The image distortion model

Although TD already compensates for small global changes, it is highly sensitive to local image transformations. We therefore propose the following image distortion model. When calculating the distance between two images $x$ and $\mu$ local deformations are allowed, i.e. the `best fitting' pixel in the reference image within a certain neighborhood $R_{ij}$ is regarded instead of computing the squared error between $x_{ij}$ and $\mu_{ij}$. Fig. 2 shows a 1D example for the IDM (left) where individual pixel displacements are independent, in comparison to TD (right), where displacements are coupled forming an affine transformation (here scaling). The resulting distance is

$$D_{\mbox{\tiny IDM}}(x,\mu)= \sum_{i,j} \mathop{\mbox{min}... ...{i^{\prime}j^{\prime}} \Vert + C_{iji^{\prime}j^{\prime}} \}$$ (4)

The cost function $C \geq 0$ represents the cost for deforming a pixel $x_{ij}$ in the input image to a pixel ${\mu}_{i^{\prime}j^{\prime}}$ in the reference image and is introduced to compensate for the fact that in an unrestricted distortion model (i.e. with $C\equiv 0$) wanted as well as unwanted transformations can be modeled. With growing neighborhood $R$ the admissible transformations may violate the assumption that they respect class-membership, but an appropriate choice of $R$ leads to a significant improvement of radiograph classification even when the cost function is disregarded. To determine the cost function $C$, one may want to learn it from the training data or choose it empirically, e.g. by using a weighted Euclidean distance between the corresponding pixel locations. This leads to a preference of local over long-range transformations.