PICCS: Prior Image Constrained Compressed Sensing

• PICCS is a prior-based compressed sensing framework that leverages a high-quality prior image to guide the reconstruction of undersampled target images.
• It employs convex optimization with L1 penalties and total variation regularization to enforce sparsity in both the image and its deviation from the prior.
• Empirical studies in CT and CBCT demonstrate that PICCS reduces measurement bounds, effectively suppresses artifacts, and preserves clinically significant local changes.

Searching arXiv for PICCS and closely related prior-information compressed sensing work to ground the article in cited papers. Prior Image Constrained Compressed Sensing (PICCS) is a prior-based compressed sensing framework for reconstructing an unknown target signal or image from undersampled measurements while exploiting a known, similar signal, typically a prior image. In the synthesis setting, it fits exactly into the general problem of compressed sensing with prior information, in which one reconstructs $x$ from $y = Ax$ while incorporating a side signal $w$ through objectives such as $\|x\|_{1} + \lambda \|x-w\|_{1}$; in CT and CBCT, the same idea is commonly instantiated in the analysis or total variation (TV) domain as a joint penalty on the image and on its difference from a prior image (Mota et al., 2014). In repeated scanning procedures, especially under-sampled CBCT with only local changes relative to an earlier high-quality scan, PICCS uses that prior image to suppress undersampling artifacts and preserve genuine changes such as a new tumor or a surgical tool (Hastings et al., 10 Sep 2025).

1. Definition and problem setting

PICCS addresses inverse problems of the form

$y = A x^{\star}$

in the noiseless case, or

$$\mathbf{A}\boldsymbol{x} + \boldsymbol{e} = \boldsymbol{b}$$

in the noisy CBCT formulation, where $A$ is the sensing or system matrix, $x^{\star}$ is the true target image, and $w$ or $x_p$ is a known similar image available beforehand (Mota et al., 2014). The defining feature is that reconstruction is constrained not only by sparsity of the target but also by sparsity of the deviation from the prior.

In the formulation analyzed as compressed sensing with prior information, the target is sparse or analysis-sparse, and the prior is informative when it is similar to the target in support, sign pattern, or transform domain (Mota et al., 2014). In CBCT repeated scanning, the relevant regime is one in which a sequence of under-sampled scans is acquired on the same object, the anatomy is largely unchanged, and only local and sparse changes occur. The assumed prior image is a good initial reconstruction from a previous over-sampled or high-dose scan and is well aligned to the current geometry (Hastings et al., 10 Sep 2025).

PICCS is therefore not merely a denoising prior around a reference image. Its canonical form enforces sparsity both of the current image and of the change relative to the prior. In CT practice, this is typically implemented with TV penalties, so that genuine, salient differences are retained while spurious differences induced by undersampling are discouraged (Hastings et al., 10 Sep 2025).

2. Convex formulations

In the synthesis setting, the central noiseless convex programs are the $y = Ax$0-$y = Ax$1 and $y = Ax$2-$y = Ax$3 formulations: $y = Ax$4 and

$y = Ax$5

or, in the quadratic prior form that is analyzed,

$y = Ax$6

The noisy constrained and Lagrangian versions replace the exact data-consistency constraint by either $y = Ax$7 or a quadratic data term $y = Ax$8 plus regularization (Mota et al., 2014).

PICCS in CT is the analysis-domain counterpart of the $y = Ax$9-$w$0 template. In the notation used for analysis or TV regularization,

$w$1

where $w$2 is an analysis operator such as the discrete gradient for TV, $w$3 is the CT system matrix, and $w$4 is the prior image (Mota et al., 2014).

The 2025 CBCT implementation uses the unconstrained quadratic form

$w$5

with 3D isotropic discrete TV

$w$6

That work also evaluates a PIPLE-style variant,

$w$7

which replaces the TV penalty on the difference by an $w$8 prior term (Hastings et al., 10 Sep 2025).

Several implementation details delimit the scope of that CBCT formulation. No explicit non-negativity constraint is enforced, no spatial masks or adaptive weighting are used, and alignment of the prior image is assumed adequate. Parameters $w$9 and $\|x\|_{1} + \lambda \|x-w\|_{1}$0 are treated as given and tuned empirically to balance artifact suppression and change preservation (Hastings et al., 10 Sep 2025).

3. Geometric recovery theory and measurement bounds

The theoretical analysis of PICCS-type reconstruction is geometric. For a convex objective $\|x\|_{1} + \lambda \|x-w\|_{1}$1, the descent cone at $\|x\|_{1} + \lambda \|x-w\|_{1}$2 is

$\|x\|_{1} + \lambda \|x-w\|_{1}$3

where $\|x\|_{1} + \lambda \|x-w\|_{1}$4. Exact recovery occurs when

$\|x\|_{1} + \lambda \|x-w\|_{1}$5

For Gaussian sensing matrices, the probability of success is controlled by the Gaussian width

$\|x\|_{1} + \lambda \|x-w\|_{1}$6

with $\|x\|_{1} + \lambda \|x-w\|_{1}$7 (Mota et al., 2014).

The general high-probability recovery condition is

$\|x\|_{1} + \lambda \|x-w\|_{1}$8

In the noisy constrained case, if

$\|x\|_{1} + \lambda \|x-w\|_{1}$9

then every solution $y = A x^{\star}$0 satisfies

$y = A x^{\star}$1

with high probability when $y = A x^{\star}$2 (Mota et al., 2014).

For classical compressed sensing with plain $y = A x^{\star}$3 minimization, the bound is

$y = A x^{\star}$4

so

$y = A x^{\star}$5

suffices for an $y = A x^{\star}$6-sparse signal (Mota et al., 2014).

The $y = A x^{\star}$7-$y = A x^{\star}$8 theory introduces support and quality parameters that quantify how the prior aligns with the target: $y = A x^{\star}$9 It distinguishes sign-based good and bad components on $$\mathbf{A}\boldsymbol{x} + \boldsymbol{e} = \boldsymbol{b}$$0: $$\mathbf{A}\boldsymbol{x} + \boldsymbol{e} = \boldsymbol{b}$$1 with counts

$$\mathbf{A}\boldsymbol{x} + \boldsymbol{e} = \boldsymbol{b}$$2

It also uses

$$\mathbf{A}\boldsymbol{x} + \boldsymbol{e} = \boldsymbol{b}$$3

and the balance parameter

$$\mathbf{A}\boldsymbol{x} + \boldsymbol{e} = \boldsymbol{b}$$4

For $$\mathbf{A}\boldsymbol{x} + \boldsymbol{e} = \boldsymbol{b}$$5, assuming $$\mathbf{A}\boldsymbol{x} + \boldsymbol{e} = \boldsymbol{b}$$6 and that there exists $$\mathbf{A}\boldsymbol{x} + \boldsymbol{e} = \boldsymbol{b}$$7 with $$\mathbf{A}\boldsymbol{x} + \boldsymbol{e} = \boldsymbol{b}$$8, the simplified bound is

$$\mathbf{A}\boldsymbol{x} + \boldsymbol{e} = \boldsymbol{b}$$9

The dominant logarithmic term scales with $A$0 rather than $A$1. Since $A$2, good prior information can dramatically reduce the measurement requirement (Mota et al., 2014).

By contrast, the $A$3-$A$4 bounds depend on magnitudes as well as supports and signs, through parameters such as $A$5, $A$6, $A$7, and $A$8. Those bounds are explicitly described as looser and often track classical compressed sensing closely, which explains why $A$9-$x^{\star}$0 brings limited benefit relative to $x^{\star}$1-$x^{\star}$2 unless the target is extremely sparse and the prior is very special (Mota et al., 2014).

4. Why PICCS helps in the TV or analysis setting

The geometric interpretation carries directly into analysis-sparse and TV-based PICCS. Adding the $x^{\star}$3 or $x^{\star}$4 term tightens the relevant sublevel set around directions that preserve similarity to the prior. When the prior has favorable sign alignment, the tangent or descent cone shrinks drastically, and only the bad components keep that cone wide (Mota et al., 2014).

For CT, this means that the benefit of PICCS is not determined solely by pixelwise similarity of the prior image. What matters is similarity in the sparsifying domain, typically the discrete gradient or another analysis operator. When $x^{\star}$5 and $x^{\star}$6 share support with favorable sign alignment, the number of projections required for accurate reconstruction can be substantially reduced relative to plain TV-regularized compressed sensing (Mota et al., 2014).

This point is central to the distinction between PICCS and quadratic prior methods. A TV or $x^{\star}$7 penalty on the difference encodes sparsity of changes, whereas an $x^{\star}$8 penalty around the prior acts more like averaging around a reference. The geometric analysis shows that the latter leaves the descent cone relatively wide, especially in high dimensions, so the measurement advantage is much smaller (Mota et al., 2014).

In repeated CBCT, the assumptions that make PICCS effective are explicit: a good prior image exists, the current scan is strongly under-sampled in angle, changes are local and sparse, and the prior is aligned to the current geometry. Under those conditions, PICCS suppresses undersampling artifacts and noise while preserving clinically relevant changes (Hastings et al., 10 Sep 2025).

5. Parameterization, robustness, and algorithms

For $x^{\star}$9-$w$0 reconstruction, the theoretical bounds are minimized at $w$1, equivalently $w$2 in the noiseless constrained form. This choice is described as universal, because it does not depend on unknown quantities in $w$3 or $w$4, and phase transition curves for $w$5-$w$6 typically achieve their best performance near $w$7 (Mota et al., 2014). For $w$8-$w$9, the bound-minimizing $x_p$0 depends on unknown magnitudes in $x_p$1 and $x_p$2 and is therefore impractical in inverse problems; small $x_p$3 avoids the severe degradation seen for large $x_p$4 (Mota et al., 2014).

In noisy problems, the constrained form $x_p$5 yields a stability guarantee under the Gaussian-width condition stated above. In practice, $x_p$6 or $x_p$7 remains the recommended default for $x_p$8-$x_p$9, while the Lagrangian weight $y = Ax$00 can be tuned by standard discrepancy or cross-validation rules (Mota et al., 2014). In the CBCT implementation, practical guidance is more application-specific: increase $y = Ax$01 with higher noise or undersampling to suppress streaking and noise; increase $y = Ax$02 when prior fidelity is high and changes are sparse and local; decrease $y = Ax$03 when misregistration or large anatomical changes are suspected; and keep the smoothing parameter $y = Ax$04 small but nonzero (Hastings et al., 10 Sep 2025).

All of the formulations are convex. For the synthesis problems, the basic proximal operators are explicit: $y = Ax$05 and

$y = Ax$06

For the quadratic prior form,

$y = Ax$07

Suitable solvers include proximal gradient or FISTA for Lagrangian forms, ADMM for constrained and analysis or TV formulations, and primal-dual methods such as Chambolle–Pock for TV or analysis regularization (Mota et al., 2014).

The 2025 real-time CBCT work implements PICCS through an iteratively reweighted norm majorization-minimization scheme. Non-smooth TV penalties are replaced by quadratically weighted $y = Ax$08 terms with weights updated across outer iterations, and the inner least-squares problem is solved by LSQR or, equivalently, CGLS; that work uses CGLS because of its short-recurrence, low-memory, and GPU-friendly nature (Hastings et al., 10 Sep 2025). At outer iteration $y = Ax$09, the weights are computed from the current iterate,

$y = Ax$10

and the quadratic subproblem is

$y = Ax$11

This is cast as an augmented least-squares problem and solved without explicitly building the matrix, using forward projection, backprojection, finite-difference stencils, and diagonal weighting operations on the GPU through the TIGRE toolbox (Hastings et al., 10 Sep 2025).

The computational profile is dominated by one forward projection and one backprojection per inner CGLS iteration. IRN converges to the minimizer of the smoothed objective, whereas CGLS can exhibit semi-convergence if iterated too long, so early stopping is used (Hastings et al., 10 Sep 2025).

6. Empirical behavior, variants, and limitations

The empirical results separate three issues: the effect of prior information itself, the difference between $y = Ax$12-$y = Ax$13 and $y = Ax$14-$y = Ax$15-type priors, and the effect of algorithmic implementation. In a representative synthetic example with $y = Ax$16 and $y = Ax$17, with prior $y = Ax$18 having about $y = Ax$19 relative $y = Ax$20 error and mixed support overlap, the $y = Ax$21-$y = Ax$22 bound predicted successful recovery with $y = Ax$23 measurements versus $y = Ax$24 for classical compressed sensing, and the empirical phase transition confirmed a drastic reduction. $y = Ax$25-$y = Ax$26 tracked classical compressed sensing closely and provided negligible savings (Mota et al., 2014).

In the 2025 synthetic head phantom experiment with 20 projections and a prior equal to the head without tumor, FDK was qualitatively poor due to extreme undersampling, SIRT was smooth but blurry, and CGLS retained artifacts. For the prior-based methods, IRN-PICCS with 25 iterations achieved PSNR $y = Ax$27, SSIM $y = Ax$28, HaarPSI $y = Ax$29, and time $y = Ax$30 s; IRN-PICCS with 100 iterations and 4 outer cycles achieved PSNR $y = Ax$31, SSIM $y = Ax$32, HaarPSI $y = Ax$33, and time $y = Ax$34; ASD-POCS-PICCS with 20 iterations achieved PSNR $y = Ax$35, SSIM $y = Ax$36, HaarPSI $y = Ax$37, and time $y = Ax$38 (Hastings et al., 10 Sep 2025). In that synthetic setting, IRN-PIPLE produced the strongest quantitative results and preserved tumor and texture very well, whereas IRN-PICCS identified changes but was more aggressive on texture.

For the thorax phantom with real data, a metal needle, and 180, 50, and 20 projections, IRN-PICCS with 20 inner iterations yielded PSNR $y = Ax$39, SSIM $y = Ax$40, HaarPSI $y = Ax$41, and time $y = Ax$42 at 180 projections; PSNR $y = Ax$43, SSIM $y = Ax$44, HaarPSI $y = Ax$45, and time $y = Ax$46 at 50 projections; and PSNR $y = Ax$47, SSIM $y = Ax$48, HaarPSI $y = Ax$49, and time $y = Ax$50 at 20 projections (Hastings et al., 10 Sep 2025). At all undersampling levels, the proposed IRN-PIPLE and IRN-PICCS materially outperformed FDK, CGLS, and TV-only methods in PSNR and SSIM and reduced metal artifacts and noise much better. Runtime speedups over ASD-POCS-PICCS were substantial: about $y = Ax$51 at 180 projections, about $y = Ax$52 at 50 projections, and about $y = Ax$53 at 20 projections (Hastings et al., 10 Sep 2025).

These results clarify a common confusion. The geometric theory strongly favors $y = Ax$54-$y = Ax$55 over $y = Ax$56-$y = Ax$57 as a general compressed sensing design principle, because only the former replaces sparsity $y = Ax$58 by the smaller effective bad-component count $y = Ax$59 in the dominant measurement term (Mota et al., 2014). However, the CBCT study found that a PIPLE-style $y = Ax$60 prior could outperform PICCS in some heavily undersampled practical settings, particularly in clean synthetic data, where it had a more averaging effect and could be more robust when the prior was strong but changes were small (Hastings et al., 10 Sep 2025). This suggests that theoretical recovery geometry and application-level image-quality tradeoffs are related but not identical.

The strengths and limitations are correspondingly regime-dependent. PICCS excels when changes are small and localized, the prior is trustworthy, and undersampling artifacts must be suppressed without erasing true changes (Hastings et al., 10 Sep 2025). Its benefit diminishes when the prior is poor, when there are many bad components in the transform domain, or when misregistration introduces structured discrepancies (Mota et al., 2014). Large global changes or overly large $y = Ax$61 can bias the reconstruction toward the prior, and very strong metal artifacts remain challenging (Hastings et al., 10 Sep 2025). Best practices therefore include registration of the prior to the current geometry, moderate initialization of regularization weights, inspection of difference images $y = Ax$62, and comparison with TV-only baselines to verify that the prior term is not suppressing genuine changes (Hastings et al., 10 Sep 2025).