Resolution |
9 |
To avoid visible pixel structure in image display, some overlap is desirable in the distributions of light produced by neighboring display elements, as
I explained in Image structure, on page 75. Also, to avoid spatial aliasing in image capture, some overlap is necessary in the distribution of sensitivity across neighboring sensor elements. Such overlap reduces sharpness, but is beneficial to continuous-tone imagery. In this chapter, I will explain resolution, which is closely related to sharpness.
Resolution is an overloaded and ambiguous term that properly refers to spatial phenomena. It is confusing to refer to a sample as having 8-bit resolution; use precision or quantization instead. In computing, it is usual to use the term “resolution” to specify the number of columns and rows in the image matrix – that is, to express pixel count. That use disregards effects of signal processing. To preempt resolution for pixel count makes it difficult to discuss the image detail that’s actually represented or delivered to the viewer. I’ll present the details of resolution, but first I must introduce the concepts of magnitude frequency response and bandwidth.
Magnitude frequency response and bandwidth
To characterize the acquisition, processing, or display of smalll elements, rather than analyzing an element of certain (small) dimensions, we analyze a group of closely spaced identical elements, characterizing the spacing between the elements. This allows mathematical analysis using transforms, particularly the Fourier transform and the z-transform.
97
Magnitude frequency
|
Input |
|
|
Output |
|
relativeresponse, |
1.0 |
|
0.707 |
||
|
||
|
0 |
0
(-3 dB) |
FREQUENCY |
HALF-POWER |
CORNER |
|
RESOLUTION |
|
LIMITING |
Frequency, relative |
|
Figure 9.1 Magnitude frequency response of an electronic or optical system typically falls as frequency increases. Bandwidth is measured at the half-power point (-3 dB), where response has fallen to 0.707 of its value at a reference frequency (often zero frequency, or DC). Useful visible detail is obtained from signal power beyond the half-power bandwidth, that is, at depths of modulation less than 70.7%. I show limiting resolution, which might occur at about 10% response.
The top graph in Figure 9.1 shows a one-dimen- sional sine wave test signal “sweeping” from zero frequency up to a high frequency. (This could be a onedimensional function of time such as an audio waveform, or the waveform of luma from one scan line of an image.) A typical optical or electronic imaging system involves temporal or spatial dispersion, which causes the response of the system to diminish at high frequency, as shown in the middle graph. The envelope of that waveform – the system’s magnitude frequency response – is shown at the bottom. An electrical engineer may call this simply frequency response. The qualifier magnitude distinguishes it from other functions of frequency such as phase frequency response.
98 |
DIGITAL VIDEO AND HD ALGORITHMS AND INTERFACES |
There are other definitions of bandwidth than the one I present here, but this is the definition that I recommend. In magnitude squared response, the half-power point is at abcissa value 0.5.
When digital information is processed or transmitted through analog channels, bits are coded into symbols that ideally remain independent. Dispersion in this context is called intersymbol interference (ISI).
E |
|
|
F P |
|
|
T O Z |
|
|
L P E D |
|
|
P E C F D |
|
|
E D F C Z P |
|
|
F E L P O P Z D |
Figure 9.2 |
|
D E F P O T E L |
||
Snellen chart |
||
L E F O D P C T |
||
|
|
Bandwidth characterizes the range of frequencies that a system can capture, record, process, or transmit. Halfpower bandwidth (also known as 3 dB bandwidth) is specified or measured where signal magnitude has fallen 3 dB – that is, to the fraction 0.707 – from its value at a reference frequency (often zero frequency, or DC). Useful visual information is typically available at frequencies higher than the bandwidth. In image science, limiting resolution is determined visually.
The maximum rate at which an analog or digital electronic signal can change state – in an imaging system, between black and white – is limited by frequency response, and is therefore characterized by bandwidth.
Figure 9.1 shows abstract input and output signals. When bandwidth of an optical system is discussed, it is implicit that the quantities are proportional to intensity. When bandwidth of video signals is discussed, it is implicit that the input and output electrical signals are gamma-corrected.
Many digital technologists use the term bandwidth to refer to data rate; however, the terms properly refer to different concepts. Bandwidth refers to the frequency of signal content in an analog or digital signal. Data rate refers to digital transmission capacity, independent of any potential signal content. A typical studio HD luma signal has 30 MHz signal bandwidth and 74.25 MB/s data rate – the terms are obviously not interchangeable.
Visual acuity
When an optometrist measures your visual acuity, he or she uses a chart similar to the one shown in Figure 9.2 in the margin.The results of this test depend upon viewing distance. The test is standardized for a viewing distance of 20 feet. At that distance, the strokes of the letters in the 20/20 row subtend one sixtieth of
a degree (1⁄60°, one minute of arc). This is roughly the limit of angular discrimination of normal vision.
Visual angles can be estimated using the astronomers’ rule of thumb depicted in Figure 9.3: When held at arm’s length, the joint of the thumb subtends about two degrees. The full palm subtends about ten degrees, and the nail of the little finger subtends about one degree. (The angular subtense of the full moon is about half a degree.)
CHAPTER 9 |
RESOLUTION |
99 |
Figure 9.3 The astronomers’ rule of thumb allows rough measurement of subtended angles. The hand is held at arm’s length; the palm then subtends about 10°. Here
I show the palm covering
a rectangle having 4:3 aspect ratio. If that rectangle was an SD picture, the viewer would be located at roughly the optimal viewing distance.
1°
2°
10°
Viewing distance and angle
If you display a white flatfield on a display with typical pixel pitch, pixel structure is likely to be visible if the viewer is located closer than the distance where adjacent image rows (or scan lines) at the display surface subtend an angle of one minute of arc (1⁄60°) or more. To achieve viewing where pixel pitch subtends 1⁄60°,
viewing distance should be about 3400 times the distance d between image rows – that is, 3400 divided by the pixel density. For example, for pixels per inch (ppi):
distance ≈ 3400 d ≈ |
3400 |
; 3400 |
≈ |
1 |
|
Eq 9.1 |
||
sin( |
1 |
° |
||||||
|
||||||||
|
ppi |
|
|
|||||
|
|
|
|
|
) |
|
||
|
|
|
|
60 |
|
|||
So, at a distance of 3400 times the distance between image rows, there are about 60 pixels per degree. Viewing distance expressed numerically as a multiple of picture height is then 3400 divided by the number of image rows (NR):
distance ≈ |
3400 |
PH |
Eq 9.2 |
|
|||
|
L |
|
|
|
A |
|
|
SD has about 480 image rows (picture lines). An image row subtends 1⁄60° at a distance of about seven times picture height (PH), as sketched in Figure 9.4 at the top of the facing page, giving roughly 600 pixels across the picture width. Picture angle is about 11°, as shown in Figure 9.5. With your hand held at arm’s length, your palm ought to just cover the width of the picture. This
100 |
DIGITAL VIDEO AND HD ALGORITHMS AND INTERFACES |
SD, 480 image rows |
|
SD, 480 |
d=1⁄480 PH |
1’ (1⁄60°) |
image rows |
|
|
11° ( × 8°) |
1 PH |
7.1 PH |
|
|
|
|
HD, 1080 image rows |
HD, 1080 |
|
d=1⁄1080 PH |
|
image rows |
|
1’ (1⁄60°) |
32° ( × 18°) |
1 PH
3.2 PH
Figure 9.4 The viewing distance where pixels become invisible occurs approximately where the pixel pitch subtends an angle of about one minute of arc (1⁄60°) at the display surface. This is roughly the limit of angular discrimination for normal vision.
Figure 9.5 The picture angle of SD, sketched at the top, has a horizontal angle of
about 11° and a vertical angle of about 8°, where pixel structure becomes invisible. In 1920× 1080 HD, horizontal angle can increase to about 33°, and vertical angle to about 18°, preserving the pixel subtense.
5 |
3 |
4 |
|
|
|
Figure 9.6 Picture height at an aspect ratio of 4:3 is 3⁄5 of the diagonal; optimum viewing distance for conventional video is 4.25 times the diagonal. Picture height at 16:9 is about half the diagonal; optimum viewing distance for 2 Mpx HD is 1.5 times the diagonal.
distance is about 4.25 times the display diagonal, as sketched in Figure 9.6 in the margin. For HD with 1080 image rows, the viewing distance that yields the 1⁄60° pixel subtense is about 3.2 PH (see the bottom of Figure 9.4), about 1.5 times the display diagonal.
For SD, the total horizontal picture angle at that viewing distance is about 11°. Viewers tend to choose a viewing distance that renders pixel structure invisible; angular subtense of a pixel is thereby preserved. Thus, the main effect of higher pixel count is to enable viewing at a wide picture angle. For 1920× 1080 HD, horizontal viewing angle is tripled to 33° compared to the 11° of SD as sketched in Figure 9.5. The “high definition” of HD does not squeeze six times the number of pixels into the same visual angle! Instead, the entire image can potentially occupy a much larger area of the viewer’s visual field. This topic is addressed further in
Viewing distance, on page 104.
CHAPTER 9 |
RESOLUTION |
101 |