# Image Registration 圖像配準

A Probabilistic Atlas of the Human Brain in Alzheimer's Disease: Emerging Patterns of Variability, Asymmetry and Degeneration 阿爾茨海默病中人腦的概率圖譜：變異性、不對稱性和退化的新模式

## Image registration 圖像配準​

• Geometric (and Photometric ) alignment of one image with another 幾何（和光度）一幅圖像與另一幅圖像的對齊
• Implemented as the process of estimating an optimal transformation between two images. 作為估計兩幅圖像之間最佳變換的過程
• Images may be of same or different types (MR, CT, visible, fluorescence, ...) 圖像可能是相同或不同類型的（MR，CT，可見，荧光，...）

## Examples of image registration 圖像配準的例子​

• Individual 個人
• Aligning an image taken now with one taken on a previous occasion (monitor the progression of disease, discover the fact of a disease) 現在與以前拍攝的圖像對齊（監控疾病的進展，發現疾病的事實）
• Aligning two images of different sorts (e.g. MRI and CT) of the same patient (data fusion) 將兩幅不同類型的圖像（例如 MRI 和 CT）對齊同一個病人的圖像（數據融合）
• Groups 團體
• Aligning the images of patients and aligning those of normals to develop a statistical model of variation associated with a disease; e.g. Alzheimer's disease 配準病人的圖像，並將正常人的圖像對齊，以開發與疾病相關的變異的統計模型，例如阿爾茨海默病
• Aligning the images from many thousands of subjects around the world as part of a clinical/drug trial 將世界各地數千名受試者的圖像對齊作為臨床/藥物試驗的一部分

## Components of registration 配準的組件​

• The registration problem can be formulated as: 配準問題可以被公式化為：
• Find transformation T (defined by a parameter vector p ) that minimises the difference between the reference image I and target image J (defined by a parameter vector q ) 找到變換 T（由參數向量 p 定義）使參考圖像 I 和目標圖像 J（由參數向量 q 定義）之間的差異最小化

## Components of registration 配準的組件​

• Issues to consider 需要考慮的問題
• What entities do we match? Features, intensities, ... 我們匹配哪些實體？特徵，強度，...
• What class of transforms? Rigid, affine, spline warps, ... 什麼類別的轉換？剛性，仿射，樣條扭曲，...
• What similarity criterion to use? Normalised cross-correlation, ... 使用什麼相似性標準？歸一化互相關，...
• What search algorithm to find the minimum T? 找到最小 T 的搜索算法是什麼？
• What interpolation method to use? Bilinear, spline, ... 使用什麼插值方法？雙線性，樣條，...

## Reference and target datasets 參考和目標數據集​

• Landmarks / control points 地標/管制站
• Image values 圖像值
• Feature images (e.g. edge images) 特徵圖像（例如邊緣圖像）
• Combinations of the above 以上組合

## Other features 其他特徵​

• Image values (intensities) 圖像值（強度）
• Edges, contours or surfaces 邊緣，輪廓或表面
• Salient features 顯著特徵
• Corners 角落
• Centres 中心
• Points of high curvature 高曲率點
• Line intersections 線交點

## Joint histogram 聯合直方圖​

• Heuristic observation is that when images are 啟發式觀察是，當圖像是
• well aligned, the joint histogram appears "sharpest" 對齊良好，聯合直方圖顯得"最銳利"

## Transformation Model 轉換模型​

• Rigid 死板的
• Affine 仿射的
• Piecewise affine 分段仿射的
• Non-rigid or elastic 非剛性或彈性

## Rigid Transformation Model 剛性轉換模型​

• Used for within-subject registration when there is no distortion 用於無失真時的主體內配準
• Composed of 3 rotations and 3 translations 由 3 個旋轉和 3 個平移組成
• Linear – can be represented as a 4x4 matrix 線性——可以表示為 4x4 矩陣

## 3D Rigid-body Transformations 3D 剛性體轉換​

• A 3D rigid body transform is defined by: 3D 剛性體轉換由以下定義：
• 3 translations - in X, Y & Z directions 3 個平移——在 X，Y 和 Z 方向
• 3 rotations - about X, Y & Z axes 3 個旋轉——關於 X，Y 和 Z 軸
• The order of the operations matters 這些操作的順序很重要

## Affine Transformation Model 仿射轉換模型​

• Used for within-subject registration when there is global gross-overall distortion 用於全局粗略失真時的主體內配準
• More typically used as a crude approximation to fully nonrigid transformation. 通常用作完全非剛性轉換的粗略近似。
• Composed of 3 rotation, 3 translations, 3 stretches and 3 shears. 由 3 個旋轉，3 個平移，3 個拉伸和 3 個剪切組成。
• Also a linear transformation – can be represented as a 4x4 matrix 也是線性轉換——可以表示為 4x4 矩陣

## Piecewise Affine Transformation Model 分段仿射轉換模型​

• Simple extension to fully non-rigid transformation model 簡單的擴展到完全非剛性轉換模型
• Typically use different affine transformation for different parts of the image 通常對圖像的不同部分使用不同的仿射轉換

## Non-rigid (elastic) transformation model 非剛性（彈性）轉換模型​

• Model the original image as an elastic body acted upon by two types of forces 模擬原始圖像作為一個彈性體受到兩種力的影響
• External forces drive deformation 外力驅動變形
• Internal forces provide constraints 內力提供製約

## Non-rigid (elastic) transformation model 非剛性（彈性）轉換模型​

• Needed for inter-subject registration and distortion correction 需要跨主體配準和失真校正
• Non-linear i.e. no matrix representation 非線性，即沒有矩陣表示
• Many different parameterizations e.g. 有很多不同的參數化，例如
• Spline parameterizations (b-splines, thin plate splines) 樣條參數化（b 樣條、薄板樣條）
• General diffeomorphisms (e.g. fluid models) 一般的微分形變形（例如流體模型）
• Truncated basis function expansion methods (Fourier parameterizations) 截斷基函數展開方法（傅立葉參數化）

## Similarity Metrics (objective functions) 相似度指標（目標函數）​

• Feature-based Methods (i.e. using corners, edges, etc) 特徵基礎方法（即使用拐角，邊緣等）
• Geometric distance between corresponding points (e.g. CPD) 對應點之間的幾何距離（例如 CPD）
• Similarity metric between feature values - Similar curvature, etc 特徵值之間的相似度度量——相似曲率等

## Similarity Metrics (objective functions) 相似度指標（目標函數）​

• Intensity-based Methods (i.e. using image values) 強度基礎方法（即使用圖像值）
• Mean Squared Difference / Sum of Squared Differences 均方差/平方差和
• Only valid for same modality with properly normalized intensities 只適用於同一模態的正確標準化強度
• Mutual Information 互信息
• A metric which maximizes the clustering of the joint histogram. 一個指標，最大化聚類的聯合直方圖。
• Normalized Cross-Correlation 正規化互相關
• Allows for linear relationship between the intensities of the two images 允許兩個圖像的強度之間的線性關係

## Mean-squared Difference (MSD) / Sum of Squared Differences (SSD) 均方差（MSD）/平方差和（SSD）​

• Minimising MSD / SSD works for intra-modal and intra-subject registration (realignment) 最小化 MSD/SSD 適用於內模式和內主體配準（重新對齊）
• Simple relationship between intensities in one image, versus those in the other image 一個圖像中強度與另一個圖像中強度之間的簡單關係

## Mutual Information 互信息​

• Algorithms for maximising mutual information (between intensities) have been some of the most popular for medical image registration to date. 用於最大化互信息（強度）的算法至今已成為醫學影像配準的最受歡迎的方法之一。
$M I(I, J \mid T)=\sum_{i, j} p_{i, j} \log \frac{p_{i, j}}{p_i p_j}$

p i= probability of pixel having value i (from image histogram) p i=像素具有值 i 的概率（從圖像直方圖）