Other Signal and Image Processing Methods
107
As in DWT, there are several techniques for choosing a suitable threshold value
for compression, and this issue is still an open problem. As shown in the aforementioned code, in MATLAB ® , the commands “dct” and “dct2” are used for 1-D
DCT and 2-D DCT, respectively. The inverse commands “idct” and “idct2” are
used for decompression of the signals and images, respectively.
In more practical image processing applications, instead of compressing the
entire image in one shot, the image is split into a number of subimages (often using
a rectangular grid) and then each subimage is compressed using DCT separately.
In reconstruction, first, each small subimage is decompressed and then the subimages are put together to form the final reconstructed image. This will be further
explored in one of the problems in the Problems section.
6.4 INTRODUCTION TO STOCHASTIC PROCESSES
Most of the techniques described so far are applicable to “deterministic” signals and
systems. If every time a signal is recorded or registered the exact same values are
obtained, the signal is called to be deterministic. As evident from the definition, there
are very few quantities in nature that give the exact same signals in all measurements.
In other words, most of the signals, including biomedical signals, are “stochastic.”
A stochastic signal, even though statically similar in all recordings, contains some
stochastic variations in each of the recordings. A stochastic process, x(w, t), is represented by two variables where t is time and the variable w identifies a particular
outcome of the stochastic process. The variable w emphasizes the fact that each measurement of the variable x at a specific time can result in a different value.
An interesting stochastic process is the “white noise.” White noise defines a stochastic process in which the value of the signal has absolutely no relation or dependency on the value of the signal at any other times. An example of such a process is a
sequence of numbers obtained in consecutive tosses of a fair coin. In such a process,
the outcome of each toss does not depend on the past or future outcomes. As biomedical example, consider a faulty recording of an ECG machine where the electrodes
are not properly attached to the patient’s chest. Observing absolutely no pattern in
the recording signal, i.e., getting only white noise, often tells the technicians that the
electrodes are disconnected. What is really measured during such situations is the
background noise, often caused by the thermal and radiation sources of noise. It is
interesting to know that often, even when electrodes are properly connected, we still
capture some of this white noise that has to be filtered. The study of the white noise
and the techniques to detect or remove it from a signal is a dynamic field of research.
6.4.1 STATISTICAL MEASURES FOR STOCHASTIC PROCESSES
When dealing with a stochastic process and in order to make the notation shorter and
simpler, often the variable w is dropped from the notation and x(t) is used to represent
x(w, t). Hereafter, when dealing with a stochastic process, while keeping in mind the
random nature of the processes, the shorter notation is used.
The difference between a deterministic signal and a stochastic process is the
fact that once we have one recording of a deterministic signal, we know everything
about it; however, every measurement of a stochastic signal is merely one outcome
107
As in DWT, there are several techniques for choosing a suitable threshold value
for compression, and this issue is still an open problem. As shown in the aforementioned code, in MATLAB ® , the commands “dct” and “dct2” are used for 1-D
DCT and 2-D DCT, respectively. The inverse commands “idct” and “idct2” are
used for decompression of the signals and images, respectively.
In more practical image processing applications, instead of compressing the
entire image in one shot, the image is split into a number of subimages (often using
a rectangular grid) and then each subimage is compressed using DCT separately.
In reconstruction, first, each small subimage is decompressed and then the subimages are put together to form the final reconstructed image. This will be further
explored in one of the problems in the Problems section.
6.4 INTRODUCTION TO STOCHASTIC PROCESSES
Most of the techniques described so far are applicable to “deterministic” signals and
systems. If every time a signal is recorded or registered the exact same values are
obtained, the signal is called to be deterministic. As evident from the definition, there
are very few quantities in nature that give the exact same signals in all measurements.
In other words, most of the signals, including biomedical signals, are “stochastic.”
A stochastic signal, even though statically similar in all recordings, contains some
stochastic variations in each of the recordings. A stochastic process, x(w, t), is represented by two variables where t is time and the variable w identifies a particular
outcome of the stochastic process. The variable w emphasizes the fact that each measurement of the variable x at a specific time can result in a different value.
An interesting stochastic process is the “white noise.” White noise defines a stochastic process in which the value of the signal has absolutely no relation or dependency on the value of the signal at any other times. An example of such a process is a
sequence of numbers obtained in consecutive tosses of a fair coin. In such a process,
the outcome of each toss does not depend on the past or future outcomes. As biomedical example, consider a faulty recording of an ECG machine where the electrodes
are not properly attached to the patient’s chest. Observing absolutely no pattern in
the recording signal, i.e., getting only white noise, often tells the technicians that the
electrodes are disconnected. What is really measured during such situations is the
background noise, often caused by the thermal and radiation sources of noise. It is
interesting to know that often, even when electrodes are properly connected, we still
capture some of this white noise that has to be filtered. The study of the white noise
and the techniques to detect or remove it from a signal is a dynamic field of research.
6.4.1 STATISTICAL MEASURES FOR STOCHASTIC PROCESSES
When dealing with a stochastic process and in order to make the notation shorter and
simpler, often the variable w is dropped from the notation and x(t) is used to represent
x(w, t). Hereafter, when dealing with a stochastic process, while keeping in mind the
random nature of the processes, the shorter notation is used.
The difference between a deterministic signal and a stochastic process is the
fact that once we have one recording of a deterministic signal, we know everything
about it; however, every measurement of a stochastic signal is merely one outcome