MTF Revealed – Part I: The Samsung SP-A900B JKP Special Edition Projector

samsung-spa900_1Note: This article was revised 1/7/2010 so as to include more accurate JVC RS20 data which was included for comparison purposes at the bottom of this article.

November 30, 2009

A few months ago in this article, we took a close look at the sharpness differences between DLP and LCOS video projectors and in addition, we examined the differences between a projector that used a .65″ DC3 DMD (the Optoma 8200) and another that used the .95″ DC3 DMD (the Planar 8150). That article provided a good basis for general sharpness comparisons, but it stopped short of assigning an objective metric in order to fully describe the sharpness differences between these projectors. It was apparent from that article that the Planar 8150 was the sharpest projector of the 3, but readers were left to wonder how much sharper, 10%, 20% 80%? If the sharpness differences could be expressed as a single objective number then the comparisons could be made much easily. Fortunately, there is such a metric called MTF (modulation transfer function) which is well known in optical design and it is often used and discussed in camera and lens comparisons.

In this article, we remedy the shortcomings of the previous article and we apply considerable effort at measuring the MTF performance of a digital projector so that we can provide an objective, numeric, sharpness metric for this and subsequent articles. By measuring MTF, we are providing the first independent, detailed look at the MTF performance of a digital projector.  In doing so, we have found a wealth of information that can be used for comparisons between projectors and display technologies. Most importantly, we have found that it can also provide valuable information about how the sharpness of a given projector can be affected by real-world constraints and modes such as throw distances, lens shift, iris settings, lamp output, etc. Information that owners can directly apply towards maximizing the image and performance of their projector.

MTF (Modulation Transfer Function) Overview

MTF is the definitive tool used to measure and specify the sharpness of optical systems.  Unfortunately, providing a detailed explanation of MTF theory is well beyond the scope of this article and it’s something that is well covered in text books and articles on the Internet (as an example here is a good MTF introduction). Fortunately, when boiled down to its essence, MTF is actually a very simple concept that describes how well an optical system can resolve black and white images. Typically a pair of white lines on a black background (or black lines on a white background) are used and the thickness and distance of the lines are varied, which is used to glean information about the response or optical performance of the system as a function of frequency. The idea being that thick lines (low frequency) are much easier for an optical system to resolve than thin (high frequency) lines. The spacing of the line pairs is referred to as the “spatial frequency” and is usually expressed as line pairs per millimeter (lp/mm). After being passed through the optical system, the contrast of a specific group of line pairs (with a specific spatial frequency) is measured and the MTF is determined using this formula:

MTF = (Imax – Imin) / (Imax + Imin) Where Imin and Imax refer to the Intensity of light for the black (Imin) and white (Imax) portions of the line pairs – as measured after passing through the optical system.

As the spacing and thickness of the line pairs is reduced, the resultant MTF changes in response to each of these frequencies and a plot of MTF vs spatial frequency is created. When the contrast differences are high, the MTF will approach 1 (100%) and when the contrast differences are low the MTF will be low. At higher frequencies, the MTF is reduced and there becomes a point where the line pairs at this high spatial frequency can no longer be resolved and this is referred to as the limiting frequency or the Nyquist Limit. With digital 1080p projectors, the highest frequency that can be represented is single pixel on and off line pairs, and these 960 vertical line pairs (or 540 horizontal line pairs) are the limiting frequency for digital projectors. In theory it’s possible for a digital projector to have a nyquist limit below the 1080p on/off pattern but in actual practice all digital projectors seem to be able to easily resolve this test pattern. What is of most interest then is what is the measured MTF at this Nyquist Limit and if it is low what is the shape of the MTF curve at other frequencies.

In practice, two different methods are usually used to determine a MTF plot, one uses a slit or knife edge measurement to collect line spread data which can then be used in a Fourier Transform to calculate the MTF at all frequencies. This method has the benefit of being fast, but the math is complex (although tools exist for quickly performing the calculations – the FFT) and the results are relatively general because they rely on extrapolations of calculations rather than precise measurements at each specific spatial frequency. The other method that is used involves a series of line pair test patterns to measure the MTF directly and precisely for each frequency, but the downside to this method is that it is more laborious and requires many measurements and a series of test patterns. Being a glutton for punishment and after investing in a relatively expensive CCD line camera, I decided to put it to good use and follow this latter approach for this article, although subsequent articles may use the other approach (and possibly compare the results of both approaches).

It should also be mentioned that the white and black line pairs are traditionally created as a sinusoidal test pattern where the intensity of white gradually changes from black to white rather than as a step function using one level of full black and another level of full white. Using sinusoidal patterns fits well with the mathematics that are used in the Fourier calculations. Unfortunately, with the advent of digital displays, sinusoidal patterns are impossible to render at high frequencies because there aren’t enough pixels in the line pair to provide the smooth transitions between the black and white peaks. This is a problem faced by anyone applying MTF to a digital device where the pixels themselves are used to provide the stimulus. This means that the Fourier calculations used to determine MTF can break down somewhat, although in practice using on and off test patterns does not usually change the results substantially. Even still, this is another reason why we chose to measure MTF directly. It should also be noted that at larger frequencies where enough pixel resolution was available to create a smooth transition between peaks, both types of test patterns were used and the results obtained were almost identical.

MTF examples – CRT and JVC QX-1 projector

So lets take a look at an example of an MTF plot. Below is a hypothetical representation of the MTF plot of a CRT projector, the vertical line represents the maximum spatial frequency for 1920×1080 HD content and as we can see the MTF of our hypothetical CRT is 30% at this limit. Because our hypothetical CRT is an analog device it may be possible to drive it to higher frequencies than 1080p and as we do so we can see that the MTF drops off rapidly before reaching a limiting frequency (the nyquist limit) which is typically quoted as being when the MTF reaches 5%.


Hypothetical Example - CRT MTF

Next lets take a look at an actual test case of a digital projector (JVC QX-1) which was published in this SPIE white paper. According to the article the plot below was obtained from a high resolution line camera using a camera and techniques very similar to those used in this and subsequent articles.


Actual MTF Data - JVC QX1 Projector

Unlike our hypothetical CRT example, here we have a digital projector with a fixed pixel count (2048×1536 in this case). We can see that the MTF is much higher than the previous CRT example and this applies to all frequencies up to the nyquist limit (the vertical line). The key difference in the case of the QX-1 projector compared to the CRT is the QX-1 has a fixed pixel count which fundamentally limits the spatial frequency (nyquist limit) to no more than the one pixel on and one pixel off line pair pattern which is indicated by the vertical line. With this particular projector, the pixel count is higher than 1920×1080 so the frequency limit will in turn be higher. This projector is therefore capable of reaching higher spatial frequencies than those possible with 1080p source content. The white paper mentions that the MTF at the Nyquist Limit is greater than 75%, but the reader should realize that with slightly lower spatial frequencies (i.e. those used in 1920×1080 content), the MTF will be even higher than that shown by the nyquist limit. Incidentally, the MTF plot shown here continues beyond the Nyquist Limit, but the reader should realize that this represents theoretical optical performance derived from the fourier calculation and this performance will never be realized unless the pixel count of the microchip is increased or the size of the microchip is reduced. Fundamentally the MTF of this particular projector with the given microchip ends at the Nyquist Limit and all of our MTF plots will also end at this limit.

What MTF doesn’t tell us

As we will see, we have taken all sorts of MTF measurements – measurement in the horizontal and vertical planes, in individual RGB colors, as a function of light intensity and with a variety of iris settings, projector throws, lens shifts and projector modes. Despite all of these measurements there are still many measurements that we haven’t performed. For example we are not measuring the sharpness across the field and in particular at the edges. Without these measurements we can not determine coma and other optical effects that may be present. It isn’t the goal of this article to completely characterize the optical performance of projectors and fortunately, we don’t have to in order to glean important aspects of the sharpness performance of a projector. Similarly, it is well known that optical quality may vary across units and it is not our intention to make the claim that these results will hold for all projectors of a given brand or model. In this respect, these measurements are no different from any other metric such as color accuracy or contrast performance that are subject to the same unit to unit vagueries and variances and yet these other metrics are still widely reported in magazines and projector reviews.

It is also worth noting that what is being measured is the overall system MTF that includes the chip and optical system. The individual components including the projector lens will have higher MTF if measured by themselves.

Edit: It should also be obvious to the reader that sharpness is only one of many attributes that contribute to good image quality. Contrast, greyscale reproduction, brightness, color accuracy, feature sets, throw, price, etc. will all play a role in the purchase of a projector and sharpness is only one of those factors. Ultimately, the relative merit of sharpness over any other factor is an individual decision and an article like this will only help to provide insight into the sharpness factor alone.

Enter The Samsung SP-A900B JKP Special Edition Projector

Now that we’ve presented the basics on what MTF is and how to interpret the charts, lets move on to the meat of this article which is a detailed examination of the MTF of the Samsung SP-A900B projector. This projector was chosen as the first projector for this MTF series specifically because it has outstanding full field optics and is a 1-chip design with no convergence issues and little chromatic aberration. As such it sets the bar very high and gives us an idea what top-tier projectors are capable of delivering as far as sharpness. The projector used was also not a cherry picked review sample, but instead purchased and shipped at random, so hopefully the results should be close to what a buyer might receive, unit to unit variances not withstanding.

First let’s take a look at what the pixel data looks like from a high resolution CCD line camera. The image below is data taken from the Samsung projector for a single pixel on and single pixel off pattern. As we can this this provides very precise details of the light intensity for both the bright and dark pixels. As can be see the contrast differences between light and dark pixels is only around 15:1 even with this very sharp projector. The camera is capable of resolving inter-pixel contrast differences of well over 300:1, so the results can be very precise and not limited by the accuracy of the equipment. The reader should note that the x-axis is labeled “pixels” and this refers to the CCD camera pixels rather than the projector pixels. Here we have oversampled and imaged 18 projector pixels (9 on and 9 off) with 3000 camera pixels in order to achieve precise results. It should also be noted that this line scan is imaged directly off of the projector so that camera optics or screen effects are not an issue.


One Pixel on, One Pixel off Linescan - Samsung A900B (Click to Enlarge)

So taking the line scan contrast information above, we can calculate the MTF for this frequency (40 lp/mm) and do it similarly for other frequencies using on/off test patterns of various pixel widths. For the chart below we used 7 different test patterns that varied the line width and spacing from one to 7 pixels. If we plot this data we get the MTF curve for the Samsung, which is shown below. As can be seen below the Samsung has outstanding sharpness across all frequencies and with single pixel on/off patterns (the nyquist limit) it achieves a whooping 88% MTF.


Samsung MTF Plot - Vertical Line Pairs

The spatial frequency in the plot above was determined by dividing the line pairs by the chip size (.95″) which is traditionally how projector MTF data is presented. Since chip sizes vary and we would like to compare the results from many chips, the data was recharted using spatial frequencies determined by the equivalent of an 8′x4.5′ screen. This is the same way that the QX-1 data was presented in the JVC white paper mentioned above with the exception that they utilized the equivalent of a 12′ wide screen rather than an 8′ wide screen. Despite the common practice of expressing spatial frequency based on chip size, it is very convenient and desirable to plot the data based on screen size because to the user this is all that matters – in other words, what is the sharpness on a given screen independent of chip size and other factors.

At this point however, the reader should realize that larger chips like the .95″ DC4 DMD have an inherent resolution advantage over a smaller chip like the .65″ DC3 DMD if the pixel count is the same. This is because in real terms, the spatial frequency in line pairs per millimeter of the smaller chip will be higher. Another way to look at it – if the pixel pitch remained constant, but the smaller chip were increased in size to equal the larger chip, it would have a higher pixel count and the nyquist limit would be higher. This is an important consideration that the reader should keep in mind when drawing conclusions about technology differences (LCOS vs DLP for example) based on MTF numbers alone when those chips have dissimilar sizes (.95″ for DLP for example, .7″ for DILA, .6″ for SXRD). In other words, the reasons for a lower MTF may have more to do with the chip size itself (or something else like the optics) than anything to do with differences in the technology.


The plot above shows MTF with spatial frequency represented by line pairs on the equivalent of an 8′ wide screen. Note that the shape of the curve has remained unchanged as has the MTF at the nyquist limit, but now we have a uniform way of comparing MTF across chips that vary in size, which is something that we will do with subsequent articles.

Since the MTF curve above ended up being so high and relatively uninteresting across the other frequencies, all of the subsequent Samsung measurements will all be done at the Nyquist limit using single on/off pixel patterns.

MTF Horizontal vs Vertical Line Pairs

Now lets take a look at what happens if we use horizontal vs vertical on/off line pairs. This is a useful measurement for 3-panel projectors as the convergence in the horizontal and vertical directions may not be the same.

  • Horizontal MTF: 88%

  • Vertical MTF: 88%

In other words the MTF was identical in both directions. Incidentally it’s worth pointing out that the spatial frequency is the same in both directions even though there are more vertical lines than horizontal line pairs and this is because the density of lines is the same in both directions.

MTF of Individual Colors – R,G, B and White

Now let’s measure the MTF of each individual color. This will be an interesting test in future articles on 3-chip projectors because it can tell us how much the convergence of the 3 panels may be holding back the MTF of a projector. On a 1-chip (and also a 3-chip display), it also tells us which color has the best focus and if the optical system is limited because of the inability to focus one or more colors.

  • Red MTF: 83%

  • Blue MTF: 91%

  • Green MTF: 93%.

Since green is the dominant color in both Rec. 709 and 601 color standards, it is the critical color to optimize and the Samsung does an outstanding job with both this color and also blue. Red is the second most dominant color and here we see a drop-off in performance, which if corrected without harming the other colors, could allow the Samsung to hit MTF values in the low 90s.

It should also be mentioned that these measurements were obtained after optimizing the focus for white. While it’s possible to refocus when measuring each individual color which would have provided higher results for each color, this has little value because it will have harmed the focus for white and it is the best MTF for all colors displayed simultaneously that we are interested in. It should be noted however, that if the display suffers from color bleeding from misconvergence or chromatic aberration then it might be beneficial to maximize the focus (i.e. minimize the pixel spot size) of the problem color at the expense of less overall sharpness, but this situation was not the case with the Samsung and was not necessary.

Other than providing MTF for each individual color, there were no additional measurements for chromatic aberration. Measurements of pixel spot size is something else that can readily be performed with the equipment and techniques used in this article and perhaps we will include these measurements in subsequent articles.

MTF vs Lamp Setting, projector modes and iris settings.

MTF was measured for both the bright and theater modes of the Samsung and found to be:

  • Bright mode MTF: 84%

  • Theater mode MTF: 88%

Only a small loss in sharpness was found between lamp modes. Next let’s put the iris into manual mode and measure it at the smallest aperture position (iris =100).

  • MTF with Iris at largest aperture (Manual Iris mode): 88%

  • MTF with Iris at smallest aperture (Manual Iris mode): 76%

The lens was refocused in this iris setting with the thought that perhaps the iris movement caused the lens to slightly defocus, but all focus attempts did not improve on this MTF. Owners should be aware that using the projector in this mode can cause a reduction in sharpness. This is particularly important if they have reduced the iris aperture so as to throttle back the light output when attempting to compare the performance of this projector against other projectors that have less light output.

MTF vs White Intensity

So far all of the line pairs utilized 100% peak whites. Out of curiousity, let’s see if varying the intensity of white (gray) will significantly change the MTF.

  • MTF – 100% white: 88%

  • MTF – 60% white: 87%

  • MTF – 40% white: 86%

In short negligible differences in MTF are found when ramping down the white peaks from 100% to 40%. The reader might wonder why the MTF is unchanged since after all, MTF is primarily a contrast measurement and we’ve just drastically changed the brightness of the white pixels. The answer of course is that the black values representing the off pixels has changed also and by about the same amount which leaves the MTF unchanged.

MTF vs Throw Distance

MTF was also measured at maximum and minimum throw distances. Initially the MTF was seen to change significantly, but this is because the focus shifted. After refocusing the difference between longest and shortest throws was unchanged.

  • MTF – Longest throw: 88%

  • MTF – Shortest throw: 88%

MTF vs Lens Shift

All of the measurements performed so far unless otherwise noted were made close to the center of the image and with a neutral vertical lens shift (the Samsung has no horizontal lens shift adjustment). With the Samsung optical design, using a neutral lens shift causes the exit light of the projected image to remain centered in the lens and it falls on an area that is much smaller than the diameter of the lens. When lens shift is added, the exit light is shifted from center and falls closer to the edge of the lens which is why a large diameter lens is needed. Adjusting the lens shift from maximum top to maximum bottom lens shift effectively sweeps the projected image from the top of the lens through center and then to the bottom of the lens. In measuring MTF vs lens shift, the camera was at all times positioned at the center of the image while the image itself was lowered. MTF vs lens shift was measured in only one direction (down) and this would be similar in orientation to someone mounting the projector high on a shelf and using lens shift to lower the image onto a screen.

It’s been widely reported and generally assumed that using lens shift degrades the sharpness of an image and is generally something to be avoided. With some projectors it’s been reported that using some lens shift can raise ANSI contrast somewhat because reflected light from the lens elements will be off-axis and will not project directly back into the optical path. Before doing doing this measurement the assumption was that this measurement would show a degradation in sharpness and the tradeoff/downside of using lens shift. What was found was very surprising….

  • MTF – Centered Lens Shift: 88%

  • MTF – 1/2 Maximum Lens Shift: 89%

  • MTF – Maximum lens shift: 91%

These results run counter to conventional wisdom and were very surprising. Increasing lens shift actually made slight improvements in sharpness to the center of the image on this particular projector! The reasons for this are unclear and may be related to the same reasons that ANSI contrast can be improved with lens shift or perhaps it was simply because this particular lens just happened to have a sweeter focus at the edge of the lens than at the center, or perhaps it’s a combination of both factors.

Edit: It was pointed out to me by a Samsung owner that the extreme edge of the field will distort when maximum lens shift is applied even though the center (the area being measured) may still have a sharp focus. As mentioned earlier in the article, only a few pixels are being measured and there can be deviations across the full field so please keep this in mind with all of these numbers.

MTF and Intra-Image Contrast

There has been discussion about the perceptions of both intra-image contrast and color with high MTF. Unfortunately such discussions go beyond the scope of this article. However, there were some interesting discoveries relating to intra-image contrast that came out of this detailed look at MTF that are worth discussing.

Variance of White

The first is that it’s been an open question with many display types whether the light intensity of white pixels vary significantly as the area of a white region is made smaller and smaller. With reflective displays like LCOS and DLP, the large white regions in a test pattern such as the ANSI contrast pattern are pretty much unchanged throughout the rectangle (excepting relatively issues such as optical rolloff or shading uniformity). But what happens when the white regions are made very small? In my own experiments with low APL intra-image contrast, I’ve found that the white regions maintain their white points very well until the white regions become too small to be measured reliably with a light meter. If we keep making the rectangles smaller, what happens to the white level once we get down to clusters of a few pixels and even the individual pixel level? By employing the techniques used in this article it’s possible to measure the white level for a white field all the way down to a single pixel and the results are startling. With the Samsung DLP projector, single pixel white levels were 94% of what they are with full field white. In fact here is the data:


With a LCOS display (JVC RS35) this same experiment was repeated and single pixel white levels were maintained at 79% of the full field value and well into the 90% range with two pixels. So with both of these reflective technologies, there is only a small rolloff in white level as white regions are decreased to a few pixels.

This in turn makes it very easy to measure intra-image contrast as it is only the blacks that are significantly changing and need to be measured. As an example, say we take something like the ANSI contrast test pattern displayed on a reflective DLP or LCOS projector and shrink the white rectangles down to something very small and on the order of a few pixels. When we measure the black area (not in the immediate region of the small white regions) we will find almost no washout from the white pixels and we will end up measuring very close to the black level floor of the projector. Since we know now that the white levels in the small cluster of pixels are unchanged, we will have measured intra-image contrast that is very close if not equal to the native on/off contrast of the projector, which is a startling result. To be sure there are few film scenes that contain single pixels of full white in a perfectly black field but this does have important implications in very dark scenes. As an example picture a space scene with only bright stars on a black field. We now know that in such a scene the stars will be rendered by a DLP or LCOS projector at close to what is called for by the luma value of those white pixels in conjunction with the display gamma. Since the small amount of display luminance will cause little washout (the stars are small) and the black level will be very low (maybe not at the black level floor of the projector but at some low value determined by film black and how the film was transferred and encoded to video). Bottom line, the the intra-image contrast will be relatively predictable and it will be much higher than most people may have previously thought. This realization helps to validate the importance of native sequential (on/off) contrast as a performance metric in dark scenes which is something that is often overlooked.

Perceptions of Sharpness

While we are avoiding the topic of the perceptions of contrast and color due to sharpness, it is worth spending some time to discuss the perception of sharpness itself. In particular, the visibility of the pixel grid surrounding a pixel will be perceived by the HVS (Human Visual System) and will have some impact, for better or worse on the perception of sharpness. From an MTF perspective, all that matters in determining sharpness is the light intensity of on pixels and off pixels. The definition of the pixels and whether they appear as round, square or something else has no bearing and does not change the MTF results. In particular, the fill ratio and the screen door effect (SDE) which results from a visible pixel grid has no bearing on MTF results so long as it doesn’t affect the light intensity of the on or off-state pixels. To a human, however, the visibility of the pixel grid can provide the perception of improved sharpness because the eye is seeing fine details that are smaller than the pixel itself (the pixel grid). At the subconscious level, the presence and perception of this fine detail may act to increase the perception of sharpness. For the same reason, the presence of film grain in movies has sometimes been cited as providing a heightened perception of sharpness, even if this perception of sharpness does not translate into seeing more true detail in the movie itself. Both SDE and film grain then may be perceived in ways that are similar to the perceptions of edge enhancement, which is a well known technique that increases the perception of sharpness even though it provides no fundamental improvement in resolution to the content itself (and in some cases can be detrimental to true detail).

With those thoughts in mind, it’s worth looking at the pixel line scans of the Samsung DLP and examine the visible pixel grid structure from 5 “on” pixels and compare them to a similar scan from a JVC RS20 projector. As we will see in subsequent articles, the JVC RS20 projector has lower MTF and less pixel grid structure and it’s anyone’s guess how much the increased perception of sharpness of the Samsung over the JVC is based on the MTF improvement alone or the combination of the higher MTF and more perceptible pixel grid.


In the image above for the Samsung A900, the large peak represents 5 “on” pixels surrounded by 5 “off” pixels and the pixel grid can be seen as the valleys between the peaks (“fingers”) representing the 5 on pixels. By comparison, the 6 “on” pixels on the JVC RS20 as seen below are relatively flat because the pixel grid is less pronounced.

(Note:   The image below was revised 1/7/2010) with improved RS20 data.  As can be seen, the edge pixels on the RS20 are diminished and this is covered in parts II and parts III of this series of articles on MTF.

Line Scan - JVC RS20 6 on (white) pixels and 6 off (black) pixels


Independent measurements of MTF have been completely missing in digital projector reviews even though they are commonly used in reviews of other products (DSLRs, lenses, etc.). Hopefully this article shows that that there is a wealth of information that can be gleaned by closely examining MTF in digital projectors. Measuring and comparing MTF values is very useful in technology comparisons, product comparisons and most importantly, we’ve shown that it provides sharpness information on the various modes and iris settings of a specific display.  This information can be used by an owner of a display to maximize the setup of their projector for optimum viewing results. Videovantage is a hobbyist blog and we’re proud to have provided the first independent, detailed, glimpse of MTF and the inner workings of a DLP projector at the pixel level.

As we have seen, the Samsung SP-A900B is an outstanding projector from a sharpness perspective and is among the best in this respect.   It would be interesting to compare this projector to other .95″ DLP projectors like the Marantz 11S2 and the Planar 8150 and perhaps we will do that in the future. Sharpness aside, we’ve also had an opportunity to view the Samsung with various content and hope to do a full review of the projector soon. As you might have guessed from some of the data shown above, we also have performed MTF measurements on several JVC projectors including the RS35 and we will be posting these results in subsequent articles in this series. Thanks for taking the time to read through this article and we hope that you have enjoyed what you have read and hopefully learned something new in the process. Thanks to image sharpness aficionado, Mark Haflich at Soundworks Audio and Video, Kensington MD for his help as a sounding board and motivation for finishing this article.

Note:  Part II of this series of articles on MTF compares DLP with 3 successive generations of DILA projectors and can be found Here.

2 Responses to “ MTF Revealed – Part I: The Samsung SP-A900B JKP Special Edition Projector ”

  1. Ron Wolff on May 21, 2012 at 4:04 am

    Enjoyed your three part article, MTF Revealed. What lens did you use with your Canon 50D camera for taking the images? You also noted that the line scan is imaged directly off of the projector so that the camera optics or scren effects are not an issue. Can you please further explain the test setup?

  2. Ron Wolff on May 21, 2012 at 6:24 pm

    What type of line scan camera/lens/controller did you use to do your experiments?

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