In MRI, artifacts are unwanted features in the image that do not represent the true anatomy. These artifacts can be classified as coherent or incoherent, depending on how structured or random they appear. In a nutshell, incoherent artifact is like noise. 🟩 Coherent Artifacts Definition: Artifacts that appear structured, regular, and predictable. Cause: Usually caused by periodic or consistent errors, such as regular undersampling or periodic motion. Appearance: Clear patterns like ghosting or aliasing (wrap-around).

I. Definition of Adjoint Operator For any vectors (or functions) $x$ and $y$, the adjoint operator $F^H$ satisfies: $$\langle F x,\;y\rangle \;=\;\langle x,\;F^H y\rangle$$ where the discrete inner product is defined as $$\langle u,v\rangle = \sum_n u_n^*\,v_n.$$ II. General Steps for Deriving the Adjoint Combine Unknowns Merge all unknowns (or functions) into a single large vector $x$. Write Forward Operator Express the forward operator as a matrix (or operator) $F$ multiplied by $x$:

SENSE (Sensitivity Encoding) accelerates MRI by undersampling k-space, causing aliasing where signals from distinct locations ($r_1, r_2, \dots, r_R$) overlap onto a single pixel $r_p$. Coil sensitivities are used to unfold these signals. SENSE Unfolding & Noise (R-fold Undersampling) For $N_c$ coils and an acceleration factor $R$, the measured aliased signals $\mathbf{m}_{\text{aliased}}(r_p) \in \mathbb{C}^{N_c}$ at pixel $r_p$ are: $$\mathbf{m}_{\text{aliased}}(r_p) = \mathbf{S}(r_p) \cdot \mathbf{m}_{\text{true}}(r_p) + \mathbf{n}(r_p)$$ Where: $\mathbf{m}_{\text{true}}(r_p) = [m(r_1), m(r_2), \dots, m(r_R)]^T \in \mathbb{C}^{R}$ are the true image intensities at the $R$ aliasing locations.

梯度、约束优化与拉格朗日乘子 关键点：最优点要同时满足约束以及目标函数，那么它在它们的交点上；如果这个二者在这个交点不相切的话，那么，我们总可