⚡ electrical engineering

A Mixed Precision FFT with applications in MRI

This paper presents a mixed-precision Fast Fourier Transform (FFT) implementation utilizing per-block microscaling, global prescaling, and prequantized twiddles, demonstrating through MRI dataset evaluations that mantissa precision is the primary fidelity limiter while larger block sizes yield superior numerical performance.

Original authors: Nikhil Deveshwar, Abhejit Rajagopal, Peder E. Z. Larson

Published 2026-02-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Nikhil Deveshwar, Abhejit Rajagopal, Peder E. Z. Larson

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to send a massive, high-resolution photo of a human knee or brain from a hospital scanner to a doctor's tablet. To do this efficiently, the computer has to translate the raw data (which looks like a chaotic storm of radio waves) into a clear picture. This translation process is called a Fast Fourier Transform (FFT).

Think of the FFT as a giant, complex recipe for turning a bowl of mixed ingredients (the raw data) into a perfect cake (the medical image).

The Problem: The "Tiny Calculator" Dilemma

Traditionally, computers do this recipe using "double-precision" math. It's like using a super-accurate, heavy-duty calculator that costs a lot of battery power and takes up a lot of space. This is great for accuracy, but it's slow and energy-hungry, especially on portable MRI machines that need to run on batteries.

Scientists wanted to switch to "low-precision" math—using a smaller, faster, lighter calculator (like an 8-bit calculator). But there's a catch: if you use a small calculator for a huge recipe, you risk two disasters:

Overflow: The numbers get too big, and the calculator explodes (the image gets distorted).
Underflow: The numbers get too tiny, and the calculator ignores them completely (you lose the fine details of the image).

The Solution: The "Smart Scaling" Chef

This paper introduces a new way to cook the recipe using Mixed Precision. Instead of just using a small calculator for everything, they created a smart system with three special tricks:

1. The Global "Volume Knob" (Global Prescale)

Before starting the recipe, the system looks at the biggest and smallest ingredients. It turns a "volume knob" (a power-of-two scale) to make sure the biggest ingredients fit in the calculator without exploding, and the tiniest ingredients are loud enough to be heard.

Analogy: Imagine you are adjusting the volume on a radio before a song starts. You turn it up so the quiet whispers are audible, but you turn it down so the loud drums don't blast your ears. This ensures everything fits in the "listening range."

2. The "Group Huddle" (Microscaling or MX)

This is the paper's main innovation. Instead of giving every single number its own tiny calculator, the system groups numbers into small teams (blocks of 32).

How it works: The whole team shares one volume knob (a shared exponent). Inside the team, everyone uses a tiny, simplified number format (like 8-bit floating point).
Analogy: Imagine a choir. Instead of every singer having their own conductor, they split into small groups of 32. Each group has one conductor who sets the volume for the whole group. The singers inside the group might have slightly imperfect voices (low precision), but because they are all singing together under one conductor, the group sounds great. This saves a huge amount of energy because you don't need a conductor for every single note.

3. The "Safety Net" (High-Precision Accumulation)

Even though the singers are using simple voices, when they add their voices together to make the final chord, the system uses a "super-accurate" calculator (32-bit) to combine them.

Analogy: The singers (the low-precision math) do the heavy lifting, but the final mix is done by a professional audio engineer (high-precision math) to ensure no detail is lost in the final blend.

What Did They Find?

The researchers tested this new "Smart Chef" system on real MRI data from brains and knees. Here is what they discovered:

The "Fingers" Matter More Than the "Range": They found that the most important part of the number isn't how big it can get (the exponent), but how many "fingers" it has to count with (the mantissa).
- Simple version: It's better to have a number that can count up to 100 with high detail (3-bit precision) than a number that can count up to a million but only in huge, blurry steps (2-bit precision).
- Result: Formats with 3 "fingers" (MXFP8-E4M3) produced crystal clear images, while formats with only 2 "fingers" made the images look blurry and noisy.
Bigger Groups Work Better: They found that having larger groups (blocks of 32) worked better than tiny groups (blocks of 2).
- Analogy: If you have a group of 2, you need a new conductor every 2 seconds. If you have a group of 32, the conductor stays in charge longer, making the process smoother and more efficient.
The "Round Trip" Test: They tested the system by turning the data into an image and then turning that image back into data. They found that errors in the first step (making the image) couldn't be fixed in the second step. Once you lose a detail, it's gone forever.

Why Does This Matter?

This research is a bridge between old-school math and modern, energy-efficient AI hardware.

For the Future: This technique could allow portable MRI scanners (like the size of a suitcase) to produce high-quality images without needing massive, power-hungry computers.
The Bottom Line: You don't need a super-computer to do MRI if you are smart about how you group your numbers. By using "group huddles" (MX) and keeping the "fingers" (precision) sharp, we can get clear medical images using much less energy.

In short: Don't just shrink the calculator; organize the math so the small calculator can do a big job without breaking.

1. Problem Statement

The Fast Fourier Transform (FFT) is a fundamental operation in Magnetic Resonance Imaging (MRI) for converting frequency-domain data (k-space) into image space. However, deploying FFTs on edge and embedded platforms (e.g., portable MRI scanners) requires reducing memory traffic and energy consumption.

Challenge: Naive low-precision quantization in complex-valued FFTs often leads to overflow/underflow errors and significant accuracy loss.
Gap: While 8-bit floating point (FP8) and microscaling (MX) formats are emerging in AI (e.g., NVIDIA Transformer Engine, Open Compute Project), their application to complex-valued FFTs in imaging is limited. Previous studies focused on static scaling or specific ML kernels (GEMM), lacking validation on the specific dynamic range and error propagation characteristics of MRI reconstruction.

2. Methodology

The authors propose a Mixed Precision FFT implementation that integrates MX-style scaling with low-bit quantization, specifically tailored for complex butterfly operations.

A. Global Power-of-Two Prescale

To prevent overflow and underflow before the FFT computation begins:

A single global power-of-two prescale is applied to the input data.
Mechanism: The algorithm analyzes the input to find the global peak magnitude ( $a_{max}$ ) and a specific percentile of non-zero entries ( $p_\tau$ ).
It computes two candidate exponents: one to scale the peak to a target value and another to lift the "tail" (small values) above a user-defined floor.
The stricter of the two requirements is applied using the ldexp function to ensure the entire dynamic range fits within the target precision limits.

B. MX-Scaled Complex Butterfly Operations

The core FFT computation (Cooley–Tukey algorithm) is adapted using Microscaling (MX) formats:

Block Processing: Data is processed in small complex blocks (e.g., $B=32$ real elements).
Quantization Flow:
1. Encoding: Operand blocks (twiddles and data) are encoded into FP8/MX format, extracting a shared block scale and FP8 mantissas.
2. Multiplication: Complex multiplication is performed on the mantissas in higher precision (effectively treating them as integers or high-precision floats).
3. Re-normalization: If the product's magnitude exceeds the format's limit, the block is re-normalized.
4. Accumulation: The resulting products are repackaged into a new MX block. Crucially, the butterfly addition and subtraction (accumulation) are performed in FP32 to minimize error accumulation.
Twiddles: The complex twiddle factors are pre-quantized to FP8.

C. Evaluation Setup

Datasets: Two public MRI datasets were used: fastMRI (2D brain, fast spin-echo) and SKM-TEA (3D knee, gradient recalled echo).
Metrics: Peak Signal-to-Noise Ratio (PSNR), Structural Similarity Index (SSIM), and Normalized Mean Squared Error (NMSE).
Comparisons: The proposed mixed-precision FFT was compared against:
1. Standard NumPy FP32 FFT (Ground Truth).
2. A 16-bit FP16 FFT positive control (with FP32 accumulation).
3. Various MX formats (MXFP8-E4M3, MXFP8-E5M2, MXFP6-E2M3, etc.).

3. Key Contributions

Novel Architecture: First adaptation of MX-style per-block scaling and FP8 quantization specifically for complex-valued FFT butterfly operations in an MRI context.
Hybrid Precision Strategy: Introduction of a Global Prescale combined with Per-Block MX Scaling and FP32 Accumulation, bridging classic Block Floating Point (BFP) concepts with modern FP8 hardware capabilities.
Comprehensive Ablation Study: Systematic analysis of how mantissa precision, exponent range, image size, and block size affect image fidelity in both forward and round-trip (inverse) FFT scenarios.

4. Results

Format Performance:
- MXFP8-E4M3 (4 exponent bits, 3 mantissa bits) delivered the best quantitative performance among FP8 formats, closely tracking FP16 results.
- Formats with only 2 mantissa bits (e.g., E5M2, E3M2) suffered significant degradation in image quality, indicating that mantissa precision is the primary limiter under MX scaling, not the exponent range.
- MXFP6-E2M3 performed well for forward FFTs but showed limitations in round-trip scenarios compared to E4M3.
Block Size Trade-offs:
- Larger block sizes ( $B=32$ ) significantly outperformed smaller blocks ( $B=2$ ).
- Small blocks ( $B=2$ ) introduce frequent rescaling boundaries, amplifying rounding errors.
- Larger blocks ( $B=8, 32$ ) amortize scale/metadata overhead, stabilize statistics, and improve PSNR/NMSE. Diminishing returns were observed beyond $B=8$ for certain formats.
Image Size: Increasing image size (from $64\times64$ to $256\times256$ ) had a negligible impact on metrics, suggesting the quantization steps are consistent regardless of resolution.
Round-Trip Errors: Inverse transforms (Round-trip) showed compounded errors compared to forward-only transforms due to the introduction of a second quantization step and normalization ( $1/N^2$ ), highlighting the difficulty of maintaining fidelity in iterative reconstruction.

5. Significance and Conclusion

Clinical Relevance: The study demonstrates that mixed-precision FFTs can be viable for MRI reconstruction, potentially enabling faster, lower-power imaging on portable devices.
Design Guidelines: The authors conclude that for mixed-precision MX-scaled FFTs:
- Mantissa width is more critical than exponent range for retaining image quality.
- FP32 accumulation is essential.
- Block sizes should be sufficiently large (e.g., $B=32$ ) to minimize rescaling overhead.
- Preferred Format: MXFP8-E4M3 with $B=32$ offers the best balance of speed and fidelity.
Future Outlook: While not yet clinically ready for all domains, this work provides a foundation for accelerating undersampled iterative image reconstruction pipelines where multiple FFT/iFFT pairs are executed per iteration. The authors suggest fusing encoding, forward, inverse, and decoding steps to further reduce round-trip loss.