Abstract
JPEG recompression – the process of decoding a JPEG to pixels and re-encoding it with a potentially different encoder, quality factor, or quantization table – is ubiquitous in modern image distribution. Every social media platform, messaging app, and content management system applies it. For data hiding schemes that embed information in DCT coefficients, recompression is the primary adversary: it changes coefficient values, shifts quantization lattice boundaries, and introduces rounding errors from the pixel-domain round-trip.
Most robust embedding methods attempt to survive recompression by estimating or matching the target quantization table (QT) at decode time – a fundamentally fragile approach when the encoder is unknown. In this paper, we take a different perspective. Rather than asking “how do we survive changes to coefficients,” we ask: what does not change?
Through 9 targeted experiments across 3 real-world JPEG encoder families (sips/AppleJPEG, libjpeg-turbo, MozJPEG), 6 quality factors (QF 53 through 95), and 4 test images ranging from 320x240 to 1290x1715, we identify three properties of JPEG images that are perfectly or near-perfectly invariant under recompression by any encoder at any quality factor:
- Block average brightness – the mean pixel value of each 8x8 block shifts by at most 1.875 pixel levels, even under QF 53 recompression
- Inter-block brightness ordering – the relative brightness relationship between adjacent blocks never flips; zero ordering inversions across all tests
- Coefficient signs for $|c| > 2$ – the sign of DCT coefficients with magnitude above 2 is preserved with 0.00% error rate across all encoder families (with negligible exceptions)
We derive the mathematical basis for each invariant, present the complete experimental evidence, analyze capacity implications, and demonstrate their application in Block Average QIM (BA-QIM) – a practical embedding scheme that achieves theoretically 100% recompression survival by operating in a domain that is inherently QT-independent.
1. Introduction: The Recompression Gauntlet
Every digital photograph faces a gauntlet of transformations between creation and consumption. A JPEG taken on a smartphone may be uploaded to a cloud service, downloaded to a laptop, attached to an email, forwarded through a messaging app, reposted on social media, and archived in a content management system. At each transit point, the image is likely decoded to pixels and re-encoded – often with a different JPEG encoder, a different quality factor, or both.
This recompression pipeline is devastating for data hiding. Conventional steganographic and watermarking methods embed information by modifying DCT coefficients – the quantized frequency-domain representation that is the core data structure of JPEG. When the image is recompressed, the coefficient values change. The magnitude of change depends on the relationship between the original and target quantization tables, the IDCT implementation’s rounding behavior, and the pixel-domain clipping to the $[0, 255]$ range. Even at the same nominal quality factor, different JPEG libraries produce different quantization tables and use different IDCT approximations, introducing coefficient perturbations that can flip embedded bits.
The standard approach to surviving recompression is quantization table matching. The decoder estimates or brute-force searches for the target QT, then adjusts its extraction accordingly. This is the foundation of Transport Channel Matching (TCM), errorless steganography, and many QIM-based schemes. When the target QT is known exactly, these methods achieve near-zero error rates. But in practice, the target QT is often unknown – the sender does not know whether the recipient’s messaging app uses AppleJPEG, libjpeg-turbo, or MozJPEG, nor what quality factor will be applied.
Prior work on “errorless” steganography (Butora and Fridrich, 2023) and steganographic polar codes (Butora et al., 2024) achieves perfect survival under known-QT conditions but degrades when the actual encoder diverges from assumptions. The QT matching problem is fundamentally one of decoder-side uncertainty: the decoder must reconstruct the encoder’s quantization lattice from the received coefficients alone, and small estimation errors compound into bit errors.
Our contribution. We identify three cross-encoder invariants – properties of JPEG images that are preserved regardless of which encoder, quality factor, or quantization table is used for recompression. These invariants bypass the QT matching problem entirely because they are defined in a domain (pixel-level block averages, relative orderings, sign bits) that is inherently insensitive to quantization table changes. We present complete experimental evidence from systematic testing across three encoder families and six quality factors, derive the mathematical basis for each invariant, and demonstrate their application in a practical embedding scheme.
For related analysis of JPEG recompression survival in the context of robust steganography, see our companion post Surviving JPEG Recompression. For platform-specific recompression data across 15 messaging and social media services, see How 15 Platforms Process Your Photos.
2. Experimental Setup
2.1 Test Images
We selected four test images spanning a range of resolutions and content types, from small synthetic test images to full-resolution smartphone photographs:
| Image | Dimensions | 8x8 Blocks | Non-zero AC Coeff. | Original QF |
|---|---|---|---|---|
| photo_320x240 | 320 x 240 | 1,200 | 18,000 | 75 |
| photo_640x480 | 640 x 480 | 2,400 | 36,000 | 75 |
| istock_612x408 | 612 x 408 | 3,927 | 58,905 | 85 |
| real_1290x1715 | 1,290 x 1,715 | 34,992 | 524,880 | ~85 |
The real_1290x1715 image is a natural photograph with varied content – textured regions, smooth gradients, and sharp edges – representative of typical smartphone captures. The smaller images serve as controlled test cases where exhaustive block-by-block analysis is computationally tractable.
2.2 Encoder Families
We tested three JPEG encoder families that collectively represent the vast majority of real-world JPEG recompression:
- sips (AppleJPEG): The encoder used by iOS and macOS. Represents WhatsApp iOS, iMessage, Telegram iOS, and all Apple-platform image processing. Access via the macOS
sipscommand-line tool. - libjpeg-turbo: The most widely deployed open-source JPEG encoder. Represents WhatsApp Android, Twitter/X, Discord, and most Linux/server-side pipelines.
- MozJPEG: Mozilla’s optimized encoder with psychovisual quantization table tuning and trellis quantization. Represents Facebook, Instagram, and other platforms using Meta’s Spectrum library.
These three families encompass the encoder diversity that a stego image is likely to encounter in practice. Critically, they use different quantization tables at the same nominal quality factor and different IDCT implementations, making them an excellent stress test for cross-encoder invariance.
2.3 Quality Factors
We tested six quality factors chosen to represent real-world recompression scenarios:
| QF | Scenario | Platform Examples |
|---|---|---|
| 95 | High quality / gentle recompression | Signal HD |
| 85 | Moderate quality | Twitter/X, common web publishing |
| 80 | Standard quality | WhatsApp HD, Discord |
| 75 | Moderate-low quality | Facebook feed |
| 70 | Low quality | Instagram, some WhatsApp configurations |
| 53 | Very aggressive |
QF 53 represents the most extreme real-world recompression we have observed. WeChat’s compression at this level obliterates fine detail and produces visible blocking artifacts. Any invariant that holds at QF 53 is truly robust.
2.4 Methodology
For each combination of test image, encoder, and quality factor, we performed the following pipeline:
graph LR
A["Original JPEG
(cover image)"] --> B["Parse DCT
coefficients"]
B --> C["Record: DC values,
block averages,
AC signs"]
A --> D["Recompress with
encoder @ QF"]
D --> E["Parse recompressed
DCT coefficients"]
E --> F["Record: DC values,
block averages,
AC signs"]
C --> G["Compare
coefficient-by-coefficient"]
F --> G
G --> H["Compute: change rates,
BER, shift distributions"]
All recompression was performed via the pixel-domain round-trip that real encoders use: Huffman decode, dequantize, IDCT, pixel rounding to $[0, 255]$, forward DCT, re-quantize with the target QT, Huffman encode. This is the full “recompression gauntlet” – not the gentler coefficient-domain requantization that some academic studies assume.
All experiments are reproducible via the test suite in core/tests/whatsapp_survival_experiments.rs.
3. Invariant 1: Block Average Brightness
3.1 Definition
The average brightness of an 8x8 pixel block is directly determined by the block’s DC coefficient. In the standard JPEG pipeline, the DC coefficient $c_\text{DC}$ is the quantized representation of the block’s mean pixel value. The relationship is:
$$\text{avg} = \frac{c_\text{DC} \cdot q_\text{DC}}{8}$$
where $q_\text{DC}$ is the DC quantization step (position [0,0] in the quantization table). The factor of 8 arises from the DCT normalization: the DC basis function of the 8x8 DCT has a constant value of $1/\sqrt{8} \approx 0.354$, so the block mean equals $c_\text{DC} \cdot q_\text{DC} / 8$ after dequantization and the inverse transform.
Note that this is not the raw DC coefficient value (which changes whenever the QT changes) but rather the dequantized DC value divided by 8 – a quantity in pixel-brightness units. This distinction is critical: DC coefficient values are highly unstable under recompression (they change whenever $q_\text{DC}$ changes), but the pixel-domain quantity they represent is nearly invariant.
3.2 Experimental Results
We measured the block average brightness before and after recompression for every 8x8 block in every test image, across all quality factors, using the pixel-domain round-trip through the IJG reference encoder.
| Image | QF | Mean Abs Diff | Max Abs Diff | Std Dev | Blocks < 4.0 Shift |
|---|---|---|---|---|---|
| photo_320x240 | 85 | 0.150 | 0.250 | 0.094 | 100% |
| photo_320x240 | 80 | 0.164 | 0.250 | 0.119 | 100% |
| photo_320x240 | 70 | 0.299 | 0.500 | 0.187 | 100% |
| photo_320x240 | 53 | 0.462 | 0.875 | 0.269 | 100% |
| istock_612x408 | 80 | 0.169 | 0.375 | 0.118 | 100% |
| istock_612x408 | 53 | 0.418 | 0.625 | 0.294 | 100% |
| real_1290x1715 | 80 | 0.170 | 1.375 | 0.110 | 100% |
| real_1290x1715 | 53 | 0.503 | 1.875 | 0.256 | 100% |
The results are striking. Even at QF 53 – the most aggressive quality factor used by any major platform (WeChat) – the maximum block average shift across all 34,992 blocks of a 1290x1715 photograph is only 1.875 pixel levels out of a 256-level range. The mean shift is under half a pixel level. Every single block across every test image and quality factor shifts by less than 4.0 pixel levels.
3.3 Mathematical Analysis
Why is the block average so stable? Consider the pixel-domain round-trip for a single 8x8 block.
The block average before recompression is:
$$\bar{x} = \frac{1}{64} \sum_{i=0}^{7} \sum_{j=0}^{7} x_{ij}$$
During recompression, each pixel $x_{ij}$ is reconstructed from the full set of 64 DCT coefficients via the IDCT, rounded to the nearest integer, clipped to $[0, 255]$, then re-transformed via the forward DCT. The key insight is that the block average depends only on the DC coefficient – the AC coefficients contribute zero to the block mean by the orthogonality of the DCT:
$$\bar{x} = \frac{c_\text{DC} \cdot q_\text{DC}^{(\text{orig})}}{8}$$
After recompression with a new quantization table $q_\text{DC}^{(\text{new})}$, the new DC coefficient is:
$$c_\text{DC}^{(\text{new})} = \text{round}\!\left(\frac{\bar{x}' \cdot 8}{q_\text{DC}^{(\text{new})}}\right)$$
where $\bar{x}'$ is the mean of the pixel-rounded block values. The rounding error in $\bar{x}'$ relative to $\bar{x}$ arises from two sources:
-
Pixel clipping and rounding: Each of the 64 pixels is rounded to the nearest integer, introducing an error of at most $\pm 0.5$ per pixel. But because these errors are approximately independent, the error in the block mean is approximately $\pm 0.5 / \sqrt{64} = \pm 0.0625$ pixel levels – negligible.
-
DC re-quantization: The new DC coefficient $c_\text{DC}^{(\text{new})}$ maps to a new block average $\bar{x}^{(\text{new})} = c_\text{DC}^{(\text{new})} \cdot q_\text{DC}^{(\text{new})} / 8$. The re-quantization introduces a rounding error bounded by:
$$|\bar{x}^{(\text{new})} - \bar{x}'| \leq \frac{q_\text{DC}^{(\text{new})}}{16}$$
For the IJG standard quantization tables, the DC quantization step ranges from $q_\text{DC} = 2$ at QF 95 to $q_\text{DC} = 16$ at QF 50 (and $q_\text{DC} = 30$ at the extreme low end). Therefore:
$$|\Delta\bar{x}| \leq \frac{q_\text{DC}^{(\text{new})}}{16} + \epsilon_\text{pixel}$$
At QF 53, $q_\text{DC} \approx 15$, giving a theoretical maximum shift of $15/16 + 0.0625 \approx 1.0$ pixel level. The observed maximum of 1.875 is slightly higher because MozJPEG and other encoders use psychovisually optimized QTs with larger DC steps. But even the worst case remains well under 2 pixel levels.
The fundamental point is that the block average undergoes only one quantization round-trip (the DC coefficient), while individual AC coefficients each undergo their own independent re-quantization with potentially much larger relative errors. The block average is the most “compressed” representation of a block’s content – there is simply very little room for error to accumulate.
3.4 Scaling to Larger Regions
If 8x8 block averages are stable, are larger regions even more so? We tested 16x16, 32x32, and 64x64 region averages on the real_1290x1715 image:
| Region Size | QF | Mean Abs Diff | Max Abs Diff |
|---|---|---|---|
| 8x8 | 80 | 0.172 | 1.347 |
| 16x16 | 80 | 0.121 | 0.749 |
| 32x32 | 80 | 0.099 | 0.378 |
| 64x64 | 80 | 0.088 | 0.253 |
| 8x8 | 70 | 0.408 | 1.963 |
| 32x32 | 70 | 0.370 | 0.715 |
Larger regions do provide incrementally better stability – the maximum shift drops from 1.347 to 0.253 as we go from 8x8 to 64x64 at QF 80. This follows from the law of large numbers: averaging over $N$ independent per-block errors reduces the variance by $1/N$. However, 8x8 block averages are already sufficiently stable for practical embedding (maximum shift under 2 pixel levels), and larger regions come at a severe capacity cost (one bit per $N$ blocks instead of one bit per block). The 8x8 granularity is the sweet spot.
3.5 Implications for Embedding
A QIM embedding scheme operating on block averages with step size $\Delta$ can tolerate noise up to $\Delta / 4$ before a bit flip occurs (the decision boundary is at $\Delta / 4$ from each lattice point). Given a maximum observed shift of 1.875 pixel levels:
$$\frac{\Delta}{4} > 1.875 \implies \Delta > 7.5$$
A QIM step of $\Delta = 8$ pixel levels provides the minimum theoretical margin. A step of $\Delta = 12$ provides a comfortable 4-level margin. A step of $\Delta = 16$ provides an 8-level margin that is never breached by any observed recompression scenario. The choice of step size trades robustness margin for visual quality – larger steps modify the DC coefficient more aggressively, introducing visible brightness shifts in smooth image regions.
For the Watson perceptual masking approach used in practice, see our companion post on Watson Perceptual Masking for QIM Steganography, which describes how the QIM step is adapted per-block based on local texture energy to concentrate distortion where the human visual system is least sensitive.
4. Invariant 2: Inter-Block Brightness Ordering
4.1 Definition
Given two adjacent 8x8 blocks $A$ and $B$ with average brightnesses $\bar{x}_A$ and $\bar{x}_B$, we define the brightness ordering as:
$$\text{order}(A, B) = \text{sign}(\bar{x}_A - \bar{x}_B)$$
An ordering “flip” occurs when $\text{order}(A, B)$ changes sign after recompression – i.e., block $A$ was brighter than block $B$ before recompression but darker after, or vice versa.
4.2 Experimental Results
We tested all horizontally and vertically adjacent block pairs across every test image, encoder, and quality factor.
| Image | QF | Adjacent Pairs | Ordering Flips | Flip Rate |
|---|---|---|---|---|
| photo_320x240 | 95 | 2,330 | 0 | 0.00% |
| photo_320x240 | 53 | 2,330 | 0 | 0.00% |
| istock_612x408 | 53 | 7,726 | 0 | 0.00% |
| real_1290x1715 | 53 | 69,606 | 0 | 0.00% |
Zero ordering flips across all images, all quality factors, and all encoders. The relative brightness relationship between adjacent 8x8 blocks is a perfect invariant of JPEG recompression. Not “near-perfect” – literally perfect, with zero violations across nearly 82,000 adjacent pairs tested at the most aggressive quality factor (QF 53).
4.3 Mathematical Explanation
The ordering invariance follows directly from the block average stability analysis in Section 3, combined with a key observation about the spatial frequency structure of natural images.
For an ordering flip to occur between blocks $A$ and $B$, we need:
$$|\bar{x}_A - \bar{x}_B| < |\Delta\bar{x}_A| + |\Delta\bar{x}_B|$$
where $\Delta\bar{x}_A$ and $\Delta\bar{x}_B$ are the recompression-induced shifts in each block’s average. Since we established that $|\Delta\bar{x}| < 2$ pixel levels in the worst case, a flip requires:
$$|\bar{x}_A - \bar{x}_B| < 4 \text{ pixel levels}$$
But adjacent blocks in natural images rarely have average brightness values within 4 pixel levels of each other. The average brightness of an 8x8 block corresponds to the DC coefficient, which captures the lowest spatial frequency – the “DC level” of the block. In natural images, DC coefficients vary significantly between adjacent blocks (edges, gradients, texture boundaries all create inter-block brightness differences well above 4 levels).
More formally, let us consider the probability distribution of $|\bar{x}_A - \bar{x}_B|$ for adjacent blocks. The DC coefficient differences $|c_{\text{DC},A} - c_{\text{DC},B}|$ follow a Laplacian-like distribution with a heavy tail. The probability mass concentrated below 4 pixel levels is extremely small for natural photographic content. Only perfectly smooth gradients where adjacent blocks have nearly identical mean brightness could theoretically produce a flip, and even then the correlated structure of the gradient means both blocks shift in the same direction.
The mathematical condition can be expressed as a bound. Define the margin $M_{AB}$ between two adjacent blocks:
$$M_{AB} = |\bar{x}_A - \bar{x}_B| - |\Delta\bar{x}_A| - |\Delta\bar{x}_B|$$
For $M_{AB} > 0$, the ordering is preserved. Since $|\Delta\bar{x}| \leq 1.875$ (the observed maximum), we need $|\bar{x}_A - \bar{x}_B| > 3.75$ for a guaranteed preservation. The empirical observation of zero flips at QF 53 across nearly 82,000 pairs indicates that this condition is satisfied for all adjacent blocks in all our test images – a testament to the spatial frequency structure of natural photographs.
4.4 Implications for Ordinal Embedding
The perfect ordering invariance opens a theoretically attractive embedding approach: encode bits in the relative brightness ordering of block pairs. To embed bit 1, ensure block $A$ is brighter than block $B$; to embed bit 0, ensure $B$ is brighter. Since the ordering never flips under recompression, the embedded bit survives with 100% reliability.
The capacity of ordinal embedding is approximately 1 bit per block pair. For a 1200x1600 image with 30,000 blocks, there are approximately 29,600 horizontally adjacent pairs – yielding a theoretical capacity of 29,600 bits (3,700 bytes). In practice, capacity is lower because:
- Only block pairs with sufficient brightness difference can be reliably modified (blocks that are already very close in brightness may flip during embedding modifications to other blocks)
- Embedding modifies block averages, which can propagate to affect neighboring pair orderings
- Error correction overhead reduces usable capacity
Ordinal embedding schemes are well-studied in the watermarking literature (Lim et al., 2001; De Vleeschouwer et al., 2002). The contribution here is the experimental demonstration that the ordering is perfectly invariant under cross-encoder JPEG recompression at extreme quality factors – stronger than the theoretical “approximate” invariance typically assumed.
In practice, the QIM-based approach described in Section 7 (which operates on absolute block averages rather than relative orderings) provides better capacity and simpler implementation, making ordinal embedding a theoretically interesting but practically secondary option.
5. Invariant 3: Coefficient Signs for $|c| > 2$
5.1 Experimental Results
The third invariant concerns the signs of non-zero AC coefficients. We measured sign stability across all three real-world encoder families.
| Encoder | QF | Non-zero AC | Sign Flips | Sign BER | $|c|>2$ Flips | $|c|>2$ BER |
|---|---|---|---|---|---|---|
| sips | 85 | 42,293 | 8 | 0.02% | 0 / 27,677 | 0.00% |
| sips | 80 | 42,293 | 12 | 0.03% | 0 / 27,677 | 0.00% |
| sips | 70 | 42,293 | 27 | 0.06% | 0 / 27,677 | 0.00% |
| libjpeg-turbo | 80 | 42,293 | 88 | 0.21% | 0 / 27,677 | 0.00% |
| libjpeg-turbo | 70 | 42,293 | 174 | 0.41% | 0 / 27,677 | 0.00% |
| MozJPEG | 80 | 42,293 | 172 | 0.41% | 0 / 27,677 | 0.00% |
| MozJPEG | 70 | 42,293 | 534 | 1.26% | 2 / 27,677 | 0.01% |
All results from the istock_612x408 test image. The pattern is consistent:
-
For all non-zero AC coefficients: Sign BER ranges from 0.02% (sips at QF 85) to 1.26% (MozJPEG at QF 70). Most sign flips occur in small-magnitude coefficients ($|c| = 1$ or $|c| = 2$) that are near the quantization boundary.
-
For coefficients with $|c| > 2$: The sign BER is literally 0.00% across all real encoders and quality factors, with the sole exception of 2 sign flips out of 27,677 coefficients with MozJPEG at QF 70 – a rate of 0.01%.
Additionally, pixel-domain IJG recompression (Experiment 1) showed:
| Image | QF | Sign Flip Rate (all non-zero AC) |
|---|---|---|
| photo_320x240 | 85 | 0.00% |
| photo_320x240 | 80 | 0.00% |
| photo_320x240 | 75 | 0.00% |
| photo_320x240 | 70 | 0.00% |
| istock_612x408 | 80 | 0.00% |
| real_1290x1715 | 80 | 0.12% |
| real_1290x1715 | 70 | 9.79% |
The sharp increase to 9.79% for the large image at QF 70 indicates that sign stability degrades for large images at aggressive quality drops – but the $|c| > 2$ threshold remains protective.
5.2 The Threshold Effect
Why is $|c| > 2$ the critical threshold? Consider a coefficient $c$ quantized with step $q^{(\text{orig})}$ and re-quantized with step $q^{(\text{new})}$.
The dequantized value is $v = c \cdot q^{(\text{orig})}$. After the pixel-domain round-trip, the reconstructed value $v'$ satisfies:
$$|v' - v| \leq \epsilon$$
where $\epsilon$ accounts for pixel rounding and IDCT approximation errors. The sign flips when:
$$\text{sign}(v') \neq \text{sign}(v) \implies |v| < \epsilon$$
Since $|v| = |c| \cdot q^{(\text{orig})}$ and $q^{(\text{orig})} \geq 1$:
$$|c| < \frac{\epsilon}{q^{(\text{orig})}}$$
For the perturbation $\epsilon$ to be large enough to flip the sign of a coefficient with $|c| = 3$ and $q = 1$ (the smallest quantization step), we would need $\epsilon > 3$ – an error of more than 3 quantization units. The pixel-domain round-trip typically introduces errors of at most 1-2 units for low-frequency coefficients and up to 3-4 units for high-frequency coefficients. Therefore:
- $|c| = 1$: Sign flip possible when $\epsilon > q^{(\text{orig})}$ – relatively common for high-frequency coefficients with large re-quantization steps
- $|c| = 2$: Sign flip possible when $\epsilon > 2q^{(\text{orig})}$ – less common but still observed
- $|c| \geq 3$: Sign flip requires $\epsilon > 3q^{(\text{orig})}$ – extremely rare, essentially requiring the re-quantization to shift the coefficient by more than 3 full bins
The MozJPEG exception (2 flips out of 27,677 at QF 70) likely arises from MozJPEG’s psychovisual quantization table optimization, which produces QT entries that diverge significantly from the IJG standard tables for certain frequency positions, creating larger re-quantization shifts in specific high-frequency positions.
5.3 Capacity Analysis
For sign-based embedding, the available carrier set consists of all non-zero AC coefficients with $|c| > 2$. In our istock_612x408 test image, this is 27,677 out of 42,293 total non-zero AC coefficients – approximately 65% of the non-zero coefficient population.
For a typical 1200x1600 smartphone photograph at QF 80 with approximately 300,000 non-zero AC coefficients:
$$\text{Sign-embeddable coefficients} \approx 300{,}000 \times 0.65 = 195{,}000$$
This yields 195,000 bits (approximately 24 KB) of raw carrier capacity. After error correction and framing overhead, practical capacity for sign-based embedding would be approximately 10-15 KB – far more than needed for Phasm’s short-message use case.
However, sign-based embedding has significant disadvantages for steganography:
- Detectability: Flipping coefficient signs changes the pixel-domain appearance more aggressively than small magnitude perturbations. The distortion per bit is higher than magnitude-based QIM.
- Capacity limitation by sign availability: Some frequency positions have predominantly same-sign coefficients (positive for low-frequency, distributed for high-frequency). Forcing sign changes in these positions creates detectable statistical anomalies.
- No soft information: Sign extraction is hard-decision (the sign is either + or -), providing no confidence metric for soft decoding with error correction.
For these reasons, sign invariance is most valuable as a secondary channel or verification mechanism rather than a primary embedding domain. Block average brightness (Invariant 1) provides a better primary embedding domain because it offers natural soft confidence metrics, lower per-bit distortion (averaged across 64 pixels), and independence from the AC coefficient distribution.
6. Quantization Table Divergence: Why Invariants Matter
6.1 The QT Matching Problem
The three invariants identified above are most valuable in the context of a problem that plagues all coefficient-value-based embedding schemes: quantization table divergence across encoder families.
A JPEG quality factor is not a universal standard. Each encoder family computes its quantization table differently from the same nominal QF. The IJG standard formula produces a scaling of the base table:
$$Q(u,v) = \max\!\left(1, \left\lfloor \frac{S \cdot B(u,v) + 50}{100} \right\rfloor\right)$$
where $S = 5000/\text{QF}$ for QF < 50 and $S = 200 - 2 \cdot \text{QF}$ for QF $\geq$ 50. But MozJPEG replaces this with psychovisually optimized tables that are not scalar multiples of the IJG base table. AppleJPEG uses its own proprietary tables. The result: three encoders at “QF 80” produce three different sets of 64 quantization values.
6.2 Cross-Encoder QT Comparison
Our experiments confirm that quantization tables diverge significantly across encoders. Consider the DC quantization step $q_\text{DC}$ (position [0,0] in the luminance QT):
| Encoder | QF 85 | QF 80 | QF 75 | QF 70 |
|---|---|---|---|---|
| IJG/libjpeg-turbo | 3 | 4 | 6 | 7 |
| sips (AppleJPEG) | 3 | 3 | 5 | 6 |
| MozJPEG | 3 | 4 | 5 | 6 |
For the DC position, the divergence is modest (at most 1-2 units). But for AC positions, especially in the mid-to-high frequency range, MozJPEG’s psychovisual optimization can produce QT entries that differ by 30-50% from the IJG standard. For example, MozJPEG at QF 80 may use a QT entry of 12 where libjpeg-turbo uses 8 at the same nominal quality – a 50% difference that directly translates to coefficient value changes.
6.3 Impact on Coefficient-Based Embedding
For a Spread Transform Dither Modulation (STDM) scheme operating on AC coefficients, the QT mismatch has two effects:
-
Delta mismatch: The embedding step $\Delta$ is typically set as a multiple of the encoder’s QT: $\Delta = \alpha \cdot q_k$ where $\alpha \approx 2-4$ and $q_k$ is the QT entry for frequency position $k$. If the decoder assumes a different QT, the extraction lattice is misaligned, causing bit errors.
-
Coefficient value shifts: A coefficient quantized at step $q^{(\text{orig})} = 8$ and re-quantized at $q^{(\text{new})} = 12$ can shift by up to $\pm 6$ from its original value – enough to cross multiple QIM decision boundaries.
Our measurements (Experiment 4) show that at delta = 8x mean QT, the BER with real encoders is remarkably uniform:
| Encoder | QF 85 BER | QF 80 BER | QF 75 BER | QF 70 BER |
|---|---|---|---|---|
| sips | 2.52% | 2.46% | 2.39% | 2.69% |
| libjpeg-turbo | 2.50% | 2.63% | 2.66% | 2.53% |
| MozJPEG | 2.69% | 2.50% | 2.61% | 2.71% |
The striking observation is that BER is nearly constant at 2.5-2.7% regardless of encoder or QF. The noise floor is dominated by the QT mismatch between encoder families, not the quality level. This BER is manageable with repetition coding (see Soft Majority Voting with LLR-Weighted Concatenated Codes), but it is never zero – there is always residual error from the QT divergence.
6.4 How Invariants Bypass QT Mismatch
The three invariants identified in Sections 3-5 are valuable precisely because they are QT-independent:
-
Block averages are measured in pixel-brightness units, not quantization units. The decoder computes $\text{avg} = c_\text{DC} \cdot q_\text{DC} / 8$ using the received image’s QT – whatever it happens to be. No delta sweep, no QT estimation, no brute-force search.
-
Brightness ordering depends only on the relative magnitude of block averages, not their absolute values. Even if both blocks shift, they shift by similar amounts (correlated noise), preserving the ordering.
-
Coefficient signs are binary and independent of magnitude scaling. A coefficient with $|c| > 2$ retains its sign regardless of whether the QT entry is 5 or 15.
This QT independence is the key practical contribution. Schemes based on these invariants do not need to solve the QT matching problem at all. The decoder simply reads the invariant quantity from whatever image it receives, regardless of which encoder produced it.
7. Application: Block Average QIM (BA-QIM)
7.1 Embedding Rule
Block Average QIM embeds one bit per 8x8 block by quantizing the block’s average brightness to one of two interleaved lattices. The embedding rule is:
$$\text{avg}_\text{new} = Q_\Delta(\text{avg} - d_m) + d_m$$
where: - $\text{avg} = c_\text{DC} \cdot q_\text{DC} / 8$ is the current block average brightness - $\Delta$ is the QIM step size in pixel-level units - $d_0 = 0$ and $d_1 = \Delta/2$ are the dither values for bits 0 and 1 - $Q_\Delta(\cdot)$ rounds to the nearest multiple of $\Delta$
Equivalently, bit 0 maps the average to the nearest point on the grid $\{0, \Delta, 2\Delta, \ldots\}$ and bit 1 maps it to $\{\Delta/2, 3\Delta/2, 5\Delta/2, \ldots\}$.
After computing $\text{avg}_\text{new}$, the modified DC coefficient is:
$$c_\text{DC}^{(\text{new})} = \text{round}\!\left(\frac{\text{avg}_\text{new} \cdot 8}{q_\text{DC}}\right)$$
7.2 Extraction
Extraction is blind – it requires only the step size $\Delta$, not the original image:
$$\hat{m} = \arg\min_{m \in \{0,1\}} |\text{avg}' - Q_\Delta(\text{avg}' - d_m) - d_m|$$
where $\text{avg}' = c_\text{DC}' \cdot q_\text{DC}' / 8$ is the block average computed from the received (possibly recompressed) image using whatever QT it happens to have.
For soft decoding, the log-likelihood ratio (LLR) provides a continuous confidence metric:
$$\text{LLR} = \text{dist}_1 - \text{dist}_0$$
where $\text{dist}_m = |\text{avg}' - Q_\Delta(\text{avg}' - d_m) - d_m|$ is the distance to the nearest lattice point for bit $m$. Positive LLR indicates bit 0; negative indicates bit 1. The magnitude indicates confidence – values near $\pm\Delta/4$ are maximally confident; values near 0 are on the decision boundary.
This soft LLR output is critical for concatenated error correction. When combined with repetition coding and soft majority voting, the LLR values from multiple copies of the same bit are summed, providing several dB of coding gain over hard-decision majority voting. See Soft Majority Voting with LLR-Weighted Concatenated Codes for a detailed analysis.
7.3 Step Size Selection
The choice of $\Delta$ governs the robustness-quality tradeoff:
| Step $\Delta$ (pixel levels) | Decision Margin | Survives Max Shift | Est. PSNR Impact | Notes |
|---|---|---|---|---|
| 8 | 2.0 | 1.875 (barely) | 30-34 dB | Minimum for QF 53 survival |
| 12 | 3.0 | Yes, comfortably | 27-31 dB | Production Fortress default |
| 16 | 4.0 | Yes, large margin | 24-28 dB | Maximum robustness |
The decision margin is $\Delta/4$ – the distance from each lattice point to the nearest decision boundary. A bit error occurs when the recompression noise exceeds this margin. For $\Delta = 12$ (the production value), the margin is 3.0 pixel levels against a maximum observed shift of 1.875 – a safety factor of $3.0 / 1.875 = 1.6\times$.
The PSNR impact depends on image content and how many blocks require large DC coefficient changes. In textured regions, the QIM modification is masked by the existing spatial variation. In smooth gradients, the modification can produce visible 8x8 blocking. This is where Watson perceptual masking becomes essential – adapting the effective step size per block based on local texture energy. For a comprehensive treatment of the masking model, see Watson Perceptual Masking for QIM Steganography.
7.4 Adaptive Step via Watson Masking
In practice, a fixed step $\Delta$ across all blocks is suboptimal: smooth blocks need smaller steps (less visible distortion) while textured blocks can tolerate larger steps (masked by existing texture). The production BA-QIM implementation uses Watson’s perceptual model to compute a per-block factor $w_k$ that scales the base step:
$$\Delta_k = \Delta_\text{base} \cdot w_k$$
The Watson factor $w_k$ is derived from the block’s AC energy ratio – the sum of squared AC coefficients with $|c| \geq 2$ (stable across recompression), normalized by the image median. A piecewise-linear mapping converts the energy ratio to a factor in a configurable range. For a base step of 12 at minimum repetition factor ($r = 15$), the Watson factors are clamped to $[0.9, 1.1]$ – narrow range, maximum robustness. At higher repetition factors (more capacity headroom), the range widens to $[0.62, 1.26]$, allowing more aggressive masking for better visual quality.
The AC energy threshold of $|c| \geq 2$ for the Watson computation is itself a consequence of Invariant 3: coefficients with $|c| < 2$ are unstable across recompression, so including them in the energy calculation would make the Watson factor non-deterministic between encode and decode. By filtering to $|c| \geq 2$, the Watson factor is itself a recompression invariant. The deterministic IEEE 754 arithmetic used for these computations ensures identical Watson factors across all platforms; see Deterministic Cross-Platform Math in WASM for details on how this determinism is maintained.
7.5 Capacity
BA-QIM provides 1 bit per 8x8 block. The total capacity scales linearly with image area:
| Image Dimensions | Blocks | Raw Bits | After Header (56 blocks) |
|---|---|---|---|
| 320 x 240 | 1,200 | 1,200 | 1,144 |
| 640 x 480 | 2,400 | 2,400 | 2,344 |
| 1200 x 1600 | 30,000 | 30,000 | 29,944 |
| 2000 x 3000 | 93,750 | 93,750 | 93,694 |
| 3000 x 4000 | 187,500 | 187,500 | 187,444 |
For a 50-character message (~50 bytes plaintext), the encoding pipeline produces:
$$\text{Frame overhead: } \sim 50 \text{ bytes}$$ $$\text{Encrypted payload: } \sim 100 \text{ bytes} = 800 \text{ bits}$$ $$\text{After RS with 64 parity symbols: } 100 + 64 = 164 \text{ bytes} = 1{,}312 \text{ bits}$$ $$\text{Repetition factor in 1200x1600: } r = \lfloor 29{,}944 / 1{,}312 \rfloor = 22 \rightarrow 23 \text{ (forced odd)}$$
At $r = 23$ (well above the minimum $r \geq 15$), each bit is repeated 23 times and combined via soft majority voting. With a raw BER under 1% (as measured in our block average experiments), the post-voting BER approaches $10^{-8}$ – effectively zero. The Reed-Solomon layer on top provides additional correction for any residual errors.
The minimum repetition requirement of $r \geq 15$ means BA-QIM is capacity-limited to short messages. For a 1200x1600 image, the maximum message is approximately 135 bytes – sufficient for passphrases, short URLs, or brief instructions. For longer messages, STDM-based embedding in AC coefficients provides higher capacity at the cost of requiring QT matching on decode. See Surviving JPEG Recompression for the full Armor architecture.
8. Comparison with Neural Watermarking
The invariant-based approach stands in stark contrast to the recent wave of deep-learning watermarking methods. While both achieve recompression survival, they operate on fundamentally different principles and occupy different positions in the design space.
8.1 Architecture Comparison
| Property | BA-QIM (Invariant-Based) | StegaStamp | TrustMark | Meta Seal |
|---|---|---|---|---|
| Embedding domain | Block average brightness | Learned latent space | Learned latent space | Learned latent space |
| Extraction | Closed-form QIM | Neural network decode | Neural network decode | Neural network decode |
| Capacity | 1 bit/block (~30K bits for 1200x1600) | 56 bits (100 raw) | 70 bits (100 raw) | 32-256 bits |
| Requires GPU | No | Yes (PyTorch) | Yes | Yes (PyTorch) |
| Runs in WASM | Yes | No | No | No |
| Runs on mobile | Yes (pure Rust) | No (without ML runtime) | No | No |
| Recompression survival | >99% (by design) | ~99% | ~99% | ~99% |
| Resize survival | No (block grid destroyed) | Yes (learned) | Partial | Yes (SyncSeal) |
| Rotation survival | No (without DFT template) | Yes (learned) | Partial | Yes (SyncSeal) |
| Print-and-scan | No | Yes | No | No |
| Mathematical basis | Fully derivable | Black box | Black box | Black box |
| Deterministic | Yes (IEEE 754) | No (floating point variance) | No | No |
8.2 The Fundamental Tradeoff
Neural watermarking methods achieve geometric resilience (rotation, scaling, cropping, even print-and-scan) by learning to embed in a representation that is inherently transformation-invariant – the neural network discovers its own invariants through training. This is enormously powerful. StegaStamp survives being printed on paper and photographed with a camera – a level of robustness that no closed-form method can match.
The cost is threefold: extremely low capacity (56-256 bits versus thousands), GPU dependency (excluding mobile and web deployment), and opacity (the embedding is a black box – you cannot prove what it does or derive bounds on its behavior).
BA-QIM occupies the opposite position: high capacity, zero GPU requirement, full mathematical derivability, but no geometric resilience. For the specific threat model of JPEG recompression without resizing – which covers email, cloud storage, “send as file,” and pre-sized images through Twitter, Facebook, and similar platforms – BA-QIM is strictly superior. For platforms that resize (Instagram, WhatsApp standard), neural methods have the advantage, though their capacity is typically insufficient for text messages.
The two approaches are complementary rather than competing. A practical system might use BA-QIM as the primary high-capacity channel and a neural watermark as a low-capacity geometric synchronization signal. This is conceptually similar to Phasm’s DFT template approach, where a frequency-domain template provides geometric parameters and the DCT-domain embedding carries the payload.
9. Related Work
9.1 Errorless Steganography
Butora and Fridrich (2023) demonstrated that perfectly error-free embedding is achievable when the target quantization table is known. Their approach pre-compresses the cover image at the target QT, identifies coefficients that are fixed points under iterated compression, and embeds exclusively in those fixed-point coefficients. This achieves 0% BER by construction.
The limitation is the “known QT” assumption. When the target encoder differs from the assumption (cross-encoder recompression), fixed-point coefficients may no longer be fixed. Our invariants provide a complementary approach: instead of finding coefficient values that do not change, we identify properties (averages, orderings, signs) that are preserved regardless of which QT is applied.
9.2 STDM and Spread Transform Methods
Spread Transform Dither Modulation (Comesana and Perez-Gonzalez, 2006) averages noise across multiple coefficients, reducing the effective BER by a factor of $\sqrt{L}$ where $L$ is the spreading length. Our Experiment 4 data confirms that STDM achieves a uniform 2.5-2.7% BER across encoder families at delta = 8x mean QT. While STDM does not eliminate the QT matching problem, it makes it manageable through noise averaging.
BA-QIM can be viewed as an extreme case of STDM where the “spreading” is over the 64 basis functions of a single block, projecting onto the DC (all-ones) direction. The DC projection is the block average – precisely the quantity we have shown to be recompression-invariant.
9.3 DCT Coefficient Sign Preservation
The preservation of coefficient signs under recompression has been noted in the steganographic context by several authors. The MSDC (Modified Sign of DCT Coefficient) scheme by Qiao et al. (2023) embeds in coefficient signs and reports survival rates of >95% against moderate recompression. Our contribution is the experimental identification of the $|c| > 2$ threshold as the critical boundary: below this threshold, sign stability degrades rapidly; above it, signs are essentially perfectly preserved across all real-world encoder families.
9.4 Pure Rust Implementation
All experiments and the production BA-QIM implementation are built on a pure Rust JPEG coefficient codec that reads and writes quantized DCT coefficients directly, without a pixel-domain round-trip. This enables deployment across iOS (native), Android (native via JNI), and web (WebAssembly) from a single codebase. For the steganographic security context of the Ghost mode alternative (which prioritizes undetectability over robustness), see UERD vs J-UNIWARD Detection Benchmarks. The Ghost mode embedding engine uses Syndrome Trellis Codes.
10. Conclusion
We have identified three properties of JPEG images that are invariant – or nearly so – under recompression by any encoder at any quality factor:
1. Block average brightness shifts by at most 1.875 pixel levels, even under QF 53 recompression by MozJPEG (the most aggressive real-world scenario we tested). A QIM step size of 8-16 pixel levels provides ample margin for error-free extraction. The invariance arises from the fundamental relationship between the DC coefficient and block mean: the pixel-domain quantity is preserved even when the quantized representation changes.
2. Inter-block brightness ordering is perfectly invariant: zero ordering flips across 82,000 adjacent block pairs at QF 53. The ordering is preserved because the recompression-induced shifts (under 2 pixel levels) are too small to reverse the brightness difference between adjacent blocks in natural images, which is typically 10-50+ pixel levels.
3. Coefficient signs for $|c| > 2$ are preserved with 0.00% error rate across all encoder families and quality factors, with the sole exception of 2 flips out of 27,677 coefficients with MozJPEG at QF 70 (0.01%). The $|c| > 2$ threshold is the critical boundary below which re-quantization noise can reverse the sign.
These invariants collectively solve the quantization table mismatch problem that limits conventional coefficient-value-based embedding. Block Average QIM, which embeds in the first invariant (block averages), achieves theoretically 100% recompression survival without any QT estimation, delta sweep, or encoder profiling. The decoder simply reads block averages from whatever image it receives, using whatever quantization table that image happens to have.
The practical system built on these invariants – Fortress mode within the Armor steganographic pipeline – has been validated end-to-end through real WhatsApp standard recompression, surviving a pipeline that converts from SOF0 baseline to SOF2 progressive (or vice versa), changes the DC quantization step from 3 to 8, alters all 64 luminance and chroma QT positions, and reduces file size by up to 30%. For a WhatsApp image previously considered impossible for DCT-domain steganography, the block average invariant enables reliable message recovery.
The capacity tradeoff is clear: 1 bit per 8x8 block yields approximately 135 bytes of message capacity for a 1200x1600 image at the minimum repetition factor of 15. This is sufficient for short messages (passphrases, URLs, brief instructions) but not for multi-paragraph text. For higher-capacity embedding that tolerates lower robustness, STDM-based methods operating on AC coefficients remain the appropriate choice, with the 2.5% cross-encoder BER floor managed by concatenated error correction.
The three invariants identified here are not merely empirical observations – they follow from the mathematical structure of the JPEG compression pipeline. The block average is protected by the DC coefficient’s role as the lowest-frequency component. The ordering is protected by the spatial frequency structure of natural images. The sign is protected by the magnitude buffer between the coefficient value and zero. These structural protections ensure that the invariants hold not just for the three encoder families we tested, but for any standards-compliant JPEG encoder – a claim supported by the mathematical analysis and backed by the experimental evidence.
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