[1]) # set translation as part of construction

Setting/getting 2D parameters

For getting the parameters in 2D form use the following

   1 t = Transform()
   2 t.set_params({"type":"2d","alpha":10,"scale":2.0,"mirror":True,"dx":3.4,"dy":0.0}) # no special interface required for 2D, use the same as 3D
   3 d = t.get_params("2d")
   4 s = Transform(d) # s is the same as t

A Transform multiplied by a Vec2f

The transformation of a 2D vector $$ v_{2D} = (v_x,v_y)^T $$ by a Transform is equivalent to setting the z component of 3D vector to 0. If $$ v = (v_x,v_y,0)^T $$ then the 2D transformation is equivalent to the following,

$$ Tr \mathbf{v_{2D}} = [sMR,M\mathbf{t}] (v_x,v_y,0,1)^T = sMR\mathbf{v}+ M\mathbf{t} $$

and the z component is ignored. Note that the internal implementation is efficient. 2D vector transformation can be done in Python using the following

   1 t = Transform()
   2 t.set_params({"type":"2d","alpha":10,"scale":2.0,"mirror":True,"tx":3.4})
   3 v = Vec2f(3,1)
   4 v_transformed = t*v
   5 v_transformed = t.transform(v) # can also do it this way if you prefer

A more in depth look at the dictionary constructor

The dictionary constructor, which is equivalent to calling the default constructor followed by the set_params function, takes up to 9 parameters as exemplified in the following

   1 t = Transform({"type":"eman","az":10,"alt":150,"phi":20,"scale":2.0,"mirror":True,"tx":3.4,"ty":3,"tz":2})
   2 t = Transform({"type":"spider","phi":10,"theta":150,"psi":20,"scale":2.0,"mirror":True,"tx":3.4,"ty":3,"tz":2})

If any of the angles or not specified they are implicitly set to 0

   1 t = Transform({"type":"spider","phi":10,"scale":2.0,"mirror":True,"tx":3.4,"ty":3,"tz":2}) # spider theta and psi are both 0
   2 t = Transform({"type":"spider","scale":2.0,"mirror":True,"tx":3.4,"ty":3,"tz":2}) # spider phi, theta and psi are all 0

If no euler type is specified then the rotation matrix is the identity

   1 t = Transform({"scale":2.0,"mirror":True,"tx":3.4,"ty":3,"tz":2}) # Rotation matrix is the identity

Similarly if any of the translation parameters are not specified they are implicitly set to 0

   1 t = Transform({"scale":2.0,"mirror":True,"tx":3.4,"ty":3}) # tz is 0
   2 t = Transform({"scale":2.0,"mirror":True}) # tx,ty and tz are all 0

If scale is not specified it is by default 1.0, similarly if mirror is not specified it is be default False

   1 t = Transform({"tx":3.4,"ty":3,"tz":3}) # scale is 1.0, mirror is False

A more in depth look at set_params

Constructing a Transform with some dictionary is the same as calling set_params on a Transform that is the identity.

   1 d = {"type":"eman","az":10,"alt":150,"phi":20,"scale":2.0,"mirror":True,"tx":3.4,"ty":3,"tz":2}
   2 t = Transform(d) # t is initialized as the identity and then set_params(d) is called internally
   3 s = Transform() # s is the identity
   4 s.set_params(d) # s is identical to t

The main difference between calling set_params explicitly on a Transform as opposed to constructing a Tranform with a dictionary lies in the fact that the construction method first sets the Transform object to the identity before calling set_params. So if calling the set_params function on a Transform that has already been initialized and had many things done than you should be aware of the following...

If unspecified, old parameters are retained

   1 t = Transform({"type":"eman","az":10,"alt":150,"phi":20,"scale":2.0,"mirror":True,"tx":3.4,"ty":3,"tz":2})
   2 d = {"tx":23,"ty":23,"tz":12}
   3 t.set_params(d) # t still has the same rotation, scale and mirror but it has an updated translation

For rotation and translation, setting one is equivalent to setting them all

   1 t = Transform({"tx":3.4,"ty":3,"tz":2})
   2 d = {"tx":23}
   3 t.set_params(d) # ty and tz are now 0
   4 #
   5 t = Transform({"type":"eman","az":10,"alt":150,"phi":20})
   6 d = {"type":"eman","phi":10}
   7 t.set_params(d) # alt and az are 0
   8 t.set_params({"type":eman}) # all angles are now 0

-  ⇤ ← Revision 53 as of 2009-02-04 20:38:48 → 
  Size: 9749
  Editor: DavidWoolford
  Comment:
+   ← Revision 54 as of 2009-02-04 20:45:44 → ⇥
  Size: 9673
  Editor: DavidWoolford
  Comment:
-Deletions are marked like this.
+Additions are marked like this.
 Line 5:
-Line 20:
+Line 18:
-Line 30:
+Line 27:
-Line 34:
+Line 30:
-Line 42:
+Line 37:
-Line 44:
+Line 38:
-Line 51:
+Line 44:
-Line 53:
+Line 45:
-Line 65:
+Line 55:
-Line 67:
+Line 56:
-Line 76:
+Line 64:
-Line 78:
+Line 65:
-Line 91:
+Line 77:
-For more information the behaviour of set_params see  [[#set_params|a more in depth look at set_params ]]
+For more information the behaviour of set_params see  [[#set_params|a more in depth look at set_params]]
-Line 95:
+Line 80:
-Line 109:
+Line 93:
-Line 111:
+Line 94:
-Line 117:
+Line 99:
-t.set_trans(...)
+t.set_trans( ... )
-Line 121:
+Line 103:
-Line 126:
+Line 107:
-t.set_params(...)
t.set_trans(...)
+t.set_params( ... )
t.set_trans( ... )
-Line 131:
+Line 112:
-In fact {{{e.transform(t)}}} actually calls {{{e.proceess_inplace("math.transform",{"transform":t})}}} internally on C++ side.
+In fact {{{e.transform(t)}}} actually calls {{{e.proceess_inplace("math.transform",{"transform":t})}}} internally on the C++ side.
-Line 139:
+Line 119:
-Line 147:
+Line 126:
 }}}
-Line 151:
+Line 128:
-Line 161:
+Line 137:
-Line 164:
+Line 138:
-Line 173:
+Line 146:
-Line 175:
+Line 147:
-Line 189:
+Line 160:
-Line 194:
+Line 164:
-Line 196:
+Line 165:
-Line 200:
+Line 170:
-Line 207:
+Line 176:
-Line 213:
+Line 181:
-Line 220:
+Line 187:
-Line 226:
+Line 192:
-Line 230:
+Line 194:
-= A more in depth look at set_params  =
----
Constructing a Transform with some dictionary is the same as calling set_params on a Transform that is the identity.
+= A more in depth look at set_params =
----
Constructing a Transform with some dictionary is the same as calling set_params on a Transform that is the identity.
-Line 240:
+Line 204:
-Line 243:
+Line 206:
-Line 245:
+Line 207:
-Line 251:
+Line 212:
-== For rotation and translation, setting one is equivalent to setting them all  ==
+== For rotation and translation, setting one is equivalent to setting them all ==