Cameras

Linearity

It is generally desirable that the relationship between the input physical signal (e.g. photons) and the output signal (e.g. voltage) be linear. Formally this means (as in eq. ) that if we have two images, a and b, and two arbitrary complex constants, w₁ and w₂ and a linear camera response, then:

where R{*} is the camera response and c is the camera output. In practice the relationship between input a and output c is frequently given by:

where is the gamma of the recording medium. For a truly linear recording system we must have = 1 and offset = 0. Unfortunately, the offset is almost never zero and thus we must compensate for this if the intention is to extract intensity measurements. Compensation techniques are discussed in Section 10.1.

Typical values of that may be encountered are listed in Table 8. Modern cameras often have the ability to switch electronically between various values of .

Table 8: Comparison of of various sensors

Sensitivity

There are two ways to describe the sensitivity of a camera. First, we can determine the minimum number of detectable photoelectrons. This can be termed the absolute sensitivity. Second, we can describe the number of photoelectrons necessary to change from one digital brightness level to the next, that is, to change one analog-to-digital unit (ADU). This can be termed the relative sensitivity.

Absolute sensitivity

To determine the absolute sensitivity we need a characterization of the camera in terms of its noise. If the total noise has a of, say, 100 photoelectrons, then to ensure detectability of a signal we could then say that, at the 3 level, the minimum detectable signal (or absolute sensitivity) would be 300 photoelectrons. If all the noise sources listed in Section 6, with the exception of photon noise, can be reduced to negligible levels, this means that an absolute sensitivity of less than 10 photoelectrons is achievable with modern technology

Relative sensitivity

The definition of relative sensitivity, S, given above when coupled to the linear case, eq. with = 1, leads immediately to the result:

The measurement of the sensitivity or gain can be performed in two distinct ways.

* If, following eq. , the input signal a can be precisely controlled by either "shutter" time or intensity (through neutral density filters), then the gain can be estimated by estimating the slope of the resulting straight-line curve. To translate this into the desired units, however, a standard source must be used that emits a known number of photons onto the camera sensor and the quantum efficiency ( ) of the sensor must be known. The quantum efficiency refers to how many photoelectrons are produced--on the average--per photon at a given wavelength. In general 0 <= ( ) <= 1.

* If, however, the limiting effect of the camera is only the photon (Poisson) noise (see Section 6.1), then an easy-to-implement, alternative technique is available to determine the sensitivity. Using equations , , and and after compensating for the offset (see Section 10.1), the sensitivity measured from an image c is given by:

where m_c and s_c are defined in equations and .

Measured data for five modern (1995) CCD camera configurations are given in Table 9.

Table 9: Sensitivity measurements. Note that a more sensitive camera has a lower value of S.

The extraordinary sensitivity of modern CCD cameras is clear from these data. In a scientific-grade CCD camera (C-1), only 8 photoelectrons (approximately 16 photons) separate two gray levels in the digital representation of the image. For a considerably less expensive video camera (C-5), only about 256 photoelectrons (approximately 512 photons) separate two gray levels.

SNR

As described in Section 6, in modern camera systems the noise is frequently limited by:

* amplifier noise in the case of color cameras;

* thermal noise which, itself, is limited by the chip temperature K and the exposure time T, and/or;

* photon noise which is limited by the photon production rate and the exposure time T.

Thermal noise (Dark current)

Using cooling techniques based upon Peltier cooling elements it is straightforward to achieve chip temperatures of 230 to 250 K. This leads to low thermal electron production rates. As a measure of the thermal noise, we can look at the number of seconds necessary to produce a sufficient number of thermal electrons to go from one brightness level to the next, an ADU, in the absence of photoelectrons. This last condition--the absence of photoelectrons--is the reason for the name dark current. Measured data for the five cameras described above are given in Table 10.

Table 10: Thermal noise characteristics

The video camera (C-5) has on-chip dark current suppression. (See Section 6.2.) Operating at room temperature this camera requires more than 20 seconds to produce one ADU change due to thermal noise. This means at the conventional video frame and integration rates of 25 to 30 images per second (see Table 3), the thermal noise is negligible.

Photon noise

From eq. we see that it should be possible to increase the SNR by increasing the integration time of our image and thus "capturing" more photons. The pixels in CCD cameras have, however, a finite well capacity. This finite capacity, C, means that the maximum SNR for a CCD camera per pixel is given by:

Capacity-limited photon noise -

Theoretical as well as measured data for the five cameras described above are given in Table 11.

Table 11: Photon noise characteristics

Note that for certain cameras, the measured SNR achieves the theoretical, maximum indicating that the SNR is, indeed, photon and well capacity limited. Further, the curves of SNR versus T (integration time) are consistent with equations and . (Data not shown.) It can also be seen that, as a consequence of CCD technology, the "depth" of a CCD pixel well is constant at about 0.7 ke^- / um².

Shading

Virtually all imaging systems produce shading. By this we mean that if the physical input image a(x,y) = constant, then the digital version of the image will not be constant. The source of the shading might be outside the camera such as in the scene illumination or the result of the camera itself where a gain and offset might vary from pixel to pixel. The model for shading is given by:

where a[m,n] is the digital image that would have been recorded if there were no shading in the image, that is, a[m,n] = constant. Techniques for reducing or removing the effects of shading are discussed in Section 10.1.

Pixel Form

While the pixels shown in Figure 1 appear to be square and to "cover" the continuous image, it is important to know the geometry for a given camera/digitizer system. In Figure 18 we define possible parameters associated with a camera and digitizer and the effect they have upon the pixel.

Figure 18: Pixel form parameters

The parameters X_o and Y_o are the spacing between the pixel centers and represent the sampling distances from equation . The parameters X_a and Y_a are the dimensions of that portion of the camera's surface that is sensitive to light. As mentioned in Section 2.3, different video digitizers (frame grabbers) can have different values for X_o while they have a common value for Y_o.

Square pixels

As mentioned in Section 5, square sampling implies that X_o = Y_o or alternatively X_o / Y_o = 1. It is not uncommon, however, to find frame grabbers where X_o / Y_o = 1.1 or X_o / Y_o = 4/3. (This latter format matches the format of commercial television. See Table 3) The risk associated with non-square pixels is that isotropic objects scanned with non-square pixels might appear isotropic on a camera-compatible monitor but analysis of the objects (such as length-to-width ratio) will yield non-isotropic results. This is illustrated in Figure 19.

Figure 19: Effect of non-square pixels

The ratio X_o / Y_o can be determined for any specific camera/digitizer system by using a calibration test chart with known distances in the horizontal and vertical direction. These are straightforward to make with modern laser printers. The test chart can then be scanned and the sampling distances X_o and Y_o determined.

Fill factor

In modern CCD cameras it is possible that a portion of the camera surface is not sensitive to light and is instead used for the CCD electronics or to prevent blooming. Blooming occurs when a CCD well is filled (see Table 11) and additional photoelectrons spill over into adjacent CCD wells. Anti-blooming regions between the active CCD sites can be used to prevent this. This means, of course, that a fraction of the incoming photons are lost as they strike the non-sensitive portion of the CCD chip. The fraction of the surface that is sensitive to light is termed the fill factor and is given by:

The larger the fill factor the more light will be captured by the chip up to the maximum of 100%. This helps improve the SNR. As a tradeoff, however, larger values of the fill factor mean more spatial smoothing due to the aperture effect described in Section 5.1.1. This is illustrated in Figure 16.

Spectral Sensitivity

Sensors, such as those found in cameras and film, are not equally sensitive to all wavelengths of light. The spectral sensitivity for the CCD sensor is given in Figure 20.

Figure 20: Spectral characteristics of silicon, the sun, and the human visual system. UV = ultraviolet and IR = infra-red.

The high sensitivity of silicon in the infra-red means that, for applications where a CCD (or other silicon-based) camera is to be used as a source of images for digital image processing and analysis, consideration should be given to using an IR blocking filter. This filter blocks wavelengths above 750 nm. and thus prevents "fogging" of the image from the longer wavelengths found in sunlight. Alternatively, a CCD-based camera can make an excellent sensor for the near infrared wavelength range of 750 nm to 1000 nm.

Shutter Speeds (Integration Time)

The length of time that an image is exposed--that photons are collected--may be varied in some cameras or may vary on the basis of video formats (see Table 3). For reasons that have to do with the parameters of photography, this exposure time is usually termed shutter speed although integration time would be a more appropriate description.

Video cameras

Values of the shutter speed as low as 500 ns are available with commercially available CCD video cameras although the more conventional speeds for video are 33.37 ms (NTSC) and 40.0 ms (PAL, SECAM). Values as high as 30 s may also be achieved with certain video cameras although this means sacrificing a continuous stream of video images that contain signal in favor of a single integrated image amongst a stream of otherwise empty images. Subsequent digitizing hardware must be capable of handling this situation.

Scientific cameras

Again values as low as 500 ns are possible and, with cooling techniques based on Peltier-cooling or liquid nitrogen cooling, integration times in excess of one hour are readily achieved.

Readout Rate

The rate at which data is read from the sensor chip is termed the readout rate. The readout rate for standard video cameras depends on the parameters of the frame grabber as well as the camera. For standard video, see Section 2.3, the readout rate is given by:

While the appropriate unit for describing the readout rate should be pixels / second, the term z is frequently found in the literature and in camera specifications; we shall therefore use the latter unit. For a video camera with square pixels (see Section 7.5), this means:

Table 12: Video camera readout rates

Note that the values in Table 12 are approximate. Exact values for square-pixel systems require exact knowledge of the way the video digitizer (frame grabber) samples each video line.

The readout rates used in video cameras frequently means that the electronic noise described in Section 6.3 occurs in the region of the noise spectrum (eq. ) described by > _max where the noise power increases with increasing frequency. Readout noise can thus be significant in video cameras.

Scientific cameras frequently use a slower readout rate in order to reduce the readout noise. Typical values of readout rate for scientific cameras, such as those described in Tables 9, 10, and 11, are 20 kz, 500 kz, and 1 Mz to 8 Mz.