Changes between Version 9 and Version 10 of DevWikiAffineParameters

Timestamp:

Nov 27, 2011, 2:01:42 PM (12 years ago)

Author:

bnordgren

Comment:

—

DevWikiAffineParameters

@@ -115 +115 @@
  * x' = o,,11,, i + o,,12,, j
  * y' = o,,21,, i + o,,22,, j
+
+== Calculating terms ==
+
+The linear transform matrix '''O''', calculated in the previous section, contained some terms which need to be calculated from the provided, physically significant inputs. These terms are s,,i,,, s,,j,,, k,,i,, and f. 
+
+=== Reflection term ===
+
+The coefficient ''f'' essentially represents a decision as to whether the ''j'' axis needs to be flipped. If f=1, the j axis is not flipped. If f=-1, the j axis is flipped. The decision of whether to flip the j axis is controlled by the θ,,ij,, input parameter as follows: 
+
+ * if θ,,ij,, < 0 then f = -1
+ * if θ,,ij,, >= 0 then f = 1 
+
+=== Scaling along i axis ===
+
+Scaling along the i axis is represented by the s,,i,, term. We will set it using the magnitude of the '''i,,b,,''' basis vector. This section will go about proving that this is a valid thing to do. 
+
+To simplify things and remove clutter from the equation, note that the final component of the aggregate transform (rotation) does not affect the magnitude of the basis vector. So, the i unit vector projected thru a non-rotating transform is calculated as follows:
+
+[[Image(basisvector_i.png)]]
+
+The magnitude of '''i,,b,,''' is given as follows: 
+
+[[Image(basisvectormag_i.png)]]
+
+From the above: 
+
+ * s,,i,, = magnitude of '''i,,b,,'''
+
+=== Shearing parallel to i axis ===
+
+Any value of θ,,ij,, which is not plus or minus 90 degrees indicates that shearing parallel to the i axis is required. This produces diamond shaped (non-rectangular) pixels in the output. In this section, we calculate a value for k,,i,, from the specific value of θ,,ij,,. We do this using the configuration in the following diagram: 
+
+[[Image(shearing_params.png)]]
+
+In this diagram, θ,,s,, represents the shearing angle, which is a [http://en.wikipedia.org/wiki/Complementary_angles complementary angle] to θ,,ij,,. The provided equation defines θ,,s,, = 90f - θ,,ij,, in order to have a single equation which is correct whether the j axis is flipped or not.
+
+ * k,,i,, = tan(90f - θ,,ij,,)
+
+This equation passes the basic sanity check that k,,i,,=0 when θ,,ij,, is plus or minus 90 and f is defined as above. Note that θ,,ij,, = 0 or 180 degrees is an error. This is not just a mathematical artifact. The basis vectors are not allowed to be parallel to each other because then you no longer have a grid.
+
+=== Scaling along the j axis ===
+
+Scaling along the j axis is represented by the s,,j,, term. We will set it using the magnitude of the '''j,,b,,''' basis vector and the above defined terms. This will be more complex than the corresponding calculation for the i axis, due to the possible presence of shearing.
+
+The '''j,,b,,''' basis vector is represented as follows: 
+
+[[Image(basisvector_j.png)]]
+
+The magnitude of '''j,,b,,''' is then calculated as follows:
+
+[[Image(basisvectormag_j.png)]]
+
+Substituting for k,,i,, and solving for s,,j,,: 
+
+[[Image(scalefactor_j.png)]]
+
 
 == Constructing the affine transformation ==