Projection

The Projection engine combines Z-stack slices into a single focused image using one of seven methods:

• Maximum Intensity Projection (MIP) — At each pixel position, the brightest value across all Z-slices is selected. Simple and fast, but biased toward noise and bright out-of-focus structures.
• Average Projection — Mean intensity across all Z-slices. Reduces noise but also reduces contrast of sharp features.
• Wavelet Fusion (5 variants) — Each Z-slice is decomposed into wavelet coefficients representing different spatial frequencies and locations. The algorithm selects coefficients from whichever Z-slice has the highest magnitude at each frequency-location pair. Inverse wavelet transform produces the fused image. This preserves sharp features while avoiding noise amplification.

The wavelet methods differ in their mathematical properties: Daubechies wavelets are compact and smooth; Biorthogonal wavelets provide linear phase (no edge shift); Discrete Meyer wavelets are nearly band-limited. The choice affects edge handling and noise behavior, but all five produce similar overall results for most tissue images.

The optical sectioning problem: Pawley explains that confocal microscopy acquires true optical sections — each Z-plane captures only in-focus information. But widefield microscopy integrates light from all depths simultaneously. For widefield Z-stacks, each slice contains in-focus signal (from structures at that plane) mixed with out-of-focus haze (from structures above and below). Projection methods must distinguish in-focus from out-of-focus content.

Wavelet fusion — the mathematical basis: Gonzalez & Woods describe wavelet decomposition as multi-resolution analysis: images are decomposed into approximation (low-frequency, smooth) and detail (high-frequency, edges) coefficients at multiple scales. Vetterli's nonlinear approximation theorem shows that keeping the largest-magnitude wavelet coefficients (regardless of which Z-slice they come from) produces the optimal sparse representation — the image that preserves the most important structures (edges, features) while suppressing noise. This is exactly what wavelet fusion does: at each spatial location and frequency scale, it selects the coefficient with the largest magnitude from the Z-stack.

Why wavelet outperforms MIP: Maximum Intensity Projection selects the brightest pixel, which is not necessarily the most focused. A noisy out-of-focus plane can have bright noise spikes that "win" the MIP competition at scattered pixels, introducing salt-and-pepper noise artifacts. Wavelet fusion operates in the frequency domain where in-focus content produces large high-frequency coefficients (sharp edges) while noise produces small coefficients at random positions. By selecting the largest coefficients, wavelet fusion inherently favors signal over noise.

Hanrahan's signal processing perspective: Each Z-slice is a sample of the 3D specimen at a different depth. Projection is reconstruction — combining these samples into a single representation. The choice of reconstruction method (pixel-wise max, mean, or wavelet) determines how faithfully the 3D information is preserved in 2D. Wavelet methods provide the best reconstruction because they respect the multi-scale structure of biological images.

Processing a 10 µm thick tissue section imaged with 5 Z-planes at 2 µm intervals:

1. Each Z-plane shows different structures in focus: surface cells in plane 1, deeper cells in plane 5
2. Wavelet Projection (Daubechies 2) fuses all planes → single all-in-focus image
3. Nuclei Detection on the projected image finds cells at all depths equally well

Without projection, detection on a single focal plane would miss cells above or below focus — potentially underdetecting 30-50% of nuclei in a thick section.