You’ve hit on a key difference between two common approaches to data consistency. The paper uses the pseudoinverse ($A^+$), while tools like BART often use the adjoint ($A^*$), and they are used for fundamentally different reasons.

  • *$A^$ (Adjoint) is for Gradient Descent:** Using the adjoint $A^$ (as in $x_t \leftarrow \tilde{x}_t - \alpha A^(A\tilde{x}_t - \tilde{y}_t)$) is a gradient descent step. It takes a small step in the direction that reduces the k-space error. This is common in iterative optimization algorithms that slowly converge to a solution.

  • $A^+$ (Pseudoinverse) is for Projection: The paper’s Equation (24) is not a gradient step; it’s a one-shot orthogonal projection. This operation takes the current image estimate $\tilde{x}_t$ and finds the closest possible image $x_t$ that is in perfect agreement with the measurements $\tilde{y}_t$.

Why This Paper Uses $A^+$

The method described here, which the authors relate to “range-null decomposition”, relies on this direct projection. The goal of this specific step is not just to “reduce” the error, but to enforce perfect data consistency on the noiseless estimate.

As the paper itself notes, this hard projection is only suitable for clean (noiseless) measurements. If the data $y_0$ were noisy, this step would perfectly inject all that noise into the final image. That is why they use a different “Plug-and-Play (PnP)” method (Algorithm 2) for the noisy data scenario.

This formula is the direct solution to a constrained optimization problem. Here is the derivation.

1. 🎯 The Optimization Problem

The goal is to enforce data consistency. We have a current image estimate, $\tilde{x}_t$, which is good but doesn’t perfectly match our measurements, $\tilde{y}_t$.

We want to find a new image, $x_t$, that solves the following problem:

“Find the image $x_t$ that is as close as possible to $\tilde{x}_t$ while perfectly satisfying the measurement constraint $A(x_t) = \tilde{y}_t$.”

Mathematically, this is written as:

\(\text{minimize} \quad \frac{1}{2} ||x - \tilde{x}_t||_2^2\) \(\text{subject to} \quad A(x) = \tilde{y}_t\)

(We use $\frac{1}{2}   \cdot   _2^2$ because it has the same minimum as $   \cdot   _2$ and its derivative is simpler.)

2. 📝 The Method: Lagrange Multipliers

We can solve this constrained problem by introducing a Lagrange multiplier, $\lambda$, and forming the Lagrangian function, $\mathcal{L}$:

\[\mathcal{L}(x, \lambda) = \underbrace{\frac{1}{2} ||x - \tilde{x}_t||_2^2}_{\text{Proximity to } \tilde{x}_t} + \underbrace{\lambda^T (A(x) - \tilde{y}_t)}_{\text{Constraint}}\]

To find the minimum, we must find the point where the gradients of $\mathcal{L}$ with respect to both $x$ and $\lambda$ are zero.

3. ⚙️ Solving the System

Step 1: Find the gradient with respect to $x$

We take the partial derivative of $\mathcal{L}$ with respect to $x$ and set it to zero.

(Note: The derivative of $\frac{1}{2}   x - c   _2^2$ with respect to $x$ is $(x - c)$. The derivative of $\lambda^T A(x)$ with respect to $x$ is $A^T\lambda$.)
\[\frac{\partial \mathcal{L}}{\partial x} = (x - \tilde{x}_t) + A^T\lambda = 0\]

Solving for $x$, we get: \(x = \tilde{x}_t - A^T\lambda\) This tells us the form of our solution $x_t$, but we still need to find $\lambda$.

Step 2: Use the constraint to find $\lambda$

We know our solution $x_t$ must satisfy the constraint $A(x_t) = \tilde{y}_t$. So, we plug our formula for $x$ from Step 1 into the constraint:

\[A(\tilde{x}_t - A^T\lambda) = \tilde{y}_t\]

Now we distribute $A$: \(A(\tilde{x}_t) - A(A^T\lambda) = \tilde{y}_t\)

And rearrange to solve for $\lambda$: \(A(\tilde{x}_t) - \tilde{y}_t = A A^T \lambda\)

\(\lambda = (A A^T)^+ (A(\tilde{x}_t) - \tilde{y}_t)\) (We use the pseudoinverse $(A A^T)^+$ in case $A A^T$ is not invertible).

Step 3: Put it all together

Now we substitute this expression for $\lambda$ back into our solution for $x$ from Step 1:

\[x_t = \tilde{x}_t - A^T \left[ (A A^T)^+ (A(\tilde{x}_t) - \tilde{y}_t) \right]\]

This looks complicated, but we can simplify it using a standard identity of the Moore-Penrose pseudoinverse:

Pseudoinverse Identity: $A^T(A A^T)^+ = A^+$

Applying this identity, the $A^T (A A^T)^+$ term simplifies to just $A^+$:

\[x_t = \tilde{x}_t - A^+ (A(\tilde{x}_t) - \tilde{y}_t)\]

This is Equation (24). It is the exact, analytical solution to the problem of finding the closest image to $\tilde{x}_t$ that perfectly matches the measurements $\tilde{y}_t$.