[Insight-users] Doing Shearing by using Rotation + non-homogeneous Scaling

[Dennis Wenzel]

Hello Luis,

Perhaps I should have clarified my statement.  Yes, you have shown that a 
shearing transform can be decomposed into "a specific" sequence of rotations 
and non-uniform scales.  However, for practical purposes, I am interested in 
decomposing a general affine transform into a predetermined sequence of 
rotations, scales, and translations.  So, that was the basis of my 
statement, i.e., that you could not decompose a general affine transform, 
e.g., one that contains skew, into a specific sequence of rotations, scales, 
and translations.

In practice, I am interested in registration which handles rigid body 
alignment (rotation and translation) and scale which adjusts for 
miscalibrated sampling in a scanner such as an MRI.

Dennis

----- Original Message ----- 
From: "Luis Ibanez" <luis.ibanez at kitware.com>
To: "Dennis Wenzel" <wenzeld at worldnet.att.net>
Cc: "Insight Users" <insight-users at itk.org>
Sent: Thursday, August 11, 2005 5:50 PM
Subject: Re: [Insight-users] Doing Shearing by using Rotation + 
non-homogeneous Scaling


>
> Hi Dennis,
>
>
> Here is how you can decompose a Shearing Transform into
> a combination of rotations and non-homogeneous scaling
> operations:
>
>
> 1) Apply a 45 degrees rotation transform
>
>      | r  -r |
>      | r   r |
>
>    where r = sqrt(2)/2
>
>
> 2) Apply a non-homogeneous scaling by a factor k
>
>     | 1  0 |
>     | 0  k |
>
>
> 3) Apply a rotation that will align the original Y
>    axis with the current one
>
>
>     |  k  1 |
>     |       | . 1/sqrt( 1 + k^2 )
>     | -1  k |
>
>
> 4) Correct for the differenct between the horizontal
>    and vertical expansions, using another
>    non-homogeneouos scaling:
>
>
>     | 1/(k.sqrt(2))            0        |
>     |       0         1/sqrt(2.(1+k^2)) |
>
>
>
> This sequence of operation will be equivalent to
> the shearing that you proposed:
>
>
>    |   1    0  |
>    |   S    1  |
>
>
>  as long as we choose k to satisfy the relationship
>
>
>       S = ( k^1 - 1 ) / sqrt( k^2+1)
>
>
>
>
>
> A more elegant descripcion could be done using elements
> of transformational geometry.. but unfortunately is
> harder to represent it in an email.
>
>
> For the intuitive support of the opreration you may
> consider the analogy of taking a matrix and computing
> its UL decomposition, which after all, is like applying
> two shearing operations one after another.  You may
> also want to consider the analogy of QR matrix
> decomposition.
>
>
> ...The transformational geometry equivalent is still the
> most elegant though...
>
>
>
>    Regards,
>
>
>       Luis
>
>
> -------------------
> Dennis Wenzel wrote:
>> Hello Luis,
>>
>>> I will be interested in seeing your counter examples that
>>> indicate that non-homogeneous scaling combined with
>>> rotation is not equivalent to shearing.
>>
>>
>> Ok, given the following 4x4 matrix with shear_xy
>>
>> | 1    0 0 0 |
>> | s_xy 1 0 0 |
>> | 0    0 1 0 |
>> | 0    0 0 1 |
>>
>> How do you decompose this into rotations and scale only?
>>
>>
>>>
>>> About the ScaleSkewVersor transform... if you dont want
>>> skew to happen, then simply set the skew parameters to
>>> zero...
>>
>>
>> That is fine if I want to *set* the skew parameters, but the optimizer(s) 
>> know nothing about which parameters *not* to try in a particular 
>> transform, i.e. to skip the skew parameters.  My whole concern is about 
>> the transform determined by the optimizer!
>>
>>>
>>>
>>> Otherwise, plan B, is for you to take the source code
>>> of the Similarity3DTransform or the ScaleSkewVersorTransform
>>> and modify it to suit your needs.  If you do so, we will
>>> be happy to include your new transform into the toolkit.
>>> Writing a transform is not too difficult. You mainly have
>>> to work on the "TransformPoint" method and the "GetJacobian"
>>> method.
>>
>>
>> I had begun to decide that is something I am probably going to have to 
>> do. I will start to look at Similarity3DTransform (downloaded from CVS) 
>> to see what is involved.
>>
>>>
>>> Please let us know if you need any assistance in writing
>>> these new Transform class.
>>
>>
>> Thanks.  I appreciate that.
>>
>> Dennis
>>
>>
>
>