Enhanced Visualization Methods for Computed Radiography Images

This paper focuses on the application of two image enhancement techniques for the picture archiving and communications systems imaging diagnostic workstation applied to computed radiography (CR) and digital radiography images. The first method is a contrast enhancement technique based on a class of nonlinear intensity transformations described by analytic transfer functions derived from Hurter and Driffield characteristic curves. The second method employs antialiasing techniques preventing the formation of Moiré patterns on subsampled CR images containing antiscatter grid lines, designed to achieve a good balance between artifact suppression and resolution degradation. These techniques are likely to become standard features for all high-end medical imaging workstations in the near future, and thus, we are suggesting that more powerful descriptions of these types of processing should be included in the Digital Imaging and Communications in Medicine standard.

Appendix A

Prefilter Design for Affine Warping

The reconstructed output g(x) in Figure 4 can be expressed in the general form as:

$$ \matrix {g{\left( {\mathbf{x}} \right)} = g_{{\text{c}}} \prime {\left( {\mathbf{x}} \right)} = {\int\limits_{{\mathbf{t}} \in {\Re }^{2} } {f_{{\text{c}}} {\left( {m^{{ - 1}} {\left( {\mathbf{t}} \right)}} \right)}h{\left( {{\mathbf{x}} - {\mathbf{t}}} \right)}} }\,{\text{d}}{\mathbf{t}} = {\int\limits_{{\mathbf{k}} \in {\Re }^{2} } {f_{{\text{c}}} {\left( {m^{{ - 1}} {\left( {\mathbf{t}} \right)}} \right)}} }{\sum\limits_{{\mathbf{k}} \in {\Im }^{2} } {f{\left( {\mathbf{k}} \right)}} }r{\left( {m^{{ - 1}} {\left( {\mathbf{t}} \right)} - {\mathbf{k}}} \right)}{\text{d}}{\mathbf{t}}} \\ { = {\sum\limits_{{\mathbf{k}} \in {\Im }^{2} } {f{\left( {\mathbf{k}} \right)}} }\,\rho {\left( {{\mathbf{x}},{\mathbf{k}}} \right)}} \ $$

where

$$ \rho {\left( {{\mathbf{x}},{\mathbf{k}}} \right)} = {\int\limits_{{\mathbf{t}} \in {\Re }^{2} } {h{\left( {{\mathbf{x}} - {\mathbf{t}}} \right)}} }r{\left( {m^{{ - 1}} {\left( {\mathbf{t}} \right)} - {\mathbf{k}}} \right)}{\text{d}}{\mathbf{t}} $$

is the resampling filter that specifies the weight of the input sample at location k for an output sample at location x as an output space integral. In the input space, the resampling filter is given by:

$$ \rho {\left( {{\mathbf{x}},{\mathbf{k}}} \right)} = {\int\limits_{{\mathbf{u}} \in {\Re }^{2} } {h{\left( {{\mathbf{x}} - m{\left( {\mathbf{u}} \right)}} \right)}} }r{\left( {{\mathbf{u}} - {\mathbf{k}}} \right)}{\left| {\mathbf{J}} \right|}{\text{d}}{\mathbf{u}} $$

where ∣J∣ is the determinant of the Jacobian of the mapping function m(u).

When the warping is an affine transformation, the Jacobian matrix is constant, and the resampling filter can be expressed as a convolution between the reconstruction filter and the warped prefilter as follows:

$$ \rho {\left( {{\mathbf{x}},{\mathbf{k}}} \right)} = \rho \prime {\left( {m^{{ - 1}} {\left( {\mathbf{x}} \right)} - {\mathbf{k}}} \right)}\,\,{\text{where}}\,\rho \prime {\left( {\mathbf{u}} \right)} = h\prime {\left( {\mathbf{u}} \right)} \times r{\left( {\mathbf{u}} \right)} = {\left| {\mathbf{J}} \right|}h{\left( {{\mathbf{uJ}}} \right)} \times r{\left( {\mathbf{u}} \right)} $$

The entire resampling process can then be reduced to a single convolution:

$$ g{\left( {\mathbf{x}} \right)} = {\sum\limits_{{\mathbf{k}} \in {\Im }^{2} } {f{\left( {\mathbf{k}} \right)}} }\rho \prime {\left( {m^{{ - 1}} {\left( {\mathbf{x}} \right)} - {\mathbf{k}}} \right)} = {\left( {f \times \rho \prime } \right)}{\left( {m^{{ - 1}} {\left( {\mathbf{x}} \right)}} \right)} $$

In the case of magnification, one could ignore the prefilter h because no high frequencies are introduced into the output upon magnification. On the other hand, minification introduces high frequencies but does not require any reconstruction of the input image because the input grid covers the output grid everywhere. Consequently, in this case, we can ignore the reconstruction filter r.

It is noticeable here that the reconstruction filter expressed in the input image space is invariant with respect to the parameters of the magnification transformation. In the frequency domain, this is reflected in the fact that the frequency spectrum of the magnified image does not contain frequency components higher than the input image.

In the case of minification, the output image is a subsampled version of the input image, and thus its frequency spectrum may contain higher frequencies. This is evident in the above equations by the dependency of the resampling filter on the inverse warp mapping, namely, that the Jacobian of the warp mapping is proportional to the minification factor.