The goal of the method presented in this work is to find and describe significant differences between a pair of chick embryo images. A deformable registration is used to assess local differences at every point between two images, or an image and a mirrored copy. Regions of interest are identified and features are extracted that will be used to characterize the regions.

The deformable registration determines the spatial transform mapping points from a source to homologous points on an object in a target image. The output is a dense deformation vector field in which the vector at each point describes the spatial transformation of that point. When applied to three-dimensional images of two objects, these vectors reflect the structural differences between the source and target objects. For this application, a B-spline deformable transform using a mutual information metric was chosen, since it is widely applicable and computationally efficient.

To interpret the deformation vector field in a meaningful way, it is necessary to define which differences between two images are significant. For this application, differences in a region showing organization are assumed to be significant. This may indicate areas where an underlying process is directing the difference in shape, in contrast to random fluctuations. While all vectors in such a region may not have similar values for properties such as magnitude or angle, variations should occur smoothly over the surface.

A. Low-Level Vector Field Properties

To locate regions in the deformable vector field with some form of organization, three low-level vector properties were chosen. The properties selected are:

- deformation vector magnitude,
- cosine distance between the deformation vector and the surface normal vector,
- cosine distance between the deformation vector and a predefined reference vector.

Each property can be extracted from the vector field and used as low-level features or to calculate mid-level features.

B. Mid-Level Vector Field Properties

Once the low-level features have been extracted from the vector field, they can be used to calculate mid-level features which identify areas of organization.

1) Average Neighborhood Similarity Measure The average neighborhood similarity is defined as the average difference between a point and its neighbors within a radius *r*. When the local similarity is calculated for a vector field property, the value at each point represents the difference between that point and its neighborhood average. A low value of average neighborhood difference indicates that the voxels surrounding the center point are similar in value. Groups of points that are spatially connected and have a low level of neighborhood difference indicate a region of interest.

2) Local Entropy Measure Entropy can be interpreted as a measure of disorder or unpredictability, so it is a natural choice for a metric when looking for organization in a data set. Given

*N* observations {

*x*_{1},

*x*_{2}, …,

*x*_{N}}, which occur with probabilities {

*p*_{1},

*p*_{2}, …,

*p*_{N}}, the Shannon entropy is inversely proportional to the log of the probability of observation

*i*, which indicates that the less likely the observation, the higher its entropy will be [

7]. A set of observations where all values are equally likely will have low predictability, since there are no dominant values, which results in high entropy. Conversely, a group of observations with higher predictability, which are clustered around a few values, will have lower entropy.

The local entropy measure is used at each point to calculate the entropy of a neighborhood with radius *r* centered on that point. Points with low entropy are centered on a neighborhood where the values contain a high degree of predictability or order. Spatially connected regions that have similar, low levels of entropy indicate areas of interest.

C. Clustering and Merging Similar Regions

Once low-level or mid-level features have been identified they are clustered to form regions. For low-level features, these are regions of similarity. For mid-level features, the regions represent areas with feature organization. K-means clustering was chosen for this task because it is computationally efficient and effective. One of the main challenges in using the conventional K-means algorithm is that the value of *K* needs to be estimated or known in advance. This problem was avoided by choosing a value of *K* higher than the expected value, since similar regions will be merged in a later step. Spatial constraints can be enforced so that spatially disconnected clusters are split apart and clusters with a very small number of voxels can be eliminated.

The last step of the algorithm merges clusters with different means that are part of the same spatially varying pattern. The goal is to identify neighboring clusters where the voxels on the cluster borders have a high level of similarity. To accomplish this, the edge set of voxels from each cluster border is identified. For each pair of neighboring clusters, the vector property values for neighboring voxels on each side of the border are compared to get a value expressing the similarity at the border of the two clusters. Lastly, a similarity threshold is applied to the border similarity values and this determines whether the clusters should be merged.