Computation of 2D 8×8 DCT Based on the Loeffler Factorization Using Algebraic Integer Encoding
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Bibliographic record
Abstract
This paper proposes a computational method for 2D 8×8 DCT based on algebraic integers. The proposed algorithm is based on the Loeffler 1D DCT algorithm, and it is shown to operate with exact computation—i.e., error-free arithmetic—up to the final reconstruction step (FRS). The proposed algebraic integer architecture maintains error-free computations until an entire block of DCT coefficients having size 8×8 is computed, unlike algorithms in the literature which claim to be error-free but in fact introduce arithmetic errors between the column- and row-wise 1D DCT stages in a 2D DCT operation. Fast algorithms are proposed for the final reconstruction step employing two approaches, namely, the expansion factor and dyadic approximation. A digital architecture is also proposed for a particular FRS algorithm, and is implemented on an FPGA platform for on-chip verification. The FPGA implementation operates at 360 MHz, and is capable of a real-time throughput of <inline-formula><tex-math notation="LaTeX"> $3.6\cdot 10^8$</tex-math> </inline-formula> 2D DCTs of size 8×8 every second, with corresponding pixel rate of <inline-formula> <tex-math notation="LaTeX">$2.3\cdot 10^{10}$</tex-math></inline-formula> pixels per second. The digital architecture is synthesized using 180 nm CMOS standard cells and shows a chip area of 7.41 mm <inline-formula> <tex-math notation="LaTeX">$^2$</tex-math></inline-formula> . The CMOS design is predicted to operate at 893 MHz clock frequency, at a dynamic power consumption 13.22 mW/MHz <inline-formula><tex-math notation="LaTeX">$\cdot$</tex-math></inline-formula> V <inline-formula> <tex-math notation="LaTeX">$_{sup}^2$</tex-math></inline-formula> .
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| 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 it