Complex Numbers: Apply It

  • Express square roots of negative numbers as multiples of i.
  • Plot complex numbers on the complex plane.
  • Add, subtract, and multiply complex numbers.
  • Rationalize complex denominators

Complex Numbers in Medical Imaging

Medical imaging technologies like MRI and CT scanners rely on complex numbers to transform raw signal data into the detailed images doctors use for diagnosis. These systems collect frequency information that must be mathematically processed using complex number operations to reconstruct clear, accurate medical images.

MRI Signal Processing

Magnetic Resonance Imaging (MRI) machines detect radio frequency signals emitted by hydrogen atoms in the body. These signals are naturally complex, containing both amplitude and phase information that complex numbers can represent mathematically.

An MRI scanner receives signals from a patient’s brain tissue. The raw signal from one location is [latex]S = 150 + 200i[/latex], where the real part represents the signal strength in phase with the reference, and the imaginary part represents the signal strength [latex]90^\circ[/latex] out of phase.

To determine the total signal intensity (what appears as brightness in the final image), radiologists calculate the magnitude: [latex]|S| = \sqrt{150^2 + 200^2} = \sqrt{22,500 + 40,000} = \sqrt{62,500} = 250[/latex]

Different tissues produce different phase signatures: gray matter might produce [latex]S_1 = 150 + 200i[/latex], while white matter produces [latex]S_2 = 180 + 120i[/latex], allowing doctors to differentiate brain structures.

magnitude

In medical imaging, the magnitude [latex]|z| = \sqrt{a^2 + b^2}[/latex] determines pixel brightness, while the phase angle helps identify tissue composition and detect abnormalities.

Fourier Transform Reconstruction

Medical scanners collect data in the frequency domain and use complex-valued Fourier transforms glossary: mathematical technique converting between time/space and frequency domains to reconstruct spatial images. This process requires extensive complex number arithmetic.

How to: Processing Medical Image Data

  1. Collect complex frequency data from scanner detectors
  2. Apply complex multiplication to adjust signal phases
  3. Add multiple complex signals from different detector positions
  4. Calculate magnitudes for final pixel intensities
  5. Use phase information to enhance tissue contrast
  6. Combine real and imaginary components to reconstruct spatial information

A CT scanner collects projection data from multiple angles around a patient’s chest. Three key frequency components are:

  • [latex]F_1 = 80 + 60i[/latex] (low frequency component)
  • [latex]F_2 = 40 - 30i[/latex] (medium frequency component)
  • [latex]F_3 = 20 + 15i[/latex] (high frequency component)

To reconstruct one pixel, the system combines these frequencies:\begin{aligned}
P &= F_1 + F_2 + F_3 \\
&= (80 + 60i) + (40 – 30i) + (20 + 15i) \\
&= (80 + 40 + 20) + (60 – 30 + 15)i \\
&= 140 + 45i
\end{aligned}

The pixel intensity is [latex]|P| = \sqrt{140^2 + 45^2} = \sqrt{19,600 + 2,025} = \sqrt{21,625} = 147[/latex]

This intensity value gets converted to a grayscale level in the final medical image.

When adding complex numbers, combine like terms: add all real parts together and add all imaginary parts together.

Contrast Enhancement and Filtering

Medical imaging systems use complex number multiplication to enhance contrast between healthy and diseased tissues, making diagnoses more accurate.

A mammography system applies a complex filter [latex]H = 1.2 + 0.8i[/latex] to enhance tumor detection. When this filter processes a tissue signal [latex]T = 100 + 50i[/latex]:

[latex]\begin{align} T_{enhanced} &= H \cdot T \\ &= (1.2 + 0.8i)(100 + 50i) \\ &= 1.2 \cdot 100 + 1.2 \cdot 50i + 0.8i \cdot 100 + 0.8i \cdot 50i \\ &= 120 + 60i + 80i + 40i^2 \\ &= 120 + 140i + 40(-1) \\ &= 120 + 140i - 40 \\ &= 80 + 140i \end{align}[/latex]

Original signal magnitude: [latex]|T| = \sqrt{100^2 + 50^2} = \sqrt{12,500} = 111.8[/latex]

Enhanced signal magnitude: [latex]|T_{enhanced}| = \sqrt{80^2 + 140^2} = \sqrt{26,000} = 161.2[/latex]

The filter increased signal strength by [latex]\frac{161.2}{111.8}=0.44 = 44[/latex]%, making potential abnormalities more visible to radiologists.

Phase-Sensitive Imaging Techniques

Advanced medical imaging uses phase information from complex numbers to detect subtle tissue changes that magnitude alone cannot reveal.

An experimental brain imaging technique compares signals from two adjacent regions:

  • Region A (healthy): [latex]S_A = 180 + 240i[/latex]
  • Region B (potentially damaged): [latex]S_B = 220 + 140i[/latex]

The phase difference analysis involves dividing one signal by the other: [latex]\frac{S_B}{S_A} = \frac{220 + 140i}{180 + 240i}[/latex]

Rationalizing the denominator: [latex]\frac{S_B}{S_A} = \frac{220 + 140i}{180 + 240i} \cdot \frac{180 - 240i}{180 - 240i} = \frac{(220 + 140i)(180 - 240i)}{180^2 + 240^2}[/latex]

[latex]= \frac{39,600 - 52,800i + 25,200i - 33,600i^2}{32,400 + 57,600} = \frac{39,600 - 27,600i + 33,600}{90,000}[/latex]

[latex]= \frac{73,200 - 27,600i}{90,000} = 0.813 - 0.307i[/latex]

The magnitude [latex]|0.813 - 0.307i| = 0.869[/latex]. This indicates Region B has [latex]1.000-0.869=-.13=13[/latex]% lower signal intensity, potentially indicating tissue damage.