Math and Science of Image Compression

Have you wondered about the technical aspects of image compression?

1. Sampling and Quantization: Image compression begins with sampling, where the continuous image is divided into discrete pixels. Each pixel’s color or intensity value is quantized, which involves reducing the precision or bit-depth of the color representation. For example, a grayscale image with 8 bits per pixel allows 256 levels of intensity.
2. Color Spaces: Images can be represented in different color spaces, such as RGB, CMYK, or YCbCr. RGB is commonly used for digital images, where each pixel’s color is represented by three values for red, green, and blue intensities. Other color spaces like YCbCr separate color and brightness information, which can be leveraged for compression.
3. Transform Coding: Transform coding involves converting the spatial representation of an image into a frequency representation. The most widely used transformation is the Discrete Cosine Transform (DCT). The DCT divides the image into blocks and expresses them in terms of cosine functions. High-frequency components, representing fine details, are concentrated towards the higher-frequency coefficients, making them easier to discard or approximate during compression.
4. Quantization Tables: Quantization assigns fewer bits to higher-frequency components of the transformed image, effectively reducing their precision. Quantization tables define the mapping from the transformed coefficients to quantized values. Higher compression levels involve more aggressive quantization, resulting in greater loss of detail and potentially noticeable artifacts.
5. Entropy Coding: Entropy coding exploits the statistical properties of image data to represent them more efficiently. Huffman coding and arithmetic coding are commonly used entropy coding techniques. Huffman coding assigns shorter codes to more frequently occurring values or patterns, while arithmetic coding assigns fractional values to ranges of values based on their probabilities.
6. Chroma Subsampling: Chroma subsampling reduces the amount of color information in an image, primarily targeting the human visual system’s higher sensitivity to brightness (luma) compared to color (chroma). By reducing the chroma resolution, substantial data reduction can be achieved without significant perceptible loss in quality.
7. Lossy Compression Algorithms: Lossy compression algorithms like JPEG (Joint Photographic Experts Group) are widely used for photographic images. JPEG utilizes a combination of the aforementioned techniques, including DCT, quantization, and entropy coding. The compression ratio can be adjusted by modifying quantization parameters, affecting the tradeoff between image quality and file size.
8. Lossless Compression Algorithms: Lossless compression algorithms like PNG (Portable Network Graphics) preserve all image data without any loss of quality. They typically use predictive coding techniques, such as delta encoding or Lempel-Ziv-Welch (LZW) compression, to exploit patterns and redundancies in the image data.
9. Metadata Compression: In addition to compressing pixel data, image compression techniques can also compress metadata, such as color profiles, text annotations, and EXIF data. Techniques like zlib compression or run-length encoding can be applied to reduce the size of metadata while maintaining its integrity.
10. Image Compression Standards: Several image compression standards have been established to ensure interoperability and compatibility across different systems. JPEG, JPEG 2000, and WebP are widely adopted standards for lossy compression, while PNG, GIF, and TIFF are commonly used for lossless compression.

JPEG

1. Color Space Conversion: JPEG compression typically starts with converting the image from the RGB color space to the YCbCr color space. The YCbCr color space separates the image into luminance (Y) and chrominance (Cb and Cr) components, as the human visual system is more sensitive to changes in brightness than changes in color.
2. Image Splitting and Transformation: The image is divided into 8×8 pixel blocks, and a two-dimensional Discrete Cosine Transform (DCT) is applied to each block independently. The DCT converts the spatial domain representation of the image into the frequency domain. The resulting DCT coefficients represent the frequency components (low frequency to high frequency) within each block.
3. Quantization: Quantization is a crucial step in JPEG compression, where the DCT coefficients are divided by a set of quantization values. These quantization values determine the level of compression and directly affect the quality of the reconstructed image. Higher values result in more aggressive compression but also more loss of detail.
4. Entropy Coding: After quantization, the quantized DCT coefficients are arranged in a zigzag pattern and then compressed using entropy coding techniques. Huffman coding is commonly used, where shorter variable-length codes are assigned to more frequently occurring coefficients, while longer codes are assigned to less frequent coefficients. Huffman tables are constructed based on the statistical properties of the coefficients within the image or blocks.
5. Chroma Subsampling: Chroma subsampling reduces the amount of color information while preserving the overall image quality. It involves reducing the resolution of the chrominance components (Cb and Cr) relative to the luminance component (Y). The most common subsampling ratios are 4:4:4 (no subsampling), 4:2:2, and 4:2:0, representing the number of samples for each component.
6. JPEG Quality Setting: The compression ratio and resulting image quality in JPEG compression can be adjusted using a quality parameter. This parameter determines the quantization step size, where higher values lead to more aggressive compression and potentially more visible artifacts.
7. Decompression and Reconstruction: To view or use the compressed image, it must be decompressed. During decompression, the reverse operations are performed in the same order as compression. This includes entropy decoding, inverse quantization, inverse DCT, and color space conversion back to RGB. The reconstructed image may not be an exact replica of the original due to the lossy nature of the compression algorithm.

Gif

1. Color Representation: GIF images can have a color depth of up to 8 bits, allowing a maximum of 256 different colors. Each pixel in the image is represented by an index value that refers to a color in a color lookup table (CLUT) known as the palette. The palette contains the 256 colors used in the image.
2. Image Splitting: If the GIF image contains animation, it consists of multiple frames. Each frame is a separate image that represents a specific point in the animation sequence.
3. LZW Compression: The key mathematical algorithm used in GIF compression is LZW (Lempel-Ziv-Welch) compression. LZW is a dictionary-based compression method that replaces repeated sequences of data with shorter codes.
4. Creating an Initial Dictionary: To begin the compression process, an initial dictionary is created with entries representing all possible color index values. Each entry is assigned a unique code.
5. Compression Process: The compression process involves scanning the image pixel by pixel and building a dictionary of encountered sequences. Starting with single-pixel sequences, the algorithm checks if the sequence is already present in the dictionary. If not, it adds a new entry to the dictionary for the sequence and assigns it a new code.
6. Code Output: During compression, the algorithm outputs a sequence of codes that represent the compressed image data. These codes correspond to the dictionary entries that match the encountered sequences in the image.
7. Code Size Optimization: The GIF compression algorithm dynamically adjusts the code size to accommodate the expanding dictionary. It increases the code size when the dictionary becomes larger and requires more codes.
8. Clear Code and End of Data: Special codes, known as clear code and end-of-data code, are used to reset the dictionary and indicate the end of the compressed data, respectively.
9. Decompression and Reconstruction: To display a GIF image, the compressed data is decompressed and the original pixel values are reconstructed. The decompression process involves reverse lookup in the dictionary using the sequence of codes, and the resulting color index values are used to recreate the image.
10. Animation Control: GIF images with animation contain additional information for controlling the timing and order of frames, such as delay times between frames and looping options.

Formats

1. JPEG (Joint Photographic Experts Group):
   • Pros: JPEG is widely supported, offers high compression ratios, and is suitable for complex photographic images. It supports millions of colors, making it ideal for photographs and realistic images.
   • Cons: JPEG is a lossy compression format, meaning some image quality is sacrificed to achieve higher compression ratios. It is not suitable for images with sharp edges or text, as it can introduce artifacts and loss of fine details.
   • Common Use Cases: JPEG is commonly used for photographs, web graphics, and images where a balance between file size and image quality is desired.
2. PNG (Portable Network Graphics):
   • Pros: PNG is a lossless compression format, preserving image quality without sacrificing details. It supports transparency, allowing images to be overlaid on different backgrounds. It also supports different color depths, including 8-bit and 24-bit.
   • Cons: PNG file sizes tend to be larger than JPEG, especially for complex images. It may not be suitable for highly detailed or photographic images due to larger file sizes.
   • Common Use Cases: PNG is commonly used for images that require transparency, such as logos, icons, and graphics with sharp edges. It is also suitable for web graphics and images where preservation of image quality is important.
3. GIF (Graphics Interchange Format):
   • Pros: GIF is well-suited for simple animations, as it supports multiple frames and looping. It uses lossless compression, preserving image quality without sacrificing details. It also supports transparency.
   • Cons: GIF has a limited color palette of 256 colors, which may result in color banding or reduced color fidelity, especially for complex images. It is not suitable for highly detailed or photographic images.
   • Common Use Cases: GIF is commonly used for simple animations, icons, logos, and graphics with limited colors. It is popular for web animations and for images where file size is a concern.
4. BMP (Bitmap Image File):
   • Pros: BMP is an uncompressed image format that preserves the highest quality and color fidelity. It is compatible with various platforms and software.
   • Cons: BMP files are typically very large in size, making them impractical for web use or situations where file size is a consideration.
   • Common Use Cases: BMP is commonly used for lossless storage of high-quality images, such as in professional graphics and image editing applications. It is also used as an intermediary format for editing or conversion before being saved in a more compressed format.
5. TIFF (Tagged Image File Format):
   • Pros: TIFF is a versatile format that supports lossless compression, preserving image quality and detail. It also supports different color depths and transparency.
   • Cons: TIFF files tend to be larger in size compared to compressed formats like JPEG or PNG. They may not be suitable for web use or situations where file size is a concern.
   • Common Use Cases: TIFF is commonly used in professional settings, such as print and publishing, where high-quality images and preservation of image detail are essential. It is also used in image editing applications.
6. WEBP:
   • Pros: WEBP is a relatively new image format developed by Google that offers both lossless and lossy compression. It provides smaller file sizes compared to JPEG or PNG while maintaining good image quality. It also supports transparency and animation.
   • Cons: WEBP may not be supported by all browsers and software, although its adoption is growing.
   • Common Use Cases: WEBP is increasingly used on the web for high-quality images with smaller file sizes.

Math of Sampling and Quantization

Sampling:

1. Continuous Image: A continuous image is represented as a continuous function of two variables, typically denoted as f(x, y), where x and y are spatial coordinates. This continuous image contains an infinite number of intensity values within its domain.
2. Sampling Grid: To convert the continuous image into a digital representation, a sampling grid is placed over the image. The grid consists of discrete points, typically arranged in a regular pattern, such as a rectangular grid.
3. Sampling Rate and Nyquist-Shannon Sampling Theorem: The sampling rate determines how frequently samples are taken from the continuous image. According to the Nyquist-Shannon sampling theorem, the sampling rate must be at least twice the maximum frequency component present in the image to avoid aliasing artifacts.
4. Pixel Grid and Sampled Image: The points on the sampling grid represent the locations where samples are taken from the continuous image. Each point on the grid corresponds to a pixel in the digital image.
5. Sampling Frequency and Pixel Density: The number of samples taken per unit distance in the x and y directions is known as the sampling frequency. The inverse of the sampling frequency represents the pixel density, indicating the number of pixels per unit distance.

Quantization:

1. Discretization of Intensity Levels: Quantization involves reducing the continuous range of intensity values to a finite set of discrete levels. Each sample’s intensity value is approximated by assigning it to the nearest quantization level.
2. Quantization Levels and Bit Depth: The number of quantization levels determines the precision with which intensity values are represented. It is typically determined by the bit depth of the digital image. For example, an 8-bit image allows 2^8 = 256 quantization levels.
3. Quantization Error: Due to the finite number of quantization levels, quantization introduces a quantization error. This error represents the difference between the original continuous intensity value and the quantized value.
4. Quantization Step Size: The difference between adjacent quantization levels is known as the quantization step size. It determines the level of precision and affects the overall quality of the quantized image.
5. Uniform and Non-Uniform Quantization: Uniform quantization uses equally spaced quantization levels, while non-uniform quantization can allocate more levels to areas with higher signal complexity, providing better representation for complex images.
6. Quantization and Compression: Quantization is an essential step in lossy image compression algorithms. Higher levels of quantization lead to more aggressive compression, reducing the image’s file size but also introducing more noticeable loss of detail and increased distortion.

FAQ

Q1: What is image compression? A1: Image compression is the process of reducing the file size of an image while attempting to preserve its visual quality. It aims to store or transmit images in a more efficient manner.

Q2: Why is image compression important? A2: Image compression is essential for various reasons:

• Reduced storage space: Compressed images require less storage space, making it easier to store and manage large collections of images.
• Faster transmission: Smaller image sizes allow for quicker upload and download times, improving user experience on websites and applications.
• Bandwidth optimization: Compressed images consume less bandwidth, resulting in reduced data usage and faster loading times. Less stress on image host.
• Improved user engagement: Images that load quickly and display seamlessly enhance user engagement and satisfaction.

Q3: What are the two types of image compression? A3: Image compression techniques can be broadly categorized into two types:

• Lossless compression: Lossless compression algorithms reduce file size without any loss of image quality. Examples include PNG and GIF formats.
• Lossy compression: Lossy compression algorithms achieve higher compression ratios by selectively discarding or approximating certain image details that are considered less noticeable to the human eye. Examples include JPEG and WebP formats.

Q4: How does lossless compression work? A4: Lossless compression algorithms analyze the image data to identify patterns and redundancies. They encode the image using mathematical techniques, such as run-length encoding or predictive coding, to eliminate these redundancies. Upon decompression, the original image is reconstructed without any loss of data.

Q5: How does lossy compression work? A5: Lossy compression algorithms exploit the limitations of human perception to reduce file size. They discard or approximate certain image details that are less noticeable to the human eye. This is achieved through techniques such as quantization, where the precision of pixel values is reduced, and perceptual coding, which prioritizes preserving visually significant image elements.

Q6: What are some popular image compression formats and their use cases? A6: Some popular image compression formats include:

• JPEG: Widely used for photographic images on the web due to its good compression ratios and support for millions of colors.
• PNG: Suitable for images with sharp edges, transparency, and lossless compression requirements, often used for icons, logos, and graphics.
• GIF: Commonly used for simple animations and graphics with limited colors, such as memes and emojis.
• WebP: Developed by Google, WebP offers both lossy and lossless compression, providing smaller file sizes and good image quality. It is becoming increasingly popular on the web.

Q7: Can image compression result in loss of quality? A7: Yes, lossy compression techniques can result in a loss of some image quality. The extent of quality loss depends on the compression level chosen and the specific algorithm used. Lossless compression, on the other hand, retains all image data without sacrificing quality.

Q8: Are there any trade-offs with image compression? A8: Yes, image compression involves trade-offs between file size reduction and image quality. Higher compression ratios often result in greater loss of image detail and increased artifacts. It is important to strike a balance between compression and maintaining an acceptable level of visual quality.

Q11: Can I compress images without losing any quality?
A11: Yes, lossless compression techniques like PNG or TIFF formats allow you to compress images without any loss of quality. However, the compression ratios achieved with lossless compression may not be as high as those achieved with lossy compression techniques.

Q12: How much can I compress an image?
A12: The amount of compression you can achieve depends on various factors, including the original image’s complexity, the chosen compression algorithm, and the desired quality level. Higher compression ratios often result in more noticeable loss of image quality.

Q13: Can I compress already compressed images?
A13: Compressing an image that has already been compressed (such as a JPEG image) using a lossy compression algorithm can lead to further degradation in quality. It is generally advisable to work with the original uncompressed or losslessly compressed version of an image for best results.

Q14: Are there any image compression techniques specifically for web optimization?
A14: Yes, optimizing images for the web often involves resizing them to appropriate dimensions, using the correct file format (e.g., JPEG for photographs, PNG for graphics with transparency), and adjusting compression settings to balance file size and image quality. Web-specific image compression tools and plugins are available to streamline this process.

Q15: What is the impact of image compression on website loading speed?
A15: Image compression plays a significant role in website loading speed. Smaller image file sizes result in faster page loading times, which improves user experience and can positively impact search engine rankings. Optimizing images for web display through proper compression can help reduce bandwidth usage and decrease loading times.

Q16: Can image compression affect accessibility for visually impaired users?
A16: Image compression should not impact accessibility features like alt text, which provides textual descriptions of images for screen readers. However, excessive compression that results in severe loss of detail or artifacts may hinder the ability of visually impaired users to perceive images effectively.

Q17: Are there any tools or software available for image compression?
A17: Yes, numerous image compression tools and software are available, both online and offline. These tools provide features to adjust compression settings, optimize images for the web, and bulk process multiple images. Some popular options include Adobe Photoshop, TinyPNG, JPEGmini, and Squoosh.

Image vs video compression

Image Compression:

• Lossless Compression: Lossless image compression algorithms, such as PNG and TIFF, aim to preserve the exact image data without any loss. They exploit redundancies in the image data by using techniques like run-length encoding, Huffman coding, and Lempel-Ziv-Welch (LZW) compression. These algorithms achieve compression by representing repeated patterns or sequences of data with shorter codes.
• Lossy Compression: Lossy image compression algorithms, like JPEG and WebP, sacrifice some image details to achieve higher compression ratios. The key technique used is the Discrete Cosine Transform (DCT), which converts blocks of pixel data into frequency coefficients. These coefficients are quantized, reducing their precision and resulting in loss of information. Additional techniques like chroma subsampling and entropy encoding (e.g., Huffman or arithmetic coding) are used to further reduce file size without significant visual impact.

Video Compression:

• Inter-Frame Compression: Video compression algorithms, such as MPEG-2, MPEG-4 (including H.264), and HEVC (H.265), exploit temporal redundancies between frames. They use techniques like motion estimation and compensation to encode only the differences (motion vectors) between consecutive frames. This reduces the amount of data required to represent the video sequence.
• Intra-Frame Compression: Within each frame, video compression algorithms employ similar techniques used in image compression. They utilize spatial redundancies through techniques like DCT and quantization to compress individual frames. Each frame is treated as a standalone image, allowing compression within the frame itself.
• Group of Pictures (GOP): Video compression algorithms divide the video stream into GOPs, which consist of multiple frames. The first frame in a GOP, known as the keyframe or intra-frame, is encoded independently. The subsequent frames are encoded differentially, referencing the keyframe or other frames within the same GOP.
• Bitrate Control: Video compression algorithms often include mechanisms for bitrate control to achieve desired compression levels. This can be done through techniques like variable bitrate (VBR), constant bitrate (CBR), or constrained bitrate (CBR with quality constraints). These techniques dynamically allocate bits to different parts of the video stream based on the complexity of the content.

It’s worth noting that various video codecs and formats employ different compression techniques and have their own strengths and trade-offs. For example, H.264 (AVC) is widely used for its good compression efficiency, while H.265 (HEVC) provides even higher compression ratios but requires more computational resources for encoding and decoding.

Science of image compression

1. Redundancy Exploitation: 1.1 Spatial Redundancy: Images often contain repetitive patterns or similar areas. Compression algorithms, such as run-length encoding, identify and encode these patterns efficiently. For example, in an image with a blue sky, the algorithm can store information about the color once and indicate that it repeats over a specific region. 1.2 Spectral Redundancy: In color images, the correlation between color channels can be exploited. For instance, the difference between the red and blue color values may be small in certain areas, allowing the algorithm to use fewer bits to represent those areas accurately.
2. Lossless and Lossy Compression: 2.1 Lossless Compression: Lossless algorithms preserve all image data without any quality loss. One example is the DEFLATE algorithm used in PNG files, which combines LZ77 (a sliding window compression technique) and Huffman coding for efficient encoding. 2.2 Lossy Compression: Lossy algorithms, like the popular JPEG format, sacrifice some image details to achieve higher compression ratios. They utilize techniques such as quantization and entropy coding to discard less noticeable information. The compression level can be adjusted to control the trade-off between file size and visual quality.
3. Transform Coding: 3.1 Discrete Cosine Transform (DCT): The DCT, used in JPEG compression, converts image data from the spatial domain to the frequency domain. The image is divided into blocks, and the DCT coefficients represent the frequency components. Higher-frequency coefficients, representing fine details, can be quantized more aggressively to achieve compression.
4. Quantization: 4.1 Quantization Step Size: Quantization reduces the precision of the transformed coefficients, determining the trade-off between file size and visual quality. A larger quantization step size leads to more aggressive compression and greater loss of detail. For example, in JPEG, higher quality settings use smaller step sizes to preserve more information.
5. Entropy Coding: 5.1 Huffman Coding: Huffman coding assigns variable-length codes to different symbols (e.g., coefficients). Frequently occurring symbols are assigned shorter codes, while less frequent symbols receive longer codes. Huffman coding is utilized in JPEG and other compression algorithms to reduce the overall file size by assigning shorter codes to more common coefficients.
6. Chroma Subsampling: 6.1 Subsampling Techniques: In many compression algorithms, such as JPEG, chroma subsampling reduces the amount of color information stored while preserving visual quality. The human eye is less sensitive to color differences than luminance details. For example, a popular subsampling ratio in JPEG is 4:2:0, where the chrominance channels are subsampled horizontally and vertically.
7. Adaptive Compression: 7.1 Adaptive Huffman Coding: Adaptive compression techniques dynamically adjust compression settings based on the image content. Adaptive Huffman coding, used in some compression algorithms, adaptively updates the code assignments during encoding to optimize compression efficiency for different symbol frequencies.
8. Real-World Examples: 8.1 JPEG Compression: JPEG is widely used for compressing digital photos. By adjusting the compression level, you can observe the trade-off between file size and visual quality. Higher compression levels introduce more artifacts and loss of detail, while lower levels preserve more image details but result in larger file sizes. 8.2 PNG Compression: PNG is a lossless image compression format suitable for graphics, icons, and images with transparency. It employs techniques like DEFLATE compression and adaptive filtering.

Technical aspects of quantization in depth:

1. Transform Coding: Quantization is commonly applied after transforming the image data from the spatial domain to a transformed domain, such as the Discrete Cosine Transform (DCT) in JPEG compression or the Discrete Wavelet Transform (DWT) in JPEG 2000 and other wavelet-based compression algorithms.
2. Quantization Matrix: Quantization is performed by dividing the transformed coefficients by a quantization step size and rounding them to integer values. The quantization step size determines the level of compression and directly affects the trade-off between file size and visual quality.
3. Quantization Step Size: The quantization step size controls the precision of the quantized coefficients. Larger step sizes result in more aggressive compression but also lead to greater loss of detail and visual quality. Smaller step sizes preserve more information but produce larger file sizes.
4. Quantization Tables: Quantization tables are used to determine the specific quantization step size for each coefficient within the transformed image data. Different quantization tables can be employed for luminance (Y) and chrominance (Cb and Cr) components in color images to accommodate the different perceptual sensitivity of these components.
5. Compression Artifacts: The loss of information caused by quantization leads to compression artifacts, which are visual distortions introduced during the compression process. The most common artifact is blockiness, visible as square-shaped blocks in the decompressed image. Other artifacts include color bleeding, ringing, and contouring.
6. Rate-Distortion Trade-off: Quantization plays a vital role in the rate-distortion trade-off, balancing the compression ratio (rate) and the visual quality (distortion) of the reconstructed image. By adjusting the quantization step size, it is possible to achieve different compression ratios, preserving more or less detail depending on the specific application requirements.
7. Quantization Matrices Optimization: Quantization matrices can be customized or optimized based on the characteristics of the input image content. This optimization process involves determining the ideal quantization step sizes for different frequency components to achieve better visual quality for specific types of images.
8. Quality Settings: Many image compression algorithms allow users to adjust the compression level or quality setting, which indirectly affects the quantization step size. Higher quality settings result in smaller quantization steps and better visual quality but larger file sizes, while lower quality settings produce more aggressive quantization and higher compression ratios but with visible loss of detail.

Technical aspects of entropy coding in depth:

1. Coding Efficiency: Entropy coding techniques aim to achieve coding efficiency by allocating fewer bits to more probable symbols and more bits to less probable symbols. This ensures that frequently occurring symbols are represented by shorter codes, reducing the overall average code length and, consequently, the file size.
2. Source Coding vs. Channel Coding: Entropy coding is a form of source coding, which is concerned with encoding the source data (e.g., image pixels) into a compressed representation. It is different from channel coding, which focuses on error detection and correction for reliable transmission over noisy channels.
3. Huffman Coding: Huffman coding is one of the most widely used entropy coding techniques. It constructs an optimal prefix code based on the probabilities or frequencies of the symbols in the source data.

• Construction: Huffman coding starts by assigning shorter codes to symbols with higher probabilities, ensuring that no code is a prefix of another code. This is achieved through a binary tree construction process known as the Huffman tree or Huffman coding tree.
• Encoding: To encode symbols, the Huffman tree is traversed from the root to the corresponding leaf node, with a left traversal representing a ‘0’ bit and a right traversal representing a ‘1’ bit. The resulting bit sequence is the Huffman-coded representation of the symbol.
• Decoding: To decode Huffman-coded data, the Huffman tree is traversed based on the received bit sequence until a leaf node is reached, which corresponds to a decoded symbol.

4. Arithmetic Coding: Arithmetic coding is another popular entropy coding technique that achieves higher coding efficiency compared to Huffman coding. It operates on a continuous range of probabilities rather than discrete symbols.

• Encoding: Arithmetic coding maps a symbol or a group of symbols to a subinterval within the unit interval [0, 1]. The size of the subinterval is proportional to the probability of the symbol(s) being encoded. The resulting subinterval represents the encoded data.
• Decoding: Decoding in arithmetic coding involves reversing the process by mapping the received subinterval back to the original symbol(s). This requires knowledge of the probability distribution used during encoding.

5. Adaptive Entropy Coding: Adaptive entropy coding techniques dynamically update the probability models during encoding and decoding based on previously encoded or decoded symbols. This adaptability allows them to achieve better coding efficiency and adapt to changes in the statistics of the data.

• Adaptive Huffman Coding: Adaptive Huffman coding maintains a dynamic Huffman tree that is updated after each symbol is encoded or decoded. This ensures that the coding tree always reflects the current frequency distribution of the symbols.
• Adaptive Arithmetic Coding: Adaptive arithmetic coding adjusts the probability distribution based on the observed frequencies of symbols. The probabilities are updated on the fly during the encoding and decoding processes.

Software and tools for image compression

1. Adobe Photoshop:
   • Adobe Photoshop is a powerful image editing software widely used by professionals.
   • It offers various image compression features, including options to save images in different formats (JPEG, PNG, GIF) with customizable compression settings.
   • Photoshop allows you to control the level of compression, adjust image quality, and preview the resulting file size before saving.
   • Additionally, Photoshop provides advanced image editing capabilities, allowing you to optimize images for specific use cases and export them with desired compression settings.
2. TinyPNG:
   • TinyPNG is an online image compression tool designed specifically for PNG images.
   • It uses smart lossy compression techniques to reduce the file size of PNG images while preserving high visual quality.
   • TinyPNG employs a combination of color reduction algorithms, quantization, and metadata optimization to achieve significant compression without visible quality loss.
   • It supports batch processing, allowing you to compress multiple images simultaneously, and provides a WordPress plugin for seamless integration with websites.
3. JPEGmini:
   • JPEGmini is a popular image compression software that specializes in reducing the file size of JPEG images.
   • It utilizes advanced algorithms to analyze and optimize JPEG compression settings while maintaining visual quality.
   • JPEGmini reduces file size significantly without sacrificing perceived image quality by removing unnecessary data and optimizing the encoding parameters.
   • It offers both desktop applications and online versions, allowing users to compress JPEG images with ease.
4. Squoosh:
   • Squoosh is a web-based image compression tool developed by Google.
   • It supports various image formats, including JPEG, PNG, and WebP.
   • Squoosh provides an interactive interface where you can upload images, compare different compression settings in real-time, and preview the resulting image quality and file size.
   • It offers advanced options such as adjusting chroma subsampling, quantization, and color profiles, giving users fine-grained control over the compression process.
5. Optimizilla:
   • Optimizilla is an online image compression tool that allows you to compress JPEG and PNG images.
   • It employs a lossy compression algorithm, adjusting the compression level dynamically based on the input image content.
   • Optimizilla provides a convenient slider to control the compression level and instantly displays the resulting image quality and file size.
   • It supports batch processing, enabling you to compress multiple images simultaneously, and offers a simple and user-friendly interface.