Procesado de Señales e Imágenes Médicas

Ingeniería Biomédica

Ph.D. Pablo Eduardo Caicedo Rodríguez

2026-03-26

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Procesamiento de imágenes

Importance of Frequency-Response Filters

  • Frequency-response filters are critical for enhancing specific features or reducing noise in images.
  • Widely used in MRI, CT, and ultrasound imaging.

What is a Frequency-Response Filter?

  • A frequency-response filter modifies the frequency components of a signal.
  • Applied in image processing to control which frequencies (details) pass or are suppressed.

Types of Frequency-Response Filters

  • Low-pass filters: Allow low frequencies, suppress high frequencies (smoothes image).
  • High-pass filters: Allow high frequencies, suppress low frequencies (sharpens image).
  • Band-pass filters: Allow frequencies in a certain range.

Spatial vs. Frequency Domain

  • Spatial domain: Operations on pixel values directly.
  • Frequency domain: Operations on the image’s frequency components.

Fourier Transform

  • Converts an image from the spatial domain to the frequency domain.
  • Formula: \[F\left(u,v\right) = \sum\sum f\left(x,y\right) e^{-j2\pi(\frac{ux}{M} + \frac{vy}{N})}\]

Low-Pass Filters

  • Removes high-frequency components (e.g., noise, sharp edges).
  • Example: Gaussian filter, Butterworth filter.

High-Pass Filters

  • Enhances edges and high-frequency details.
  • Example: Laplacian filter.

Band-Pass Filters

  • Allows frequencies within a specific range.
  • Useful for isolating specific image features.

Noise Reduction in MRI

  • Low-pass filters reduce noise and artifacts in MRI scans.
  • Smoothes the image without losing crucial details.

Edge Enhancement in Ultrasound Images

  • High-pass filters help in detecting tissue boundaries by enhancing edges.
  • Improves clarity of anatomical structures.

Feature Extraction in CT Scans

  • Filters can help in extracting features like tumors or vessels.
  • Band-pass filters isolate structures of interest at specific frequency ranges.

Case Study: Applying a Low-Pass Filter in MRI. Step-by-step Process

  1. Load MRI image.
  2. Apply Fourier Transform to move the image into the frequency domain.
  3. Design and apply a low-pass filter.
  4. Perform Inverse Fourier Transform to return to the spatial domain.
  5. Visualize the result.

Common Frequency-Domain Filter Transformations

Let \(D(u,v)\) be the distance from the center of the spectrum:

\[ D(u,v)=\sqrt{(u-c_x)^2+(v-c_y)^2} \]

Let \(D_0\) be the cutoff frequency, and let \(D_1 < D_2\) define a frequency band.

Transformation Table

Filter type Ideal circular mask Gaussian mask Transformation relationship
Low-pass \(\displaystyle H_{LP}(u,v)=\begin{cases}1,& D\le D_0\\0,& D>D_0\end{cases}\) \(\displaystyle H_{LP}(u,v)=\exp\left(-\frac{D^2}{2D_0^2}\right)\) Base filter
High-pass \(\displaystyle H_{HP}(u,v)=\begin{cases}0,& D\le D_0\\1,& D>D_0\end{cases}\) \(\displaystyle H_{HP}(u,v)=1-\exp\left(-\frac{D^2}{2D_0^2}\right)\) \(\displaystyle H_{HP}=1-H_{LP}\)
Band-pass \(\displaystyle H_{BP}(u,v)=1 \text{ if } D_1\le D\le D_2,\; 0 \text{ otherwise}\) \(\displaystyle H_{BP}(u,v)=H_{LP}(D_2)-H_{LP}(D_1)\) Difference of two low-pass filters
Band-stop \(\displaystyle H_{BS}(u,v)=0 \text{ if } D_1\le D\le D_2,\; 1 \text{ otherwise}\) \(\displaystyle H_{BS}(u,v)=1-H_{BP}(u,v)\) \(\displaystyle H_{BS}=1-H_{BP}\)
High-frequency emphasis Not usually written in a standard ideal form \(\displaystyle H_{HFE}(u,v)=a+b\,H_{HP}(u,v)\) Uses \(a>0\), \(b>0\) to preserve background and enhance detail

Practical Conversion Rules

Conversion Formula
Low-pass \(\rightarrow\) high-pass \(\displaystyle H_{HP}=1-H_{LP}\)
Two low-pass filters \(\rightarrow\) band-pass \(\displaystyle H_{BP}=H_{LP}(D_2)-H_{LP}(D_1)\)
Band-pass \(\rightarrow\) band-stop \(\displaystyle H_{BS}=1-H_{BP}\)
High-frequency emphasis \(\displaystyle H_{HFE}=a+b\,H_{HP}\)

Summary

  • Frequency-response filters play a crucial role in biomedical image processing.
  • Help enhance key features and suppress unwanted noise.