Deriving the Algebraic Reconstruction Technique (ART) by the Method of Projections onto Convex Sets (POCS)
Bibliographic record
Abstract
To solve the reconstruction problem iteratively, the image is defined by the vector X and the projections, obtained in N views, by the vector Y. The weights of each pixel j in projections i are entered in matrix A. The image can thus be related to its projection by: Y =A X This equation can be more conveniently written as: y =xa i jij j ∑ for each i. In computerized tomography this system of linear equations is huge and usually underdetermined (128*128 unknowns for 128*64 equations per slice is typical), so it has many solutions X. A popular method to solve it is ART. The most important contribution of this paper is to have demonstrated that ART can be derived by POCS by considering each linear equation as a convex set. Since ART is POCS, additional convex constraints or a priori knowledge can be introduced in order to reduce the solution space. The constraints we have used are the positivity constraint and the support constraint (the image is zero outside a prescribed region). Since POCS converges to the point belonging to the intersection of all the convex sets which is the closest to the initial solution, the choice of the latter is also of importance. The performance of this method will be illustrated with numerical and physical phantoms and on cardiac patients. It will also be compared to other methods (MART, SIRT, MLEM and OSEM).
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".