Tag Archives: dft

Fourier Optics, MTF, PSF, Sharpness, Spatial Resolution

Introduction to Texture MTF

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Texture MTF is a method to measure the sharpness of a digital camera and lens by capturing the image of a target of known characteristics.  It purports to better evaluate the perception of fine details in low contrast areas of the image – what is referred to as ‘texture’ – in the presence of noise reduction, sharpening or other non-linear processing performed by the camera before writing data to file.

Figure 1. Image of Dead Leaves low contrast target. Such targets are designed to have controlled scale and direction invariant features with a power law Power Spectrum.

The Modulation Transfer Function (MTF) of an imaging system represents its spatial frequency response,  from which many metrics related to perceived sharpness are derived: MTF50, SQF, SQRI, CMT Acutance etc.  In these pages we have used to good effect the slanted edge method to obtain accurate estimates of a system’s MTF curves in the past.[1]

In this article we will explore proposed methods to determine Texture MTF and/or estimate the Optical Transfer Function of the imaging system under test from a reference power-law Power Spectrum target.  All three rely on variations of the ratio of captured to reference image in the frequency domain: straight Fourier Transforms; Power Spectral Density; and Cross Power Density.  In so doing we will develop some intuitions about their strengths and weaknesses. Continue reading Introduction to Texture MTF →

Quantization, Sensors, Spatial Resolution

The Units of Discrete Fourier Transforms

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This article is about specifying the units of the Discrete Fourier Transform of an image and the various ways that they can be expressed.  This apparently simple task can be fiendishly unintuitive.

The image we will use as an example is the familiar Airy Pattern from the last few posts, at f/16 with light of mean 530nm wavelength. Zoomed in to the left in Figure 1; and as it looks in its 1024×1024 sample image to the right:

Airy Mesh and Intensity
Figure 1. Airy disc image I(x,y). Left, 1a, 3D representation, zoomed in. Right, 1b, as it would appear on the sensing plane (yes, the rings are there but you need to squint to see them).

Continue reading The Units of Discrete Fourier Transforms →

Optics, Sensors, Sharpness, Spatial Resolution

A Simple Model for Sharpness in Digital Cameras – Diffraction and Pixel Aperture

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Now that we know from the introductory article that the spatial frequency response of a typical perfect digital camera and lens (its Modulation Transfer Function) can be modeled simply as the product of the Fourier Transform of the Point Spread Function of the lens and pixel aperture, convolved with a Dirac delta grid at cycles-per-pixel pitch spacing

(1)   \begin{equation*} MTF_{Sys2D} = \left|\widehat{ PSF_{lens} }\cdot \widehat{PIX_{ap} }\right|_{pu}\ast\ast\: \delta\widehat{\delta_{pitch}} \end{equation*}

we can take a closer look at each of those components (pu here indicating normalization to one at the origin).   I used Matlab to generate the examples below but you can easily do the same with a spreadsheet.   Continue reading A Simple Model for Sharpness in Digital Cameras – Diffraction and Pixel Aperture →