Now that the components of the modulation transfer function (MTF), resolution and contrast/modulation, are defined, consider MTF itself. The MTF of a lens, as the name implies, is a measurement of its ability to transfer contrast at a particular resolution from the object to the image. In other words, MTF is a way to incorporate resolution and contrast into a single specification. As line spacing decreases (i.e. the frequency increases) on the test target, it becomes increasingly difficult for the lens to efficiently transfer this decrease in contrast; as result, MTF decreases (Figure 6).

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MTF is one of the best tools available to quantify the overall imaging performance of a system in terms of resolution and contrast. As a result, knowing the MTF curves of each imaging lens and camera sensor within a system allows a designer to make the appropriate selection when optimizing for a particular resolution.

To understand the relation between contrast and image quality, consider an imaging lens with the same resolution as the one in Figure 1 and Figure 4, but used to image an object with a greater line-pair frequency. Figure 5 illustrates that as the spatial frequency of the lines increases, the contrast of the image decreases. This effect is always present when working with imaging lenses of the same resolution. For the image to appear defined, black must be truly black and white truly white, with a minimal amount of grayscale between.

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Knowledge Center/ Application Notes/ Optics Application Notes/ Introduction to Modulation Transfer Function

In imaging applications, the imaging lens, camera sensor, and illumination play key roles in determining the resulting image contrast. The lens contrast is typically defined in terms of the percentage of the object contrast that is reproduced. The sensor's ability to reproduce contrast is usually specified in terms of decibels (dB) in analog cameras and bits in digital cameras.

The above finishes are some of many mechanical finishes which are often used in the metal finishing industry. Both the NiDi No. 9012 and previous revisions of ASME BPE specifications include a list that describes various mechanical surface finishes for stainless steel.

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As described in ASME B46.1, Ra is the arithmetic average of the absolute values of the profile height deviations from the mean line, recorded within the evaluation length. Simply put, Ra is the average of a set of individual measurements of a surfaces peaks and valleys. Reveal the Ra formula for more insight.

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Consider normalizing the intensity of a bar target by assigning a maximum value to the white bars and zero value to the black bars. Plotting these values results in a square wave, from which the notion of contrast can be more easily seen (Figure 3). Mathematically, contrast is calculated with Equation 3:

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When this same principle is applied to the imaging example in Figure 1, the intensity pattern before and after imaging can be seen (Figure 4). Contrast or modulation can then be defined as how faithfully the minimum and maximum intensity values are transferred from object plane to image plane.

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A practical way of understanding line-pairs is to think of them as pixels on a camera sensor, where a single line-pair corresponds to two pixels (Figure 2). Two camera sensor pixels are needed for each line-pair of resolution: one pixel is dedicated to the red line and the other to the blank space between pixels. Using the aforementioned metaphor, image resolution of the camera can now be specified as equal to twice its pixel size.

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As described in ASME B46.1, RMS is the root mean square average of the profile height deviations from the mean line, recorded within the evaluation length. Reveal the RMS formula for more insight.

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When optical designers attempt to compare the performance of optical systems, a commonly used measure is the modulation transfer function (MTF). MTF is used for components as simple as a spherical singlet lens to those as complex as a multi-element telecentric imaging lens assembly. In order to understand the significance of MTF, consider some general principles and practical examples for defining MTF including its components, importance, and characterization.

Ra and RMS are both representations of surface roughness, but each is calculated differently. Ra is calculated as the Roughness Average of a surfaces measured microscopic peaks and valleys. RMS is calculated as the Root Mean Square of a surfaces measured microscopic peaks and valleys. Each value uses the same individual height measurements of the surfaces peaks and valleys, but uses the measurements in a different formula. The formulas are shown below. One can infer from examination of the formulas, that a single large peak or flaw within the microscopic surface texture will effect (raise) the RMS value more than the Ra value.

Under the same testing parameters, it is clear to see that #59-871 (with a better MTF curve) yields better imaging performance compared to #54-855 (Figures 11 – 12). In this real-world example with these particular 1951 USAF elements, a higher modulation value at higher spatial frequencies corresponds to a clearer image; however, this is not always the case. Some lenses are designed to be able to very accurately resolve lower spatial frequencies, and have a very low cut-off frequency (i.e. they cannot resolve higher spatial frequencies). Had the target been group -1, elements 5-6, the two lenses would have produced much more similar images given their modulation values at lower frequencies.

Figure 6 plots the MTF of an aberration-free image with a rectangular pupil. As can be expected, the MTF decreases as the spatial resolution increases. It is important to note that these cases are idealized and that no actual system is completely aberration-free.

In traditional system integration (and less crucial applications), the system's performance is roughly estimated using the principle of the weakest link. The principle of the weakest link proposes that a system's resolution is solely limited by the component with the lowest resolution. Although this approach is very useful for quick estimations, it is actually flawed because every component within the system contributes error to the image, yielding poorer image quality than the weakest link alone.

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Every component within a system has an associated modulation transfer function (MTF) and, as a result, contributes to the overall MTF of the system. This includes the imaging lens, camera sensor, image capture boards, and video cables, for instance. The resulting MTF of the system is the product of all the MTF curves of its components (Figure 7). For instance, a 25mm fixed focal length lens and a 25mm double gauss lens can be compared by evaluating the resulting system performance of both lenses with a Sony monochrome camera. By analyzing the system MTF curve, it is straightforward to determine which combination will yield sufficient performance. In some metrology applications, for example, a certain amount of contrast is required for accurate image edge detection. If the minimum contrast needs to be 35% and the image resolution required is $ 30 \tfrac{\text{lp}}{\text{mm}} $, then the 25mm double gauss lens is the best choice.

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The electropolishing process may improve a surface finish by up to 50%. The electropolishing reaction removes material while it improves surface roughness. Because of the material removal, process runtimes are often limited to maintain dimensional tolerances. Limited runtimes result in real world surface roughness improvements from 10 to 35%. Remember, electropolishing improves a surface on the microscopic level. If a material has a texture or surface scratch, electropolishing will only result in a lustrous texture or lustrous scratch. Mechanical polishing is utilized to remove macroscopic texture or blemishes.

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A greater area under the MTF curve does not always indicate the optimal choice. A designer should decide based on the resolution of the application at hand. As previously discussed, an MTF graph plots the percentage of transferred contrast versus the frequency (cycles/mm) of the lines. A few things should be noted about the MTF curves offered by Edmund Optics®:

Conceptually, MTF can be difficult to grasp. Perhaps the easiest way to understand this notion of transferring contrast from object to image plane is by examining a real-world example. Figures 8 - 12 compare MTF curves and images for two 25mm fixed focal length imaging lenses: #54-855 Finite Conjugate Micro-Video Lens and #59-871 Compact Fixed Focal Length Lens. Figure 8 shows polychromatic diffraction MTF for these two lenses. Depending upon the testing conditions, both lenses can yield equivalent performance. In this particular example, both are trying to resolve group 2, elements 5 -6 (indicated by the red boxes in Figure 10) and group 3, elements 5 – 6 (indicated by the blue boxes in Figure 10) on a 1951 USAF resolution target (Figure 9). In terms of actual object size, group 2, elements 5 – 6 represent 6.35 – $ 7.13\tfrac{\text{lp}}{\text{mm}} $ (14.03 - 15.75μm) and group 3, elements 5 – 6 represent 12.70 – $ 14.25 \tfrac{\text{lp}}{\text{mm}} $ (7.02 - 7.87μm). For an easy way to calculate resolution given element and group numbers, use our 1951 USAF Resolution Edmund Optics Tech Tool.

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A theoretical modulation transfer function (MTF) curve can be generated from the optical prescription of any lens. Although this can be helpful, it does not indicate the actual, real-world performance of the lens after accounting for manufacturing tolerances. Manufacturing tolerances always introduce some performance loss to the original optical design since factors such as geometry and coating deviate slightly from an ideal lens or lens system. For this reason, in our manufacturing sites, Edmund Optics® invests in optical test and measurement equipment for quantifying MTF. This MTF test and measurement equipment allow for the characterization of the actual performance of both designed lenses and commercial lenses (whose optical prescription is not available to the public). As a result, precise integration - previously limited to lenses with known prescriptions - can now include commercial lenses.

Resolution is an imaging system's ability to distinguish object detail. It is often expressed in terms of line-pairs per millimeter (where a line-pair is a sequence of one black line and one white line). This measure of line-pairs per millimeter $ \small{\left(\tfrac{\text{lp}}{\text{mm}}\right)} $ is also known as frequency. The inverse of the frequency yields the spacing in millimeters between two resolved lines. Bar targets with a series of equally spaced, alternating white and black bars (i.e. a 1951 USAF target or a Ronchi ruling) are ideal for testing system performance. For a more detailed explanation of test targets, view Choosing the Correct Test Target. For all imaging optics, when imaging such a pattern, perfect line edges become blurred to a degree (Figure 1). High-resolution images are those which exhibit a large amount of detail as a result of minimal blurring. Conversely, low-resolution images lack fine detail.

Correspondingly, object resolution is calculated using the camera resolution and the primary magnification (PMAG) of the imaging lens (Equations 1 – 2). It is important to note that these equations assume the imaging lens contributes no resolution loss.

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Modulation transfer function (MTF) is one of the most important parameters by which image quality is measured. Optical designers and engineers frequently refer to MTF data, especially in applications where success or failure is contingent on how accurately a particular object is imaged. To truly grasp MTF, it is necessary to first understand the ideas of resolution and contrast, as well as how an object's image is transferred from object to image plane. While initially daunting, understanding and eventually interpreting MTF data is a very powerful tool for any optical designer. With knowledge and experience, MTF can make selecting the appropriate lens a far easier endeavor - despite the multitude of offerings.

To properly define the modulation transfer function, it is necessary to first define two terms required to truly characterize image performance: resolution and contrast.

For an aberration-free image with a circular pupil, MTF is given by Equation 4, where MTF is a function of spatial resolution $\small{\left( \xi \right)}$, which refers to the smallest line-pair the system can resolve. The cut-off frequency $\small{\left( \xi _c \right)}$ is given by Equation 6.